A Resource-Optimal Canonical Form for Single-qubit Quantum Circuits

Alex Bocharov, Krysta M. Svore

Published Jan 1, 2012• 6 citations

Abstract

Abstract. Determining the optimal implementation of a quantum gate is critical for designing a quantum computer. We consider the crucial task of efficiently decomposing a general single-qubit quantum gate into a sequence of fault-tolerant quantum operations. For a given single-qubit circuit, we construct an optimal gate sequence consisting of fault-tolerant Hadamard (H) and π/8 rotations (T). Our scheme is based on a novel canonical form for single-qubit quantum circuits and the corresponding rules for exactly reducing a general single-qubit circuit to our canonical form. The result is optimal in the number of T gates. We demonstrate that a precomputed ɛ-net of canonical circuits in combination with our scheme lowers the depth and number of T gates of approximation circuits by up to three orders of magnitude compared to previously reported results.

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OpenAlex

Abstract

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Optimal Hadamard gate count for Clifford

+T

synthesis of Pauli rotations sequences

The Clifford

+T

gate set is commonly used to perform universal quantum computation. In such setup the

T

gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of

T

gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial

T

-count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of

T

gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of

π/4

Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last

T

gate of the circuit.

We develop efficient algorithms for compiling single-qubit unitary gates into circuits over the universal

V

basis. The

V

basis is an alternative universal basis to the more commonly studied basis consisting of Hadamard and $\ensuremath{\pi}/8$ gates. We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time [in precision $\mathrm{log}(1/\ensuremath{\epsilon})$] and produces an $\ensuremath{\epsilon}$ approximation to a single-qubit unitary with a circuit depth $\ensuremath{\le}12\phantom{\rule{0.16em}{0ex}}{\mathrm{log}}_{5}(2/\ensuremath{\epsilon})$. The second algorithm performs optimized direct search and yields circuits a factor of 3 to 4 times shorter than our first algorithm, but requires time exponential in $\mathrm{log}(1/\ensuremath{\epsilon})$; however, we show that in practice the runtime is reasonable for an important range of target precisions. Decomposing into the

V

basis may offer advantages when considering the fault-tolerant implementation of quantum circuits.

T

-count that equals

3 \log_2(1/ε)

for a target precision

ε

. Magnitude Approximation can reduce the

T

-count to only

1 \log_2(1/ε)

by allowing large residual errors, which are rotations about orthogonal axes. Within a complete quantum circuit, these residual errors can then be absorbed into neighboring gates before they are approximated themselves. Determining the optimal allocation of approximation strategies within a large, multi-qubit circuit presents a significant combinatorial challenge. In this work, we present a linear-time algorithm that guarantees an optimal solution to this problem. We demonstrate that the issue of delegating Magnitude versus Diagonal approximation across a circuit maps formally to a classical 1D Ising model with a spatially varying field. By minimizing the energy of this Hamiltonian, we identify the optimal approximation configuration for each rotation without exponential overhead. Benchmarking our method against standard diagonal approximation on random quantum circuits, we observe an average reduction of 26\% in the total approximating circuit gate count, offering a significant efficiency gain for the implementation of quantum algorithms on near-term and fault-tolerant architectures.