digital quantum simulation - uni stuttgart€¦ · plaquette interaction (light red) a ... g r y y...
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Digital Quantum Simulation
Hendrik Weimer
Institute for Theoretical Physics, Leibniz University Hannover
Blaubeuren, 22 July 2014
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Leibniz University Hannover
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Course on Quantum Simulation
Leibniz Universitat Hannover, Summer 2013
Lecture notes:
http://v.gd/qsim2013
Wikiversity page:
http://en.wikiversity.org/wiki/Quantum Simulation
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Outline
1 Why quantum simulation?
2 Simulation of coherent dynamics
3 Dissipative quantum state engineering
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Outline
1 Why quantum simulation?
2 Simulation of coherent dynamics
3 Dissipative quantum state engineering
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Strongly correlated systems
Strongly correlated systemsare difficult to describetheoretically
Exponentially growingHilbert space dimension
High-temperaturesuperconductors
Quark bound states(protons, neutrons)
Frustrated quantum magnets
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Exact diagonalization
Slightly misleading: only as “exact” as your computer’s arithmeticprecision
Time-independent Schrodinger equation
H|ψ〉 = E|ψ〉
Ground state: find |ψ〉 such that 〈ψ|H|ψ〉 is minimal
Example: Transverse field Ising chain
H = g∑i
σ(i)x −
∑i
σ(i)z σ(i+1)
z
Exponential complexity: dimH = 2N
N = 40: 8 TB of memoryN = 300: more basis states than atoms in the universe!
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Quantum Monte-Carlo
Quantum-classical mapping for the partition function
Z = Tr {exp(−βH)} = Tr
{exp
[−β(g∑i
σ(i)x −
∑i
σ(i)z σ(i+1)
z
)]}
Suzuki-Trotter formula:
exp
[1
N(A+B)
]= exp
(A
N
)exp
(B
N
)+O(1/N2)
Z = limNy→∞
Tr
exp
(− βgNy
∑i
σ(i)x
)Ny
exp
(β
Ny
∑i
σ(i)z σ(i+1)
z
)Ny
Ny multiplications: additional dimension
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Quantum Monte-Carlo II
Classical partition function
Z = ΛNNyTr
exp
γ N∑i=1
Ny∑j=1
σ(i,j)z σ(i,j+1)
z +β
Ny
N∑i=1
Ny∑j=1
σ(i,j)z σ(i+1,j)
z
Classical temperature βcl = β/Ny 6= β
Solve using standard Monte Carlo methods (Metropolis algorithm)N. Metropolis et. al., J. Chem. Phys. 21, 1087 (1953)
Quantum-classical mapping does not always work
H = J∑〈ij〉
σ(i)+ σ
(i)− + H.c.
Antiferromagnetic interaction on a non-bipartite lattice: negativeprobabilities in the corresponding classical model (sign problem)
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Density-Matrix Renormalization Group
Matrix product state (MPS) ansatzU. Schollwock, Ann. Phys. 326, 92 (2011)
|ψ〉 =∑i
Tr
N∏j=1
Aj
|i〉A is a D ×D-dimensional matrix (D � dimH)How much information does an MPS contain? Entanglement entropy
S = −Tr {ρA log ρA} ≤ 2 logD
Area law of entanglement entropyJ. Eisert et al., Rev. Mod. Phys. 82, 277 (2010)
S(ρA) . A(A)
MPS are good only for one-dimensional systems with short-rangedinteractions
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Universal quantum simulator
Simulating quantum mechanics with otherquantum systemsR. P. Feynman, Int. J. Theo. Phys. 21, 467 (1982)
Universal Quantum Simulator (UQS):device simulating the dynamics of anyother quantum system with short-rangeinteractionsS. Lloyd, Science 273, 1073 (1996)
Digital quantum simulator: UQS fordiscrete time steps
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Analog vs digital
HW, M. Muller, H. P. Buchler, I. Lesanovsky, Quant. Inf. Proc. 10, 885 (2010)
Analog quantum simulators have dominant two-body interactionsStrength of three-, four-, five-body interactions decays exponentiallyTurning off two-body interactions requires enormous fine-tuningDigital quantum simulator is non-perturbative and does not requirefine-tuning
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Outline
1 Why quantum simulation?
2 Simulation of coherent dynamics
3 Dissipative quantum state engineering
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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The Toric Code
Control spins (red) and ensemble spins(blue) on a 2D lattice
Plaquette interaction (light red)
Ap = σ(i)x σ(j)
x σ(k)x σ(l)
x
Site interaction (green)
Bs = σ(i)z σ(j)
z σ(k)z σ(l)
z
Control spins only mediate the interactionsToric code HamiltonianA. Kitaev, Ann. Phys. 303, 2 (2003)
H = −E0
(∑p
Ap +∑s
Bs
)Each Ap and Bs has to eigenvalues ±1 (eightfold degenerate)All Ap and Bs commute: ground state has Ap = 1 and Bs = 1
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Anyonic excitations
Excitations are created by flipping individual spins
|ψ′〉 = σ(i)x |ψ〉
Always in pairs: excitation gap 2E0, two kinds (σx and σz)Moving them around: string operator
|ψ′′〉 = σ(j)x |ψ′〉 = σ(j)
x σ(i)x |ψ〉
But: if second excitation is present
|φ′〉 = σ(l)x σ(k)
x σ(j)x σ(i)
x |φ〉 = −|φ〉Anyonic statistics
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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From N + 1-body gates to N -body interactions
Goal: simulate an effective plaquette interactionH = Ap = σ
(i)x σ
(j)x σ
(k)x σ
(l)x
Implement the time dependent Schrodingerequation
i~d
dt|ψ〉 = H|ψ〉
Time-independent Hamiltonian
|ψ(t)〉 = exp(−iHt/~)|ψ(0)〉
Here: implement U = exp(−iHt/~) using singlequbit gates plus Controlled-NOTN
UCNOTN = |0〉〈0|cN⊗i=1
1i + |1〉〈1|cN⊗i=1
σxi
sj
sj
s
j...
..
.
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Mapping onto control atom
Map |±1〉 eigenstates of Ap onto |0〉, |1〉 of the control atom
Ry = exp(−iπσy/4) =1√2
(1 −11 1
)
|0〉 |0〉|0〉 |1〉
|+〉|+〉|−〉|−〉
G
Ry R†y
CN
OTN
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Coherent dynamics
Apply the mapping G transferring the eigenvalue of Ap onto thecontrol spin
Write a phase exp(−iφσz) onto the control spin
Undo the mapping G = G−1
|0〉 |0〉
G G†
e−iφσz
e−iHτ/~
Reset
Simulates the Hamiltonian H at discrete times t = kτ (digital)
Energy scale E0 = ~φ/τHW, M. Muller, I. Lesanovsky, P. Zoller, H. P. Buchler, Nature Phys. 6, 382 (2010)
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Scaling up to the full lattice model
Implement Bs = σ(i)z σ
(j)z σ
(k)z σ
(l)z the same way as Ap by swapping
σx for σz using local gatesTotal Hamiltonian
H = −E0
(∑p
Ap +∑s
Bs
)=∑λ
hλ
Suzuki-Trotter decomposition
U(τ) = exp(−iHτ/~) =∏λ
exp(−hλτ/~) +O(τ2)
Straightforward parallelimplementation
But: only one hλ acting oneach spin at a time
HW, Mol. Phys. 111, 1753 (2013)
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Robustness against errors
Trotterization errors due to non-commuting interactions
Errors in many-body gate
Ug = |0〉〈0|c ⊗ eiφQ + |1〉〈1|c ⊗Ap
Modified Hamiltonian h = −(Ap +Q), additional dephasing
ρ→ ρ− iφ [h, ρ]− φ2
2[h, [h, ρ]] +
φ2
2
(2QρQ−
{Q2, ρ
})Tolerable if quantum phase stable against fluctuations (no errorcorrection required)
Dephasing leads to an effective heating
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Outline
1 Why quantum simulation?
2 Simulation of coherent dynamics
3 Dissipative quantum state engineering
Hendrik Weimer (Leibniz University Hannover) Digital Quantum Simulation
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Quantum state engineering
Tailored quantum states are an importantresource (quantum simulation, quantumcommunication, quantum metrology, ...)
Previously: coherent evolution (adiabaticfollowing, quantum logic gates)
New tool: controlled dissipationS. Diehl et al., Nature Phys. 4, 878 (2008)
F. Verstraete et al., Nature Phys. 5, 633 (2009)
HW et al., Nature Phys. 6, 382 (2010)
Engineer a suitable attractor state of thedynamics
Inherently more robust
Position
Vel
ocity
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Quantum master equation
Open quantum system described by a quantum master equation(Lindblad form)
dρ
dt= −i~ [H, ρ] +
∑n
γn
(cnρc
†n −
1
2{c†ncn, ρ}
)
ρ =∑i
pi|ψi〉〈ψi| Density operator
H Hamiltonian
cn Quantum jump operators (non-Hermitian)
γn Decay rate
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Stationary state
Sufficient (and usually also necessary) condition: dρdt = 0
Special case: pure state ρ = |ψ〉〈ψ|More general: von Neumann entropy
S = −Tr {ρ log ρ}Only Hermitian jump operators⇒ Maximally mixed state (S = log d)
ρ =
1/d1/d
. . .
dρ
dt= −i~ [H, ρ] +
∑n
γn
(cnρc
†n −
1
2{c†ncn, ρ}
)
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Dissipative cooling
Goal: cool into the ground state of H = Ap (−1 eigenstate)
Use the same mapping (ensemble 7→ control) as before
Instead of writing a phase on the control spin: controlled spin flip ofone random ensemble spin j
U = |0〉〈0|c ⊗ 1 + |1〉〈1|c ⊗ exp(iφσ(j)z )
If we do a spin flip: control atom will not end in |0〉Reset spin (incoherent) from |1〉 to |0〉Discrete Markovian master equation
ρ(t+ τ) = ρ(t) + γ
(cρc† − 1
2
{c†c, ρ
})Rate γ = φ2/τ , jump operator c = σ
(j)z (1 +Ap)/2
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Scaling up
Cooling: random walk of the anyons
Averaging over 103 realizations of the dynamics
Imperfections: residual anyon density n
0
0.1
0.2
0.3
0.4
0 20 40 60t[γ−1]
n
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Linear response theory
Gate error probability ε: probability to endup in a state orthogonal to the desired one
Toric code: gate errors create anyons
0
0.4
0.8
1.2
1.6
0 0.01 0.02 0.03 0.04 0.05
T[E
0/kB]
ε
0
0.2
0.4
0.6
0 2 4 6 8 10
n
t[h/E0]
HW, Mol. Phys. 111, 1753 (2013)
Uncorrelated errors⇒Effective temperature
T ≈ − 2E0
kB log n
Anyon density n within linearresponse: n = 14ε
⇒ Effective temperaturebenchmarks the quantumsimulator
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Experimental realization
Proof of principle experiment with trapped ions
N -body Mølmer-Sørensen gateA. Sørensen and K. Mølmer, Phys. Rev. Lett. 82, 1971 (1999)
Four ensemble spins + 1 control ion
Minimal instance of a toric code Hamiltonian (1 plaquette)
J. Barreiro et al., Nature 470, 486 (2011)
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Herrenhausen Castle
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Herrenhausen Castle
Funded by:
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PhD and Postdoc Positions Available!
Freigeist project “Quantum States on Demand”
Quantum state engineering
Dissipative many-body quantum dynamics
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H = U∑o
(∑i∈o σ
(i)z
)2− J∑
pBp + V
∑pB2p
Ring-exchange Bp = σ+σ−σ+σ− + h.c via gate sequence
Low-energy sector (U � J, V ): three spins up/down on eachoctahedron
V = J : Rokhsar-Kivelson point (non-stabilizer state)
V < J : Spin liquid phase with Coulombic 1/r interactions
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2D Fermi-Hubbard model isbelieved to be realized inhigh-temperature cupratesuperconductors
H = −t∑<ij>σ
c†iσcjσ +U∑i
ni↑ni↓
Mapping fermions onto spins:Jordan-Wigner transformation
Problem in 2D: Wigner strings(highly nonlocal interactions)
Solution: Introduce auxillaryfermion fieldVerstrate, Cirac, J. Stat. Mech. 2005, P09012
(2005)
1 1’ 2 2’ 3 3’ 4 4’
5 5’6 6’7 7’8 8’
9 9’ 10 10’ 11 11’ 12 12’
13 13’14 14’15 15’16’16
HW, M. Muller, H. P. Buchler, I.
Lesanovsky, Quant. Inf. Proc. 10, 885
(2010)
Haux = −V∑{i,j}σ
Pi′,j′Pj′+1,i′−1
Pi′,j′ = (di′σ+d†i′σ)(dj′σ−d†j′σ)
Results in localsix-body interactions
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