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

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.

[1]  Aram W. Harrow,et al.  Universal Refocusing of Systematic Quantum Noise , 2016, 1602.07963.

[2]  Xue Zhang,et al.  Complexity Classification of Two-Qubit Commuting Hamiltonians , 2016, Electron. Colloquium Comput. Complex..

[3]  A. Louis,et al.  Documenta Mathematica , 1996 .

[4]  Maris Ozols,et al.  The Complexity of Translationally Invariant Spin Chains with Low Local Dimension , 2016, Annales Henri Poincaré.

[5]  Dmitri Maslov,et al.  Practical Approximation of Single-Qubit Unitaries by Single-Qubit Quantum Clifford and T Circuits , 2012, IEEE Transactions on Computers.

[6]  IEEE Transactions on Computers , Computing in Science & Engineering.

[7]  Ran Raz.  Proceedings of the 31st Conference on Computational Complexity , 2016 .

[8]  Neil J. Ross,et al.  Optimal ancilla-free Clifford+T approximation of z-rotations , 2014, Quantum Inf. Comput..

[9]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[10]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[11]  D. Aharonov,et al.  Polynomial Quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane Preliminary Version , 2008 .

[12]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[13]  Thomas Zeugmann.  Epsilon Nets , 2010, Encyclopedia of Machine Learning.

[14]  A. Kitaev.  Quantum computations: algorithms and error correction , 1997 .

[15]  Stephen Piddock,et al.  The Complexity of Translationally-Invariant Low-Dimensional Spin Lattices in 3D , 2017, ArXiv.

[16]  B. Recht,et al.  Efficient discrete approximations of quantum gates , 2001, quant-ph/0111031.

[17]  P. Sarnak,et al.  Super-Golden-Gates for PU(2) , 2017, 1704.02106.

[18]  Jean Bourgain,et al.  A Spectral Gap Theorem in $SU(d)$ , 2011, 1108.6264.

[19]  Adam Bouland,et al.  Complexity Classification of Conjugated Clifford Circuits , 2017, CCC.

[20]  Michael A. Nielsen,et al.  The Solovay-Kitaev algorithm , 2006, Quantum Inf. Comput..

[21]  David McKinnon,et al.  Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets , 2015, 1501.04944.

[22]  P. Varjú.  Random walks in compact groups , 2012, Documenta Mathematica.

[23]  Maris Ozols,et al.  Trading Inverses for an Irrep in the Solovay-Kitaev Theorem , 2017, TQC.

[24]  Greg Kuperberg,et al.  How Hard Is It to Approximate the Jones Polynomial? , 2009, Theory Comput..

[25]  October I.  Physical Review Letters , 2022 .

[26]  Adam Sawicki,et al.  Epsilon-Nets, Unitary Designs, and Random Quantum Circuits , 2020, IEEE Transactions on Information Theory.