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The Solovay-Kitaev algorithm

Dawson, C M, Michael A. Nielsen

Published May 6, 2005• 251 citations
OpenAlex

Abstract

This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $\\pi / 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\\sc not}, and $\\pi / 8$ gates. The algorithm runs in $O(\\log^{2.71}(1/\\epsilon))$ time, and produces as output a sequence of $O(\\log^{3.97}(1/\\epsilon))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $\\epsilon > 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from $SU(d)$.

Related papers

The Solovay-Kitaev algorithm

Christopher M. Dawson, Michael A. Nielsen

This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $π/ 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and $π/ 8$ gates. The algorithm runs in $O(\log^{2.71}(1/ε))$ time, and produces as output a sequence of $O(\log^{3.97}(1/ε))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $ε> 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from $SU(d)$.

arXiv:quant-ph/0505030•May 6, 2005•8 citations

The Solovay-Kitaev algorithm

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This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $\pi / 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and $\pi / 8$ gates. The algorithm runs in $O(\log^{2.71}(1/\epsilon))$ time, and produces as output a sequence of $O(\log^{3.97}(1/\epsilon))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $\epsilon > 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d).

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Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets

Anuradha Mahasinghe, Dulmi Fernando, Kaushika De Silva

The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy ε from a sequence of O(log8.62(1/ε)) orthogonal gates taken from an inverse-free set.

International Online Conference on Mathematics and Applications (IOCMA)•Jun 13, 2023•0 citations

Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem

H. F. Chau

Arbitrarily accurate fault-tolerant (FT) universal quantum computation can be carried out using the Clifford gates Z, S, CNOT plus the non-Clifford T gate. Moreover, a recent improvement of the Solovay-Kitaev theorem by Kuperberg implies that to approximate any single-qubit gate to an accuracy of $ε> 0$ requires $\text{O}(\log^c[1/ε])$ quantum gates with $c > 1.44042$. Can one do better? That was the question asked by Nielsen and Chuang in their quantum computation textbook. Specifically, they posted a challenge to efficiently approximate single-qubit gate, fault-tolerantly or otherwise, using $Ω(\log[1/ε])$ gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using $\text{O}(\log[1/ε] \log\log[1/ε] \log\log\log[1/ε] \cdots)$ FT gates chosen from a finite set depending on the value of $ε$. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy $ε> 0$. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem.

arXiv:2406.04846•Jun 7, 2024•0 citations

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

Adam Bouland, Tudor Giurgică-Tiron

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 inverse-free 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 $\text{SU}(d)$ and $\text{SL}(d, \mathbb{C})$. The algorithm works by showing that approximate gate implementations of the generalized Pauli group can self-correct their errors.

arXiv:2112.02040•Dec 3, 2021•7 citations

Efficient quantum approximation : examining the efficiency of select universal gate sets in approximating 1-qubit quantum gates.

Brent A Mode

Quantum computation is of current ubiquitous interest in physics, computer science, and the public interest. In the not-so-distant future, quantum computers will be relatively common pieces of research equipment. Eventually, one can expect an actively quantum computer to be a common feature of life. In this work, I study the approximation efficiency of several common universal quantum gate sets at short sequence lengths using an implementation of the Solovay-Kitaev algorithm. I begin by developing from almost nothing the relevant formal mathematics to rigorously describe what one means by the terms universal gate set and covering efficiency. I then describe some interesting results on the asymptotic covering properties of certain classes of universal gate sets and discuss the theorem which the Solovay-Kitaev algorithm is based on. Moving from mathematical introduction to experimental method, I then describe how sets will be compared. I use the commonly studied sets H+T, Pauli+V, V, and Clifford+T to determine which is the most efficient at approximating randomly generated unitaries. By doing so, we get an understanding of how well each set would perform in the context of a general quantum computer processor. This was accomplished by using the same implementation of the Solovay-Kitaev algorithm throughout, with roughly equal-sized preprocessed libraries formed from each gate set, over approximations for 10,000 randomly generated unitary matrices at algorithm depth n=5. Ultimately, the Pauli+V and V sets were the most efficient and had similar performance qualities. On average the Pauli+V set produced approximations of length 15,491 and accuracy 0.0002686. The V basis produced approximations of average sequence length 16,403 and accuracy 0.0001465. This performance is about equal given this particular implementation of the Solovay-Kitaev algorithm. We conclude that this result is somewhat surprising as the general behavior and efficiency of these particular choices of gate set are expected to be similar. It is possible though that the asymptotic efficiencies of these gate sets vary by a relatively wide margin and this has effected the experiment. It is also possible that some aspect of a naive implementation of the Solovay-Kitaev algorithm resulted in the Hadamard gate based sets performing more poorly than the V basis sets overall. Due to constraints on computational power, this result could also be limited to this particular accuracy regime and could even out as tolerance 𝜀 is taken to be arbitrarily small. Further possibilities of this result as well as further work are then discussed.

•Jan 1, 2018•0 citations
This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of pi/2k\\pi / 2^kpi/2k, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and pi/8\\pi / 8pi/8 gates. The algorithm runs in O(log2.71(1/epsilon))O(\\log^{2.71}(1/\\epsilon))O(log2.71(1/epsilon)) time, and produces as output a sequence of O(log3.97(1/epsilon))O(\\log^{3.97}(1/\\epsilon))O(log3.97(1/epsilon)) quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within epsilon>0\\epsilon > 0epsilon>0. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d)SU(d)SU(d).
This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of π/2kπ/ 2^kπ/2k, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and π/8π/ 8π/8 gates. The algorithm runs in O(log⁡2.71(1/ε))O(\log^{2.71}(1/ε))O(log2.71(1/ε)) time, and produces as output a sequence of O(log⁡3.97(1/ε))O(\log^{3.97}(1/ε))O(log3.97(1/ε)) quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within ε>0ε> 0ε>0. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d)SU(d)SU(d).
This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of π/2k\pi / 2^kπ/2k, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and π/8\pi / 8π/8 gates. The algorithm runs in O(log⁡2.71(1/ϵ))O(\log^{2.71}(1/\epsilon))O(log2.71(1/ϵ)) time, and produces as output a sequence of O(log⁡3.97(1/ϵ))O(\log^{3.97}(1/\epsilon))O(log3.97(1/ϵ)) quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within ϵ>0\epsilon > 0ϵ>0. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d).
The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy ε from a sequence of O(log8.62(1/ε)) orthogonal gates taken from an inverse-free set.
Arbitrarily accurate fault-tolerant (FT) universal quantum computation can be carried out using the Clifford gates Z, S, CNOT plus the non-Clifford T gate. Moreover, a recent improvement of the Solovay-Kitaev theorem by Kuperberg implies that to approximate any single-qubit gate to an accuracy of ε>0ε> 0ε>0 requires O(log⁡c[1/ε])\text{O}(\log^c[1/ε])O(logc[1/ε]) quantum gates with c>1.44042c > 1.44042c>1.44042. Can one do better? That was the question asked by Nielsen and Chuang in their quantum computation textbook. Specifically, they posted a challenge to efficiently approximate single-qubit gate, fault-tolerantly or otherwise, using Ω(log⁡[1/ε])Ω(\log[1/ε])Ω(log[1/ε]) gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using O(log⁡[1/ε]log⁡log⁡[1/ε]log⁡log⁡log⁡[1/ε]⋯ )\text{O}(\log[1/ε] \log\log[1/ε] \log\log\log[1/ε] \cdots)O(log[1/ε]loglog[1/ε]logloglog[1/ε]⋯) FT gates chosen from a finite set depending on the value of εεε. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy ε>0ε> 0ε>0. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem.
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 inverse-free 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)\text{SU}(d)SU(d) and SL(d,C)\text{SL}(d, \mathbb{C})SL(d,C). The algorithm works by showing that approximate gate implementations of the generalized Pauli group can self-correct their errors.
Quantum computation is of current ubiquitous interest in physics, computer science, and the public interest. In the not-so-distant future, quantum computers will be relatively common pieces of research equipment. Eventually, one can expect an actively quantum computer to be a common feature of life. In this work, I study the approximation efficiency of several common universal quantum gate sets at short sequence lengths using an implementation of the Solovay-Kitaev algorithm. I begin by developing from almost nothing the relevant formal mathematics to rigorously describe what one means by the terms universal gate set and covering efficiency. I then describe some interesting results on the asymptotic covering properties of certain classes of universal gate sets and discuss the theorem which the Solovay-Kitaev algorithm is based on. Moving from mathematical introduction to experimental method, I then describe how sets will be compared. I use the commonly studied sets H+T, Pauli+V, V, and Clifford+T to determine which is the most efficient at approximating randomly generated unitaries. By doing so, we get an understanding of how well each set would perform in the context of a general quantum computer processor. This was accomplished by using the same implementation of the Solovay-Kitaev algorithm throughout, with roughly equal-sized preprocessed libraries formed from each gate set, over approximations for 10,000 randomly generated unitary matrices at algorithm depth n=5. Ultimately, the Pauli+V and V sets were the most efficient and had similar performance qualities. On average the Pauli+V set produced approximations of length 15,491 and accuracy 0.0002686. The V basis produced approximations of average sequence length 16,403 and accuracy 0.0001465. This performance is about equal given this particular implementation of the Solovay-Kitaev algorithm. We conclude that this result is somewhat surprising as the general behavior and efficiency of these particular choices of gate set are expected to be similar. It is possible though that the asymptotic efficiencies of these gate sets vary by a relatively wide margin and this has effected the experiment. It is also possible that some aspect of a naive implementation of the Solovay-Kitaev algorithm resulted in the Hadamard gate based sets performing more poorly than the V basis sets overall. Due to constraints on computational power, this result could also be limited to this particular accuracy regime and could even out as tolerance 𝜀 is taken to be arbitrarily small. Further possibilities of this result as well as further work are then discussed.