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Simulating Quantum Chemistry with a Quantum Computer Improved algorithms & analysis David Poulin Équipe de Recherche sur la Physique de l’Information Quantique Département de Physique Université de Sherbrooke University of New Mexico Albuquerque, NM, September 2014 David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 1 / 30

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Simulating Quantum Chemistrywith a Quantum Computer

Improved algorithms & analysis

David Poulin

Équipe de Recherche sur la Physique de l’Information QuantiqueDépartement de PhysiqueUniversité de Sherbrooke

University of New MexicoAlbuquerque, NM, September 2014

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 1 / 30

Outline

1 Background

2 Molecules

3 Improvements

4 New simulation algorithm

5 Conclusion

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 2 / 30

Background

Outline

1 Background

2 Molecules

3 Improvements

4 New simulation algorithm

5 Conclusion

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 3 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

Quantum dynamics

A quantum state |ψ〉 is a vector in a D-dimensional complex vectorstate.D scales exponentially with the number of particles in the system.|ψ〉 evolves in time following Schrödinger’s equationddt |ψ〉 = −iH|ψ〉The Hamiltonian H is a D × D matrix.The solution to Schrödinger’s equation is |ψ(t)〉 = e−iHt |ψ(0)〉.

You need to understand how quantum state evolve toKnow how electricity flows in a piece of material.Know how two molecules will react when put into contact.Know what comes out of a high-energy collision between twoparticles.etc.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 4 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

General problem of interest

INPUTSA quantum register prepared in initial state |ψ〉.An efficiently specifiable Hamiltonian H.A time t .

OUTPUTA quantum register in state close to |ψ(t)〉 = e−iHt |ψ〉

Use to simulate dynamics.Combined to QPE to measure energy.Used as sub-routine to prepare physically relevant state |ψ〉:

Adiabatic state preparation.Quantum Metropolis sampling.

Compute response functions 〈σi(0)σj(t)〉.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 5 / 30

Background

Setting

Classical computers need to diagonalize the matrix H, whose sizeis exponential with the number of particles (size of the lattice |Λ|).In general, quantum computers are good at simulating thetime-evolution generated by sparse Hamiltonians.Special case: k-body couplings: H =

∑α⊂Λ Hα

Λ is a lattice, with a quantum particle living at each vertex.Each particle has a finite dimensional Hilbert space (spin).α are subsets of particlesHα = 0 when |α| > k (k = 2 in nature).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 6 / 30

Background

Setting

Classical computers need to diagonalize the matrix H, whose sizeis exponential with the number of particles (size of the lattice |Λ|).In general, quantum computers are good at simulating thetime-evolution generated by sparse Hamiltonians.Special case: k-body couplings: H =

∑α⊂Λ Hα

Λ is a lattice, with a quantum particle living at each vertex.Each particle has a finite dimensional Hilbert space (spin).α are subsets of particlesHα = 0 when |α| > k (k = 2 in nature).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 6 / 30

Background

Setting

Classical computers need to diagonalize the matrix H, whose sizeis exponential with the number of particles (size of the lattice |Λ|).In general, quantum computers are good at simulating thetime-evolution generated by sparse Hamiltonians.Special case: k-body couplings: H =

∑α⊂Λ Hα

Λ is a lattice, with a quantum particle living at each vertex.Each particle has a finite dimensional Hilbert space (spin).α are subsets of particlesHα = 0 when |α| > k (k = 2 in nature).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 6 / 30

Background

Setting

Classical computers need to diagonalize the matrix H, whose sizeis exponential with the number of particles (size of the lattice |Λ|).In general, quantum computers are good at simulating thetime-evolution generated by sparse Hamiltonians.Special case: k-body couplings: H =

∑α⊂Λ Hα

Λ is a lattice, with a quantum particle living at each vertex.Each particle has a finite dimensional Hilbert space (spin).α are subsets of particlesHα = 0 when |α| > k (k = 2 in nature).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 6 / 30

Background

Setting

Classical computers need to diagonalize the matrix H, whose sizeis exponential with the number of particles (size of the lattice |Λ|).In general, quantum computers are good at simulating thetime-evolution generated by sparse Hamiltonians.Special case: k-body couplings: H =

∑α⊂Λ Hα

Λ is a lattice, with a quantum particle living at each vertex.Each particle has a finite dimensional Hilbert space (spin).α are subsets of particlesHα = 0 when |α| > k (k = 2 in nature).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 6 / 30

Background

Setting

Classical computers need to diagonalize the matrix H, whose sizeis exponential with the number of particles (size of the lattice |Λ|).In general, quantum computers are good at simulating thetime-evolution generated by sparse Hamiltonians.Special case: k-body couplings: H =

∑α⊂Λ Hα

Λ is a lattice, with a quantum particle living at each vertex.Each particle has a finite dimensional Hilbert space (spin).α are subsets of particlesHα = 0 when |α| > k (k = 2 in nature).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 6 / 30

Background

Setting

Classical computers need to diagonalize the matrix H, whose sizeis exponential with the number of particles (size of the lattice |Λ|).In general, quantum computers are good at simulating thetime-evolution generated by sparse Hamiltonians.Special case: k-body couplings: H =

∑α⊂Λ Hα

Λ is a lattice, with a quantum particle living at each vertex.Each particle has a finite dimensional Hilbert space (spin).α are subsets of particlesHα = 0 when |α| > k (k = 2 in nature).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 6 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Trotter-Suzuki decomposition

eA+B = eAeB + O([A,B]).eA+B = {eA/Nt +B/Nt}Nt = {eA/Nt eB/Nt + O([A/Nt ,B/Nt ])}Nt

= {eA/Nt eB/Nt + O([A,B])/N2t }Nt = {eA/Nt eB/Nt}Nt + O([A,B])/Nt .

For H =∑

α⊂Λ Hα, we use this formula recursively and get

Exact time-evolution operator U(∆t ) = e−iH∆t .Approximate time-evolution operator UTS(∆t ) =

∏α⊂Λ e−iHα∆t .

Trotter-Suzuki error δTS := ‖U(∆t )− UTS(∆t )‖

δTS ≤m∑α=1

∥∥∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]

∥∥∥∥∥∆3t

where H>α =∑

β>α Hβ

The TS error δTS will be the object of central interest in this talk.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 7 / 30

Background

Solovay-Kitaev Theorem

Each term e−iHα∆t appearing in the TS decomposition UTS is atmost a k -body gate.Solovay-Kitaev: Given a universal gate set, e−iHα∆t can besimulated to accuracy ε using O(loga(1/ε)) gates.In this talk, we will not pay attention to this error (But seeDuclos-Cianci & D.P. arXiv:1403.5280).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 8 / 30

Background

Solovay-Kitaev Theorem

Each term e−iHα∆t appearing in the TS decomposition UTS is atmost a k -body gate.Solovay-Kitaev: Given a universal gate set, e−iHα∆t can besimulated to accuracy ε using O(loga(1/ε)) gates.In this talk, we will not pay attention to this error (But seeDuclos-Cianci & D.P. arXiv:1403.5280).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 8 / 30

Background

Solovay-Kitaev Theorem

Each term e−iHα∆t appearing in the TS decomposition UTS is atmost a k -body gate.Solovay-Kitaev: Given a universal gate set, e−iHα∆t can besimulated to accuracy ε using O(loga(1/ε)) gates.In this talk, we will not pay attention to this error (But seeDuclos-Cianci & D.P. arXiv:1403.5280).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 8 / 30

Background

Total TS error

We want to simulate time evolution for some constant time T to someconstant accuracy.

Need Nt = T/∆t time TS steps, so the total error will be NtδTS.

With δTS ≤∑m

α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t , we need

Nt ∝

(m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥) 1

2

Since ‖Hα‖ = O(1) and ‖H>α‖ = O(m), we need Nt ∝ m32 TS

time steps.In a single time-step, there are m steps, i.e.,UTS(∆t ) =

∏mα=1 e−iHα∆t .

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 9 / 30

Background

Total TS error

We want to simulate time evolution for some constant time T to someconstant accuracy.

Need Nt = T/∆t time TS steps, so the total error will be NtδTS.

With δTS ≤∑m

α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t , we need

Nt ∝

(m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥) 1

2

Since ‖Hα‖ = O(1) and ‖H>α‖ = O(m), we need Nt ∝ m32 TS

time steps.In a single time-step, there are m steps, i.e.,UTS(∆t ) =

∏mα=1 e−iHα∆t .

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 9 / 30

Background

Total TS error

We want to simulate time evolution for some constant time T to someconstant accuracy.

Need Nt = T/∆t time TS steps, so the total error will be NtδTS.

With δTS ≤∑m

α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t , we need

Nt ∝

(m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥) 1

2

Since ‖Hα‖ = O(1) and ‖H>α‖ = O(m), we need Nt ∝ m32 TS

time steps.In a single time-step, there are m steps, i.e.,UTS(∆t ) =

∏mα=1 e−iHα∆t .

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 9 / 30

Background

Total TS error

We want to simulate time evolution for some constant time T to someconstant accuracy.

Need Nt = T/∆t time TS steps, so the total error will be NtδTS.

With δTS ≤∑m

α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t , we need

Nt ∝

(m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥) 1

2

Since ‖Hα‖ = O(1) and ‖H>α‖ = O(m), we need Nt ∝ m32 TS

time steps.In a single time-step, there are m steps, i.e.,UTS(∆t ) =

∏mα=1 e−iHα∆t .

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 9 / 30

Background

Total TS error

We want to simulate time evolution for some constant time T to someconstant accuracy.

Need Nt = T/∆t time TS steps, so the total error will be NtδTS.

With δTS ≤∑m

α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t , we need

Nt ∝

(m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥) 1

2

Since ‖Hα‖ = O(1) and ‖H>α‖ = O(m), we need Nt ∝ m32 TS

time steps.In a single time-step, there are m steps, i.e.,UTS(∆t ) =

∏mα=1 e−iHα∆t .

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 9 / 30

Background

Total TS error

We want to simulate time evolution for some constant time T to someconstant accuracy.

Need Nt = T/∆t time TS steps, so the total error will be NtδTS.

With δTS ≤∑m

α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t , we need

Nt ∝

(m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥) 1

2

Since ‖Hα‖ = O(1) and ‖H>α‖ = O(m), we need Nt ∝ m32 TS

time steps.In a single time-step, there are m steps, i.e.,UTS(∆t ) =

∏mα=1 e−iHα∆t .

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 9 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Fermions

There are N fermion modes p = 1,2, . . .N, each can be occupiedor empty: dim(H) = 2N .

c†p creates a fermion in mode p while cp annihilates a fermion.Pauli exclusion principle cpcq = −cqcp if p 6= q.Free fermion operator: given N × N matrix h, define operator onH: H(h) :=

∑p,q hpqc†pcq.

Lie algebra: [H(h),H(h′)] = H([h,h′]).Evolution e−iH(h)tcpeiH(h)t = eiht~c.Simulation: Using Householder transformation, free fermionevolution can be simulated exactly on a QC, i.e. no need for TSdecomposition.Norm: ‖H(h)‖ = max{E+,E−} where E+ is the sum of the positiveeigenvalues of h.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 10 / 30

Background

Jordan-Wigner transform

Operators acting on different qubits commute.Operators acting on different fermion modes anti-commute.Map: γp = σzσz . . . σz︸ ︷︷ ︸

p−1

σx and these anti-commute.

Z Z Z Z Z Z X I I I IZ Z Z Z Z Z Z Z Z X I

Simulation: e−iθ(c†p cq+cpc†

q )t requires cascade of CNOTs

Simulating a single term e−iHα∆t in a single TS step costs O(N).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 11 / 30

Background

Jordan-Wigner transform

Operators acting on different qubits commute.Operators acting on different fermion modes anti-commute.Map: γp = σzσz . . . σz︸ ︷︷ ︸

p−1

σx and these anti-commute.

Z Z Z Z Z Z X I I I IZ Z Z Z Z Z Z Z Z X I

Simulation: e−iθ(c†p cq+cpc†

q )t requires cascade of CNOTs

Simulating a single term e−iHα∆t in a single TS step costs O(N).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 11 / 30

Background

Jordan-Wigner transform

Operators acting on different qubits commute.Operators acting on different fermion modes anti-commute.Map: γp = σzσz . . . σz︸ ︷︷ ︸

p−1

σx and these anti-commute.

Z Z Z Z Z Z X I I I IZ Z Z Z Z Z Z Z Z X I

Simulation: e−iθ(c†p cq+cpc†

q )t requires cascade of CNOTs

Simulating a single term e−iHα∆t in a single TS step costs O(N).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 11 / 30

Background

Jordan-Wigner transform

Operators acting on different qubits commute.Operators acting on different fermion modes anti-commute.Map: γp = σzσz . . . σz︸ ︷︷ ︸

p−1

σx and these anti-commute.

Z Z Z Z Z Z X I I I IZ Z Z Z Z Z Z Z Z X I

Simulation: e−iθ(c†p cq+cpc†

q )t requires cascade of CNOTs

Simulating a single term e−iHα∆t in a single TS step costs O(N).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 11 / 30

Background

Jordan-Wigner transform

Operators acting on different qubits commute.Operators acting on different fermion modes anti-commute.Map: γp = σzσz . . . σz︸ ︷︷ ︸

p−1

σx and these anti-commute.

Z Z Z Z Z Z X I I I IZ Z Z Z Z Z Z Z Z X I

Simulation: e−iθ(c†p cq+cpc†

q )t requires cascade of CNOTs

Simulating a single term e−iHα∆t in a single TS step costs O(N).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 11 / 30

Background

Jordan-Wigner transform

Operators acting on different qubits commute.Operators acting on different fermion modes anti-commute.Map: γp = σzσz . . . σz︸ ︷︷ ︸

p−1

σx and these anti-commute.

Z Z Z Z Z Z X I I I IZ Z Z Z Z Z Z Z Z X I

Simulation: e−iθ(c†p cq+cpc†

q )t requires cascade of CNOTs

Simulating a single term e−iHα∆t in a single TS step costs O(N).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 11 / 30

Background

Jordan-Wigner transform

Operators acting on different qubits commute.Operators acting on different fermion modes anti-commute.Map: γp = σzσz . . . σz︸ ︷︷ ︸

p−1

σx and these anti-commute.

Z Z Z Z Z Z X I I I IZ Z Z Z Z Z Z Z Z X I

Simulation: e−iθ(c†p cq+cpc†

q )t requires cascade of CNOTs

Simulating a single term e−iHα∆t in a single TS step costs O(N).

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 11 / 30

Molecules

Outline

1 Background

2 Molecules

3 Improvements

4 New simulation algorithm

5 Conclusion

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 12 / 30

Molecules

Why quantum simulate the dynamics of molecules

Relevant in many areas of technologies:Catalysis in chemical reactions, industrial applications.Drug design.

Intractable on classical computers, ∼60 spin orbitals max.There is a business model for a quantum computer with 100 goodqubits.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 13 / 30

Molecules

Why quantum simulate the dynamics of molecules

Relevant in many areas of technologies:Catalysis in chemical reactions, industrial applications.Drug design.

Intractable on classical computers, ∼60 spin orbitals max.There is a business model for a quantum computer with 100 goodqubits.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 13 / 30

Molecules

Why quantum simulate the dynamics of molecules

Relevant in many areas of technologies:Catalysis in chemical reactions, industrial applications.Drug design.

Intractable on classical computers, ∼60 spin orbitals max.There is a business model for a quantum computer with 100 goodqubits.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 13 / 30

Molecules

Why quantum simulate the dynamics of molecules

Relevant in many areas of technologies:Catalysis in chemical reactions, industrial applications.Drug design.

Intractable on classical computers, ∼60 spin orbitals max.There is a business model for a quantum computer with 100 goodqubits.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 13 / 30

Molecules

Why quantum simulate the dynamics of molecules

Relevant in many areas of technologies:Catalysis in chemical reactions, industrial applications.Drug design.

Intractable on classical computers, ∼60 spin orbitals max.There is a business model for a quantum computer with 100 goodqubits.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 13 / 30

Molecules

Mathematical model

Using, e.g., Hartree-Fock calculations, find N most relevant spinorbitals φp(x).Energy = Kinetic + Coulomb, project onto space spanned bychosen orbitals:

H =∑pq

hpqc†pcq +∑pqrs

hpqrsc†pcqc†r cs.

Using the notation H =∑m

α=1 Hα, we see that there arem = O(N4) terms in the Hamiltonian.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 14 / 30

Molecules

Mathematical model

Using, e.g., Hartree-Fock calculations, find N most relevant spinorbitals φp(x).Energy = Kinetic + Coulomb, project onto space spanned bychosen orbitals:

H =∑pq

hpqc†pcq +∑pqrs

hpqrsc†pcqc†r cs.

Using the notation H =∑m

α=1 Hα, we see that there arem = O(N4) terms in the Hamiltonian.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 14 / 30

Molecules

Mathematical model

Using, e.g., Hartree-Fock calculations, find N most relevant spinorbitals φp(x).Energy = Kinetic + Coulomb, project onto space spanned bychosen orbitals:

H =∑pq

hpqc†pcq +∑pqrs

hpqrsc†pcqc†r cs.

Using the notation H =∑m

α=1 Hα, we see that there arem = O(N4) terms in the Hamiltonian.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 14 / 30

Molecules

Problem

There are m = O(N4) terms in H.

We’ve seen that we need Nt ∝ m32 = N6 TS steps to achieve a

constant-accuracy constant-time evolution.Each TS steps comprises m elementary element e−iHα∆t .Realizing one such elements requires O(N) gates due to theWigner-Jordan transform.

Ng = O(N11)

Quantum chemistry may have a modest memory requirement (100qubits), it has a prohibitive gate-count based on existing techniquesand analysis.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 15 / 30

Molecules

Problem

There are m = O(N4) terms in H.

We’ve seen that we need Nt ∝ m32 = N6 TS steps to achieve a

constant-accuracy constant-time evolution.Each TS steps comprises m elementary element e−iHα∆t .Realizing one such elements requires O(N) gates due to theWigner-Jordan transform.

Ng = O(N11)

Quantum chemistry may have a modest memory requirement (100qubits), it has a prohibitive gate-count based on existing techniquesand analysis.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 15 / 30

Molecules

Problem

There are m = O(N4) terms in H.

We’ve seen that we need Nt ∝ m32 = N6 TS steps to achieve a

constant-accuracy constant-time evolution.Each TS steps comprises m elementary element e−iHα∆t .Realizing one such elements requires O(N) gates due to theWigner-Jordan transform.

Ng = O(N11)

Quantum chemistry may have a modest memory requirement (100qubits), it has a prohibitive gate-count based on existing techniquesand analysis.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 15 / 30

Molecules

Problem

There are m = O(N4) terms in H.

We’ve seen that we need Nt ∝ m32 = N6 TS steps to achieve a

constant-accuracy constant-time evolution.Each TS steps comprises m elementary element e−iHα∆t .Realizing one such elements requires O(N) gates due to theWigner-Jordan transform.

Ng = O(N11)

Quantum chemistry may have a modest memory requirement (100qubits), it has a prohibitive gate-count based on existing techniquesand analysis.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 15 / 30

Molecules

Problem

There are m = O(N4) terms in H.

We’ve seen that we need Nt ∝ m32 = N6 TS steps to achieve a

constant-accuracy constant-time evolution.Each TS steps comprises m elementary element e−iHα∆t .Realizing one such elements requires O(N) gates due to theWigner-Jordan transform.

Ng = O(N11)

Quantum chemistry may have a modest memory requirement (100qubits), it has a prohibitive gate-count based on existing techniquesand analysis.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 15 / 30

Molecules

Problem

There are m = O(N4) terms in H.

We’ve seen that we need Nt ∝ m32 = N6 TS steps to achieve a

constant-accuracy constant-time evolution.Each TS steps comprises m elementary element e−iHα∆t .Realizing one such elements requires O(N) gates due to theWigner-Jordan transform.

Ng = O(N11)

Quantum chemistry may have a modest memory requirement (100qubits), it has a prohibitive gate-count based on existing techniquesand analysis.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 15 / 30

Molecules

This was the state of affairs in early 2013.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 16 / 30

Improvements

Outline

1 Background

2 Molecules

3 Improvements

4 New simulation algorithm

5 Conclusion

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 17 / 30

Improvements

1) Wigner-Jordan

When Hpqrs and Hpqrs+1 are simulated in sequence, the CNOTscascades “almost” cancel.With a few corrections to fix commutation relations, they do.

M. B. Hastings, D. Wecker, B. Bauer and M. Troyer, arXiv:1403.1539

Average cost of implementing e−iHα∆t goes from O(N) to O(1).

O(N11)→ O(N10)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 18 / 30

Improvements

1) Wigner-Jordan

When Hpqrs and Hpqrs+1 are simulated in sequence, the CNOTscascades “almost” cancel.With a few corrections to fix commutation relations, they do.

M. B. Hastings, D. Wecker, B. Bauer and M. Troyer, arXiv:1403.1539

Average cost of implementing e−iHα∆t goes from O(N) to O(1).

O(N11)→ O(N10)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 18 / 30

Improvements

1) Wigner-Jordan

When Hpqrs and Hpqrs+1 are simulated in sequence, the CNOTscascades “almost” cancel.With a few corrections to fix commutation relations, they do.

M. B. Hastings, D. Wecker, B. Bauer and M. Troyer, arXiv:1403.1539

Average cost of implementing e−iHα∆t goes from O(N) to O(1).

O(N11)→ O(N10)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 18 / 30

Improvements

1) Wigner-Jordan

When Hpqrs and Hpqrs+1 are simulated in sequence, the CNOTscascades “almost” cancel.With a few corrections to fix commutation relations, they do.

M. B. Hastings, D. Wecker, B. Bauer and M. Troyer, arXiv:1403.1539

Average cost of implementing e−iHα∆t goes from O(N) to O(1).

O(N11)→ O(N10)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 18 / 30

Improvements

1) Wigner-Jordan

When Hpqrs and Hpqrs+1 are simulated in sequence, the CNOTscascades “almost” cancel.With a few corrections to fix commutation relations, they do.

M. B. Hastings, D. Wecker, B. Bauer and M. Troyer, arXiv:1403.1539

Average cost of implementing e−iHα∆t goes from O(N) to O(1).

O(N11)→ O(N10)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 18 / 30

Improvements

1) Wigner-Jordan

When Hpqrs and Hpqrs+1 are simulated in sequence, the CNOTscascades “almost” cancel.With a few corrections to fix commutation relations, they do.

M. B. Hastings, D. Wecker, B. Bauer and M. Troyer, arXiv:1403.1539

Average cost of implementing e−iHα∆t goes from O(N) to O(1).

O(N11)→ O(N10)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 18 / 30

Improvements

1) Wigner-Jordan

When Hpqrs and Hpqrs+1 are simulated in sequence, the CNOTscascades “almost” cancel.With a few corrections to fix commutation relations, they do.

M. B. Hastings, D. Wecker, B. Bauer and M. Troyer, arXiv:1403.1539

Average cost of implementing e−iHα∆t goes from O(N) to O(1).

O(N11)→ O(N10)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 18 / 30

Improvements

2) Trotter-Suzuki error

Bound Nt ∝ N6 is obtained from counting number of O(1) terms in

m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥.

Many of these terms are 0:Operator H>α contains O(m) = O(N4) terms.For a given Hα = Hpqrs, there are only O(N3) terms that don’tcommute with Hα (one index pqrs must be repeated).

O(N10)→ O(N9)

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 19 / 30

Improvements

2) Trotter-Suzuki error

Bound Nt ∝ N6 is obtained from counting number of O(1) terms in

m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥.

Many of these terms are 0:Operator H>α contains O(m) = O(N4) terms.For a given Hα = Hpqrs, there are only O(N3) terms that don’tcommute with Hα (one index pqrs must be repeated).

O(N10)→ O(N9)

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 19 / 30

Improvements

2) Trotter-Suzuki error

Bound Nt ∝ N6 is obtained from counting number of O(1) terms in

m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥.

Many of these terms are 0:Operator H>α contains O(m) = O(N4) terms.For a given Hα = Hpqrs, there are only O(N3) terms that don’tcommute with Hα (one index pqrs must be repeated).

O(N10)→ O(N9)

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 19 / 30

Improvements

2) Trotter-Suzuki error

Bound Nt ∝ N6 is obtained from counting number of O(1) terms in

m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥.

Many of these terms are 0:Operator H>α contains O(m) = O(N4) terms.For a given Hα = Hpqrs, there are only O(N3) terms that don’tcommute with Hα (one index pqrs must be repeated).

O(N10)→ O(N9)

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 19 / 30

Improvements

2) Trotter-Suzuki error

Bound Nt ∝ N6 is obtained from counting number of O(1) terms in

m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥.

Many of these terms are 0:Operator H>α contains O(m) = O(N4) terms.For a given Hα = Hpqrs, there are only O(N3) terms that don’tcommute with Hα (one index pqrs must be repeated).

O(N10)→ O(N9)

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 19 / 30

Improvements

2) Trotter-Suzuki error

Bound Nt ∝ N6 is obtained from counting number of O(1) terms in

m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥.

Many of these terms are 0:Operator H>α contains O(m) = O(N4) terms.For a given Hα = Hpqrs, there are only O(N3) terms that don’tcommute with Hα (one index pqrs must be repeated).

O(N10)→ O(N9)

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 19 / 30

Improvements

3) Numerical estimate

This O(N9) is only an upper bound.Triangle inequality ‖A + B‖ ≤ ‖A‖+ ‖B‖ may not be tight.For small enough molecules (20ish spin orbitals), we cannumerically simulate the quantum simulation, compute the exacttime-evolution operator, and evaluate the error.There aren’t many interesting small enough molecules, so choosecouplings hpqrs at random.

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

O(N9)→ O(N8)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 20 / 30

Improvements

3) Numerical estimate

This O(N9) is only an upper bound.Triangle inequality ‖A + B‖ ≤ ‖A‖+ ‖B‖ may not be tight.For small enough molecules (20ish spin orbitals), we cannumerically simulate the quantum simulation, compute the exacttime-evolution operator, and evaluate the error.There aren’t many interesting small enough molecules, so choosecouplings hpqrs at random.

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

O(N9)→ O(N8)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 20 / 30

Improvements

3) Numerical estimate

This O(N9) is only an upper bound.Triangle inequality ‖A + B‖ ≤ ‖A‖+ ‖B‖ may not be tight.For small enough molecules (20ish spin orbitals), we cannumerically simulate the quantum simulation, compute the exacttime-evolution operator, and evaluate the error.There aren’t many interesting small enough molecules, so choosecouplings hpqrs at random.

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

O(N9)→ O(N8)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 20 / 30

Improvements

3) Numerical estimate

This O(N9) is only an upper bound.Triangle inequality ‖A + B‖ ≤ ‖A‖+ ‖B‖ may not be tight.For small enough molecules (20ish spin orbitals), we cannumerically simulate the quantum simulation, compute the exacttime-evolution operator, and evaluate the error.There aren’t many interesting small enough molecules, so choosecouplings hpqrs at random.

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

O(N9)→ O(N8)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 20 / 30

Improvements

3) Numerical estimate

This O(N9) is only an upper bound.Triangle inequality ‖A + B‖ ≤ ‖A‖+ ‖B‖ may not be tight.For small enough molecules (20ish spin orbitals), we cannumerically simulate the quantum simulation, compute the exacttime-evolution operator, and evaluate the error.There aren’t many interesting small enough molecules, so choosecouplings hpqrs at random.

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

O(N9)→ O(N8)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 20 / 30

Improvements

3) Numerical estimate

This O(N9) is only an upper bound.Triangle inequality ‖A + B‖ ≤ ‖A‖+ ‖B‖ may not be tight.For small enough molecules (20ish spin orbitals), we cannumerically simulate the quantum simulation, compute the exacttime-evolution operator, and evaluate the error.There aren’t many interesting small enough molecules, so choosecouplings hpqrs at random.

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

O(N9)→ O(N8)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 20 / 30

Improvements

3) Numerical estimate

This O(N9) is only an upper bound.Triangle inequality ‖A + B‖ ≤ ‖A‖+ ‖B‖ may not be tight.For small enough molecules (20ish spin orbitals), we cannumerically simulate the quantum simulation, compute the exacttime-evolution operator, and evaluate the error.There aren’t many interesting small enough molecules, so choosecouplings hpqrs at random.

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings and M. Troyer, arXiv:1312.1695

O(N9)→ O(N8)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 20 / 30

Improvements

This was the state of affairs in early 2014.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 21 / 30

Improvements

4) Numerically tractable bound

Using Jacobi identity & triangle inequality, get simpler bound onTS error

δTS ≤m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t

≤ 4∑α

‖Hα‖

(∑β

′‖Hβ‖

)2

∆3t

Each term ‖Hpqrs‖ = ‖hpqrsc†pcqc†r cs‖ = |hpqrs| is a known number.Error δTS can be upper bounded efficiently and rigorouslynumerically.Can evaluate for real molecules.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 22 / 30

Improvements

4) Numerically tractable bound

Using Jacobi identity & triangle inequality, get simpler bound onTS error

δTS ≤m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t

≤ 4∑α

‖Hα‖

(∑β

′‖Hβ‖

)2

∆3t

Each term ‖Hpqrs‖ = ‖hpqrsc†pcqc†r cs‖ = |hpqrs| is a known number.Error δTS can be upper bounded efficiently and rigorouslynumerically.Can evaluate for real molecules.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 22 / 30

Improvements

4) Numerically tractable bound

Using Jacobi identity & triangle inequality, get simpler bound onTS error

δTS ≤m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t

≤ 4∑α

‖Hα‖

(∑β

′‖Hβ‖

)2

∆3t

Each term ‖Hpqrs‖ = ‖hpqrsc†pcqc†r cs‖ = |hpqrs| is a known number.Error δTS can be upper bounded efficiently and rigorouslynumerically.Can evaluate for real molecules.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 22 / 30

Improvements

4) Numerically tractable bound

Using Jacobi identity & triangle inequality, get simpler bound onTS error

δTS ≤m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t

≤ 4∑α

‖Hα‖

(∑β

′‖Hβ‖

)2

∆3t

Each term ‖Hpqrs‖ = ‖hpqrsc†pcqc†r cs‖ = |hpqrs| is a known number.Error δTS can be upper bounded efficiently and rigorouslynumerically.Can evaluate for real molecules.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 22 / 30

Improvements

4) Numerically tractable bound

Using Jacobi identity & triangle inequality, get simpler bound onTS error

δTS ≤m∑α=1

∥∥∥[[Hα,H>α],Hα] + [[H>α,Hα],H>α]∥∥∥∆3

t

≤ 4∑α

‖Hα‖

(∑β

′‖Hβ‖

)2

∆3t

Each term ‖Hpqrs‖ = ‖hpqrsc†pcqc†r cs‖ = |hpqrs| is a known number.Error δTS can be upper bounded efficiently and rigorouslynumerically.Can evaluate for real molecules.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 22 / 30

Improvements

Scaling for real molecules

101 102100

101

102

103

104

105

106

107

Number of Spin orbitals N

Num

ber

ofT

Sst

eps

1/�

t

Real Molecules

Artific

ialM

olecu

lesFu

ll

N4.81

N1.70

N4.83

Artific

ialM

olecu

lesSp

arse

DP, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, arXiv:1406.4920

O(N8)→ O(N5.5−6)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 23 / 30

Improvements

Scaling for real molecules

101 102100

101

102

103

104

105

106

107

Number of Spin orbitals N

Num

ber

ofT

Sst

eps

1/�

t

Real Molecules

Artific

ialM

olecu

lesFu

ll

N4.81

N1.70

N4.83

Artific

ialM

olecu

lesSp

arse

DP, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, arXiv:1406.4920

O(N8)→ O(N5.5−6)

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 23 / 30

New simulation algorithm

Outline

1 Background

2 Molecules

3 Improvements

4 New simulation algorithm

5 Conclusion

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 24 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Alternative TS decomposition

Free (quadratic) fermion Hamiltonians are easy to simulate.Some quartic Hamiltonians are easy to simulate, e.g. H = H2

Free.Viewing the list of coefficients hpqrs as a N2 × N2 matrixMµν = h(pq),(rs) and performing a spectral decompositionM = VDV †, we can write any quartic Hamiltonian

H =N2∑µ=1

dµ(HFreeµ )2.

Can perform TS decomposition in terms of HFreeµ .

To simulate e−i(HFreeµ )2∆t :

Rotate into eigenmodes of HFreeµ (mode transform = free evolution).

Apply phase gates to simulate diagonal Hamiltonian squared.

Terms HFreeµ are no sparser than H.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 25 / 30

New simulation algorithm

Advantages

TS decomposition now has only N2 terms instead of N4.Each term has norm O(N2) instead of O(1).In principle these things cancel out.In terms of a bound, decomposition with fewer terms makes feweruses of triangle inequality.

Expect to get a tighter bound.‖∑N

j=1 Xj‖ ∼√

N ≤∑N

j−1 ‖Xj‖.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 26 / 30

New simulation algorithm

Advantages

TS decomposition now has only N2 terms instead of N4.Each term has norm O(N2) instead of O(1).In principle these things cancel out.In terms of a bound, decomposition with fewer terms makes feweruses of triangle inequality.

Expect to get a tighter bound.‖∑N

j=1 Xj‖ ∼√

N ≤∑N

j−1 ‖Xj‖.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 26 / 30

New simulation algorithm

Advantages

TS decomposition now has only N2 terms instead of N4.Each term has norm O(N2) instead of O(1).In principle these things cancel out.In terms of a bound, decomposition with fewer terms makes feweruses of triangle inequality.

Expect to get a tighter bound.‖∑N

j=1 Xj‖ ∼√

N ≤∑N

j−1 ‖Xj‖.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 26 / 30

New simulation algorithm

Advantages

TS decomposition now has only N2 terms instead of N4.Each term has norm O(N2) instead of O(1).In principle these things cancel out.In terms of a bound, decomposition with fewer terms makes feweruses of triangle inequality.

Expect to get a tighter bound.‖∑N

j=1 Xj‖ ∼√

N ≤∑N

j−1 ‖Xj‖.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 26 / 30

New simulation algorithm

Advantages

TS decomposition now has only N2 terms instead of N4.Each term has norm O(N2) instead of O(1).In principle these things cancel out.In terms of a bound, decomposition with fewer terms makes feweruses of triangle inequality.

Expect to get a tighter bound.‖∑N

j=1 Xj‖ ∼√

N ≤∑N

j−1 ‖Xj‖.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 26 / 30

New simulation algorithm

Advantages

TS decomposition now has only N2 terms instead of N4.Each term has norm O(N2) instead of O(1).In principle these things cancel out.In terms of a bound, decomposition with fewer terms makes feweruses of triangle inequality.

Expect to get a tighter bound.‖∑N

j=1 Xj‖ ∼√

N ≤∑N

j−1 ‖Xj‖.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 26 / 30

New simulation algorithm

Scaling for real molecules

101 102100

101

102

103

104

105

106

107

Number of Spin orbitals N

Num

ber

ofT

Sst

eps

1/�

t

Real Molecules

Artific

ialM

olecu

lesFu

ll

N4.81

N1.70

N4.83

Artific

ialM

olecu

lesSp

arse

DP, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, arXiv:1406.4920

Case-by-case, but both upper bounds can be computed efficiently.David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 27 / 30

New simulation algorithm

Scaling for real molecules

101 102100

101

102

103

104

105

106

107

Number of Spin orbitals N

Num

ber

ofT

Sst

eps

1/�

t

Real Molecules

Artific

ialM

olecu

lesFu

ll

N4.81

N1.70

N2.56

N4.83 N

5.21

N4.71

Artific

ialM

olecu

lesSp

arse

DP, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, arXiv:1406.4920

Case-by-case, but both upper bounds can be computed efficiently.David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 27 / 30

New simulation algorithm

Scaling for real molecules

101 102100

101

102

103

104

105

106

107

Number of Spin orbitals N

Num

ber

ofT

Sst

eps

1/�

t

Real Molecules

Artific

ialM

olecu

lesFu

ll

N4.81

N1.70

N2.56

N4.83 N

5.21

N4.71

Artific

ialM

olecu

lesSp

arse

DP, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, arXiv:1406.4920

Case-by-case, but both upper bounds can be computed efficiently.David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 27 / 30

Conclusion

Outline

1 Background

2 Molecules

3 Improvements

4 New simulation algorithm

5 Conclusion

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 28 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

Practically relevant quantum chemistry simulations could berealize on a quantum computer containing only hundreds ofqubits.Despite vast literature, prior analysis mainly focused on spacerather than time complexity.

Recent studies show that time complexity is a big issue with thisapplication.

Room for plenty of improvementsAlgorithms.Analysis.Not shown are many other tricks that don’t change scaling butreduce complexity by large constant factor.

TS decomposition not only based on sparsity.Applications in other algorithms?

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 29 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30

Conclusion

How many Piano tuners in Chicago?

Iron sulfur Fe2S2

Bases with 112-168 spin orbitals.Outside range of good classical algorithms (50 orbitals).Considered as good test case (IARPA)

107 − 108 gate depts per TS step.Use 103 − 104 TS steps to estimate ground state energy tomilliHartree accuracy.Each gate compiled from universal set: 100. (distillation is offline)Circuit optimization drop by 10− 1000 (coalescing, angleminimizing compiling, more offline)100 nanosecond per gate?1 minute - 10 days.

David Poulin (Sherbrooke) Quantum Chemistry UNM’ 14 30 / 30