An Arbitrary Two-qubit Computationin 23 Elementary Gates or Less

Stephen S. Bullockand Igor L. Markov

University of Michigan

Departments of Mathematics and EECS

Outline

Introduction A crash-course in quantum circuits

Prior work Synthesis by matrix factorizations

Our contribution Entanglers and disentanglers Recognizing tensor products Circuit decompositions

Conclusions and on-going work

Introduction

Abstract synthesis of logic circuits Input: a function that represents a computation Output: a circuit that implements that function Minimize circuit size

Our focus: two-qubit quantum computation Quantum states are complex vectors Computations and gates are 4x4-unitary matrices

Existing commercial applications Secure communication: quantum key distribution

(circuits are small, but gates are very expensive) Future applications: quantum computing

Quantum Information

Most popular carriers … Polarization of an individual photon Spin of an individual nucleus or electron Energy-level of an individual electron

Common features “Basis” states, e.g., and or and Linear combinations are allowed

|0>+|1>; ||2+||2=1; , are complex

Fundamental change from classical info

Special sessiontomorrow : Cambridge, NISTLos Alamos, and Michigan

Classical Models Quantum Models

0-1 strings

E.g., one bit{0,1}

Bool. Functions Gates & circuits

Primary outputs

Lin. combinationsof 0-1 strings E.g., one qubit

|0>+|1> Matrices

Gates & circuits Probabilistic

measurement Mostly ignored here

From Classical to Quantum

All quantum computations M must be unitary M x M-conjugate-transpose = I M-1 (recall “reversible computations”)

A conventional reversible gate/computationcan be extended by linearity E.g., a quantum inverter swaps |0> and |1> Maps the state (|0>+|1>)/2 to itself

Can apply an inverter on one of two qubits E.g., (|00>+i|11>)/2 (|01>+i|10>)/2

0 11 0

Quantum Circuits

Can apply an inverter on one of two qubits E.g., (|00>+i|11>)/2 (|01>+i|10>)/2

How do we describe this computation? Tensor product: IdentityNOT More generally: AB 0 1 0 0

1 0 0 00 0 0 10 0 1 0

0 0 1 00 0 0 11 0 0 00 1 0 0

CNOT gatex

y

xyx

Elementary Gates Q. Computation

Technology-indep. Synthesis

Input: Unitary 4x4-matrix M Generic quantum computation on 2 qubits

Output: circuit in terms of elem. gates that implements M up to a constant

Minimize: circuit cost E.g., gate count or (gate costs)

Solutions existiff the gate library is universal

Phase can be ignored

Gate 1 Gate 2 Gate 3

Previous Work

Proof of universality is constructive [Barenco et. al `95] in Phys. Rev. A Can be interpreted as a synthesis algorithm However, no attempt to minimize #gates

Can be viewed as matrix factorization [Cybenko `01] M=QR with unitary Q & upper-triangular R

(M unitary R diagonal) We count gates, and the answer is 61

Our Work (1)

Two new synthesis procedures (2 qubits) Based on a different matrix factorization

(related to SVD and KAK decompositions) Also reduce generic synthesis to diag synth Small circuits for diagonal computations Smaller overall circuits than QR (Cybenko)

Constructive worst-case bounds Any circuit in gates or less Any circuit with 15 non-constant gates

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Our Work (2)

Lower bounds two-qubit computations (most of them)

that require at least 17 elementary gates At least 15 non-const gates At least 2 CNOTs

Bounds are not constructive and not tight,except for “15 non-const gates”

We never use “temporary storage” qubitsbut this could lead to smaller gate counts

The Entangler and Disentangler

“Computational basis” |00>,|01>,|10> and |11>

The “entangler” computation maps|00> to (|00>+|11>)/2, etc.

The “disentangler”is E-1=E*

Key lemma If U=AB, then EUE* has only real entries An efficient way to recognize tensor products

Circuits For E and E*

A specific circuit for the entangler E

S=diag(1,i) counts as one elementary gate The Hadamard gate H counts as two E* is implemented by reversing the diagram

Change S to S-1=diag(1,-i)

7 elem. gates

The “canonical decomposition” for 2-qubit computations:

U K1,K2 and such that U=K1K2

EE* is diagonal (5 gates) K1,K2 have only real entries

The terms K1, K2 and can be found explicitly Numerical analysis: polar and spectral decompositions

Reduce K1 and K2 to tensor products using entanglers EUE*=E(AB)E*EE*E(CD)E* A,B,C and D are one-qubit computations: 3 gates each

Note that E and E* are the same for any input

Our (Key) Synthesis Procedure

Rz RzRz

Rz

Details (1)

After the initial “divide-and-conquer”many gate cancellations can be made

This brings down max #gates to 28 Only 15 of them depend on input,

which matches an a priori lower bound Further reductions from the analysis

of E(AB)E* and E(CD)E* Max #gates reduced to However, 19 gates depend on the input

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Details (2)

The structure of the 23-gate circuit

For additional details, see Our paper in Physical Review A http://xxx.lanl.gov/abs/quant-ph/0211002

Validation of Our Synthesis Algo

Implementation in C++ Produces 23 gates for randomly-

generated 4x4-unitary matrices Can capture structure:

several examples in quant-ph/0211002 Optimal results for any AB circuit

(QR decomposition typically 61 gates) For 2-qubit Fourier transform:

a circuit with minimal # of CNOT gates

Conclusions and On-going Work

First generic synthesis algorithm to capture circuit structure, e.g., AB

On-going work Lower and upper bounds of gates (almost done) Solved synthesis of n-qubit diagonal computations

(produce circuits within 2x from optimal)

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Thank you!

Questions are welcome

Quantum dots Nuclear Magnetic Resonance

Ion Traps

Thank you!