Efficient Universal Quantum Compilation: An Inverse-free Solovay-Kitaev Algorithm

Abstract

The Solovay-Kitaev algorithm is a fundamental result in quantum computation. It gives an algorithm for efficiently compiling arbitrary unitaries using universal gate sets: any unitary can be approximated by short gates sequences, whose length scales merely poly-logarithmically with accuracy. As a consequence, the choice of gate set is typically unimportant in quantum computing. However, the Solovay-Kitaev algorithm requires the gate set to be inverse-closed. It has been a longstanding open question if efficient algorithmic compilation is possible without this condition. In this work, we provide the first inversefree Solovay-Kitaev algorithm, which makes no assumption on the structure within a gate set beyond universality, answering this problem in the affirmative, and providing an efficient compilation algorithm in the absence of inverses for both SU(d) and SL(d, C). The algorithm works by showing that approximate gate implementations of the generalized Pauli group can self-correct their errors.

Efficient and practical quantum compiler towards multi-qubit systems with deep reinforcement learning

“…That is, in the single-qubit situation, the sequence length for the inverse-free universal basis sets scales with O(log c (1/ε)) with c = 8.62. In conjunction with the achieved results and the conclusion of [34], our proposal provides certain empirical evidence for the existence of a more efficient inverse-free Solovay-Kitaev theorem. We defer the discussion under spectral norm [34] to Appendix D.…”

“…In conjunction with the achieved results and the conclusion of [34], our proposal provides certain empirical evidence for the existence of a more efficient inverse-free Solovay-Kitaev theorem. We defer the discussion under spectral norm [34] to Appendix D.…”

“…According to the achieved simulation results, our protocol allows a lower exponent c than this deterministic solution. A recent study proposed an inverse-free Solovay-Kitaev theorem [34]. That is, in the single-qubit situation, the sequence length for the inverse-free universal basis sets scales with O(log c (1/ε)) with c = 8.62.…”

“…To this end, an important line of research is deriving the optimal sequence length in the inverse-free setting. A recent study [34] has proved an inverse-free version of Solovay-Kitaev theorem, which states that the sequence * Corresponding authors length scales with O(log c (1/ε)) but has a larger c = 8.62 compared to the inverse-closed setting. Nevertheless, it is still an open question whether the value of c could be further reduced.…”

“…[50,51] investigated the application of temporal planners in the problem of quantum circuits synthesis; Ref. [52][53][54] proposed the quantum-assisted compilers to convert the quantum compiling problem into a complex optimization problem, [35] Fibonacci anyons Single-qubit O 2.24 log 1.43 (1/ε) Inverse-closed S-K theorem [35] Fibonacci anyons Single-qubit O log 5.18 (1/ε) Inverse-closed S-K theorem [36] -Single-qubit O log 3 (1/ε) Inverse-free S-K theorem [34] -Single-qubit O log 8.62 (1/ε) Inverse-free algorithm [37] Inverse-free diffusive set Single-qubit O log 1.585 (1/ε) Our RL-based compiler Inverse-free diffusive set Single-qubit O 2.683 log 0.9735 (1/ε) Other RL-based compiler [35] Fibonacci anyons Single-qubit O 1.55 log 1.6 (1/ε) Our RL-based compiler Fibonacci anyons Single-qubit O 0.737 log 1.52 (1/ε) Other RL-based compiler [38] HRC efficient universal set Single-qubit O log 1.25 including discrete parameters (e.g., quantum circuit layouts) and continuous parameters (e.g., the angles in rotational quantum gates). However, these optimizationbased compilers fail to show any prominent advantage in the inference time compared with those deterministic quantum compilers.…”