ROS: Resource-constrained Oracle Synthesis for Quantum Computers

@martinquantum

martinro@microsoft.com

microsoft.com/quantum

Motivation: oracle synthesis

6/13/2019 QPL 2019 2

Arithmetic:Factoring: needs “constant” modular arithmeticECC dlogs: need generic modular arithmetic,

need to execute non-trivial straight-line programsHHL: need sparse matrix access, need integer inverses; Newton type methods

Amplitude amplification: Implementation of “oracles” / subroutines, e.g., for search, collision etc. Implementation of walk operators on data structures

Quantum simulation: Addressing/indexing functions for sparse matricesComputing Hamiltonian terms on the fly

k-LUT based oracle synthesis

Logicnetwork

k-LUT network

Reversible network

Quantum circuit

k-LUTmapping

mappinginto qubits

Graysynthesis

6/13/2019 QPL 2019 3

Logic Network

Hierarchical methods are based on logic networks rather than on less scalable logic representations, i.e. truth tables.

A logic network is a graph, in which each node represents a Boolean function and edges define data dependency.

A network is characterized by the operations that it supports.

x1 x2 x3 x4

f

AND

AND

AND

And-Inverter Graph (AIG) Xor-And-inverter Graph (XAG)

x1 x2

f

AND

XOR

x3

Majority-Inverter Graph (MIG)

x1 x2

f

Maj

x3

Maj

1

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k-LUT mapping

Hierarchical methods use k-LUT mapping to decompose the original network.

A k-LUT is a subnetwork with maximum k inputs.

The decomposition is not unique.

6/13/2019 QPL 2019 5

Mapping into qubits

This step exploits the one-to-one correspondence between a LUT node and a single-target gate.

f

abcdeg

abcdeg ⊕ f(a, b, c, d)

𝑇𝑐 𝑥1, … , 𝑥𝑘 , 𝑥𝑘+1𝑥1, … , 𝑥𝑘 is a set of control lines𝑥𝑘+1 is the target line𝑓: 𝐵𝑘 → 𝐵 is the control function

x1 x2 x3 x4

g

n3

n1 n2

𝑛1 = x1 ∧ x2𝑛2 = 𝑥3⨁ ҧ𝑥4𝑛3 = (𝑛1 ∧ 𝑛2) ∨ 𝑥3

n1

n2 n2

n1

n3

x1

x2

x3

x4

x1

x2

x3

x4

g

Each auxiliary qubit stores one intermediate result.

The order in which operations are computed affects the number of qubits and the number of gates.

6/13/2019 QPL 2019 6

Single target gate (STG)

k-LUT based oracle synthesis

Graphk-LUT

networkReversible network

Quantum circuit

k-LUTmapping

mappinginto qubits

Graysynthesis

During this step we define:• Number of nodes in the k-LUT network• Complexity of the function performed by each node

6/13/2019 QPL 2019 7

Gray synthesis: Phase polynomial

M. Amy, P. Azimzadeh, and M. Mosca, “On the CNOT-complexity of CNOT-phase circuits,” arXiv:1712.01859, 2017

6/13/2019 QPL 2019 8

A {CNOT, RZ} circuit has associated unitary:

𝑈𝐶 =

𝒙∈𝐹2𝑛

𝑒2𝜋𝑖𝑓(𝒙)ȁ ۧ𝐴𝒙 ۦ ȁ𝒙

𝑓 𝒙 =

𝒚∈𝐹2𝑛

መ𝑓(𝒚) (𝑥1𝑦1⨁𝑥2𝑦2⨁…⨁𝑥𝑛𝑦𝑛)

The expression 𝑓(𝑥) is a weighted sum of parities in 𝒙.

Computing a minimal parity network for a particular set of parity functions is shown to be equivalent to finding a CNOT-optimal circuit for a particular phase polynomial

𝐴: ȁ ۧ𝑥1 ȁ ۧ𝑥2 ȁ ۧ𝑥3 → ȁ ۧ𝑥1 ȁ ۧ𝑥1⨁𝑥2 ȁ ۧ𝑥1⨁𝑥2⨁𝑥3

𝑓 𝑥 =𝜋

4𝑥1⨁𝑥2 +

7

4𝜋(𝑥1⨁𝑥2⨁𝑥3)

Example

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Synthesis of single-target gates

When a STG performing ȁ ۧ𝑥1 ȁ ۧ𝑥2 ȁ ۧ𝑥3 → ȁ ۧ𝑥1 ȁ ۧ𝑥2 ȁ ۧ𝑥1⨁𝑓1(𝑥1, 𝑥2)

is synthesized into a {CNOT, RZ} circuit, the CNOTs are used to generate linear combinations of the inputs.

The rotation angles are proportional to the Rademacher-Walsh spectral coefficients of the function

ො𝑔 = 𝑥3 ∧ 𝑓1(𝑥1, 𝑥2)

𝜃𝑖 =𝜋𝑠𝑖2𝑛

A function with many zeros as spectral coefficients will be translated into less gates.

6/14/2019 QPL 2019 10

Space-time tradeoffs: pebbling

Example:

Rules of the game: [Bennett, SIAM J. Comp., 1989]• n boxes, labeled i = 1, …, n• in each move, either add or remove a pebble• a pebble can be added or removed in i=1 at any time• a pebble can be added of removed in i>1 if and only if there is a pebble in i-1.

1 2 3 4

# i

1 12 23 34 45 36 27 1

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Space-time tradeoffs: pebbling

Example: (n=3, S=3)

Imposing resource constraints:• only a total of S pebbles are allowed• corresponds to reversible algorithm with at most S auxiliary qubits

1 2 3 4

# i

1 12 23 34 15 46 37 18 29 1

Mapping into qubits: pebbling strategy

This step defines the number of auxiliary qubits.It can be optimized by changing the order in which the graph is traversed.

a b

f

e

dc

x1 x2 x3x4

y1

y2 inputs

6 ancillae

outputs

x1

x2

x3

x4

ȁ ۧ0

ȁ ۧ0

ȁ ۧ0

ȁ ۧ0

a ab b

c cd

e

d

f

ȁ ۧ0

ȁ ۧ0 y2

y1

ȁ ۧ0

x1

x2

x3

x4

ȁ ۧ0

ȁ ۧ0

ȁ ۧ0

6/13/2019 QPL 2019 12

Mapping into qubits: pebbling strategy

a b

f

e

dc

x1 x2 x3x4

y1

y2 inputs

5 ancillae

x1

x2

x3

x4

y2

y1

ȁ ۧ0

ȁ ۧ0

ȁ ۧ0

ȁ ۧ0

a ab b

c cd

e

d

f

ȁ ۧ0

x1

x2

x3

x4

ȁ ۧ0ȁ ۧ0

ȁ ۧ0

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This step defines the number of auxiliary qubits.It can be optimized by changing the order in which the graph is traversed.

Mapping into qubits: pebbling strategy

a b

f

e

dc

x1 x2 x3x4

y1

y2 inputs

4 ancillae

x1

x2

x3

x4

y2

y1

ȁ ۧ0

ȁ ۧ0

ȁ ۧ0

ȁ ۧ0

a a

b bc d

e

df

a

b

a

cb

ȁ ۧ0

x1

x2

x3

x4

ȁ ۧ0

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This step defines the number of auxiliary qubits.It can be optimized by changing the order in which the graph is traversed.

Results

6/13/2019 QPL 2019 15

github.com/microsoft/quantum/

docs.microsoft.com/quantum/

microsoft.com/quantum