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Quantum Information and Computation

Efficient Clifford+T approximation of single-qubit operators

Peter Selinger

Published Jan 1, 2015• 94 citations
Publisher

Abstract

We give an efficient randomized algorithm for approximating an arbitrary element of $\SU(2)$ by a product of Clifford+$T$ operators, up to any given error threshold $\epsilon>0$. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in $\log(1/\epsilon)$. If the operator to be approximated is a $z$-rotation, the resulting gate sequence has $T$-count $K+4\log_2(1/\epsilon)$, where $K$ is approximately equal to $10$. We also prove a worst-case lower bound of $K+4\log_2(1/\epsilon)$, where $K=-9$, so that our algorithm is within an additive constant of optimal for certain $z$-rotations. For an arbitrary member of $\SU(2)$, we achieve approximations with $T$-count $K+12\log_2(1/\epsilon)$. By contrast, the Solovay-Kitaev algorithm achieves $T$-count $O(\log^c(1/\epsilon))$, where $c$ is approximately $3.97$.

DOI
10.26421/qic15.1-2-10

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Efficient Clifford+T approximation of single-qubit operators

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We give an efficient randomized algorithm for approximating an arbitrary element of $SU(2)$ by a product of Clifford+$T$ operators, up to any given error threshold $ε>0$. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in $\log(1/ε)$. If the operator to be approximated is a $z$-rotation, the resulting gate sequence has $T$-count $K+4\log_2(1/ε)$, where $K$ is approximately equal to $10$. We also prove a worst-case lower bound of $K+4\log_2(1/ε)$, where $K=-9$, so that our algorithm is within an additive constant of optimal for certain $z$-rotations. For an arbitrary member of $SU(2)$, we achieve approximations with $T$-count $K+12\log_2(1/ε)$. By contrast, the Solovay-Kitaev algorithm achieves $T$-count $O(\log^c(1/ε))$, where $c$ is approximately $3.97$.

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quant-pharXiv:2603.14202•Mar 17, 2026•0 citations
We give an efficient randomized algorithm for approximating an arbitrary element of \SU(2)\SU(2)\SU(2) by a product of Clifford+TTT operators, up to any given error threshold ϵ>0\epsilon>0ϵ>0. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in log⁡(1/ϵ)\log(1/\epsilon)log(1/ϵ). If the operator to be approximated is a zzz-rotation, the resulting gate sequence has TTT-count K+4log⁡2(1/ϵ)K+4\log_2(1/\epsilon)K+4log2​(1/ϵ), where KKK is approximately equal to 101010. We also prove a worst-case lower bound of K+4log⁡2(1/ϵ)K+4\log_2(1/\epsilon)K+4log2​(1/ϵ), where K=−9K=-9K=−9, so that our algorithm is within an additive constant of optimal for certain zzz-rotations. For an arbitrary member of \SU(2)\SU(2)\SU(2), we achieve approximations with TTT-count K+12log⁡2(1/ϵ)K+12\log_2(1/\epsilon)K+12log2​(1/ϵ). By contrast, the Solovay-Kitaev algorithm achieves TTT-count O(log⁡c(1/ϵ))O(\log^c(1/\epsilon))O(logc(1/ϵ)), where ccc is approximately 3.973.973.97.
We give an efficient randomized algorithm for approximating an arbitrary element of SU(2)SU(2)SU(2) by a product of Clifford+TTT operators, up to any given error threshold ε>0ε>0ε>0. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in log⁡(1/ε)\log(1/ε)log(1/ε). If the operator to be approximated is a zzz-rotation, the resulting gate sequence has TTT-count K+4log⁡2(1/ε)K+4\log_2(1/ε)K+4log2​(1/ε), where KKK is approximately equal to 101010. We also prove a worst-case lower bound of K+4log⁡2(1/ε)K+4\log_2(1/ε)K+4log2​(1/ε), where K=−9K=-9K=−9, so that our algorithm is within an additive constant of optimal for certain zzz-rotations. For an arbitrary member of SU(2)SU(2)SU(2), we achieve approximations with TTT-count K+12log⁡2(1/ε)K+12\log_2(1/ε)K+12log2​(1/ε). By contrast, the Solovay-Kitaev algorithm achieves TTT-count O(log⁡c(1/ε))O(\log^c(1/ε))O(logc(1/ε)), where ccc is approximately 3.973.973.97.
Decomposing unitaries into a sequence of elementary operations is at the core of quantum computing. Information theoretic arguments show that approximating a random unitary with precision $\ensuremath{\epsilon}$ requires $\ensuremath{\Omega}\mathbf{(}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ gates. Prior to our work, the state of the art in approximating a single qubit unitary included the Solovay-Kitaev algorithm that requires $O\mathbf{(}{log}^{3+\ensuremath{\delta}}(1/\ensuremath{\epsilon})\mathbf{)}$ gates and does not use ancillae and the phase kickback approach that requires $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathrm{log}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ gates but uses $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathbf{)}$ ancillae. Both algorithms feature upper bounds that are far from the information theoretic lower bound. In this Letter, we report an algorithm that saturates the lower bound, and as such it guarantees asymptotic optimality. In particular, we present an algorithm for building a circuit that approximates single qubit unitaries with precision $\ensuremath{\epsilon}$ using $O\mathbf{(}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ Clifford and TTT gates and employing up to two ancillary qubits. We connect the unitary approximation problem to the problem of constructing solutions corresponding to Lagrange's four-square theorem, and thereby develop an algorithm for computing an approximating circuit using an average of $O\mathbf{(}{log}^{2}(1/\ensuremath{\epsilon})\mathrm{log}\mathrm{log}(1/\ensuremath{\epsilon})\mathbf{)}$ operations with integers.
We consider the problem of approximating arbitrary single-qubit z-rotations by ancillafree Clifford+T circuits, up to given epsilon. We present a fast new probabilistic algorithm for solving this problem optimally, i.e., for finding the shortest possible circuit whatsoever for the given problem instance. The algorithm requires a factoring oracle (such as a quantum computer). Even in the absence of a factoring oracle, the algorithm is still near-optimal under a mild number-theoretic hypothesis. In this case, the algorithm finds a solution of T-count m + O(log(log(1/ε))), where m is the T-count of the second-to-optimal solution. In the typical case, this yields circuit approximations of Tcount 3 log2 (1/ε) + O(log(log(1/ε))). Our algorithm is efficient in practice, and provably efficient under the above-mentioned number-theoretic hypothesis, in the sense that its expected runtime is O(polylog(1/ε)).
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.
We study the Clifford+T synthesis of quantum multiplexers U=⨁i=12nUiU=\bigoplus_{i=1}^{2^n}U_iU=⨁i=12n​Ui​: an nnn-qubit control register selects which single-qubit gate Ui∈SU(2)U_i\in\mathrm{SU}(2)Ui​∈SU(2) acts on the target qubit. For each block UiU_iUi​, consider the minimum even T-count needed to approximate it within diamond distance ε\varepsilonε without ancillae, and let mmm be the largest of these costs. For an arbitrary multiplexer UUU, we construct an approximation of UUU within the same distance using at most m+O(2n(m+1))m+O(\sqrt{2^n(m+1)})m+O(2n(m+1)​) TTT gates. With one control qubit, the cost is m+O(1)m+O(1)m+O(1) and no ancillae are required. We also prove a lower bound that allows the Clifford+T circuit to use any number of clean ancillae initialized to zero and returned exactly to zero. For fixed nnn and independent Haar-random blocks, the minimum T-count among these circuits lies between (3−δ)log⁡2(1/ε)(3-δ)\log_2(1/\varepsilon)(3−δ)log2​(1/ε) and (3+δ)log⁡2(1/ε)(3+δ)\log_2(1/\varepsilon)(3+δ)log2​(1/ε) for every fixed δ>0δ>0δ>0, except with probability O(εcn,δ)O(\varepsilon^{c_{n,δ}})O(εcn,δ​) as ε→0\varepsilon\rightarrow0ε→0. The optimal leading coefficient is therefore the same as for a single Haar-random SU(2)\mathrm{SU}(2)SU(2) gate. In the lower-bound proof, unitarity forces pairs of off-diagonal blocks describing transitions in opposite directions to satisfy cancellation relations with residuals of order ε2\varepsilon^2ε2. We combine these relations with the arithmetic of Clifford+T circuits to bound the Haar measure of targets compatible with each pair of off-diagonal blocks. As an application, we obtain ancilla-free approximations of Haar-random SU(4)\mathrm{SU}(4)SU(4) gates within diamond distance ε\varepsilonε, using at most (9+δ)log⁡2(1/ε)(9+δ)\log_2(1/\varepsilon)(9+δ)log2​(1/ε) TTT gates for every fixed δ>0δ>0δ>0, except with probability O(εcδ)O(\varepsilon^{c_δ})O(εcδ​) with cδ>0c_δ>0cδ​>0.