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Quantum Information and Computation

Optimal ancilla-free Clifford+V approximation of z-rotations

Neil J. Ross

Published Sep 1, 2015• 72 citations
Publisher

Abstract

We describe a new efficient algorithm to approximate $z$-rotations by ancilla-free Clifford+$V$ circuits, up to a given precision $\epsilon$. Our algorithm is optimal in the presence of an oracle for integer factoring: it outputs the shortest Clifford+$V$ circuit solving the given problem instance. In the absence of such an oracle, our algorithm is still near-optimal, producing circuits of $V$\!-count $m + O(\log(\log(1/\epsilon)))$, where $m$ is the $V$\!-count of the third-to-optimal solution. A restricted version of the algorithm approximates $z$-rotations in the Pauli+$V$ gate set. Our method is based on previous work by the author and Selinger on the optimal ancilla-free approximation of $z$-rotations using Clifford+$T$ gates and on previous work by Bocharov, Gurevich, and Svore on the asymptotically optimal ancilla-free approximation of $z$-rotations using Clifford+$V$ gates.

DOI
10.26421/qic15.11-12-4

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We describe a new efficient algorithm to approximate zzz-rotations by ancilla-free Clifford+VVV circuits, up to a given precision ϵ\epsilonϵ. Our algorithm is optimal in the presence of an oracle for integer factoring: it outputs the shortest Clifford+VVV circuit solving the given problem instance. In the absence of such an oracle, our algorithm is still near-optimal, producing circuits of VVV!-count m+O(log⁡(log⁡(1/ϵ)))m + O(\log(\log(1/\epsilon)))m+O(log(log(1/ϵ))), where mmm is the VVV!-count of the third-to-optimal solution. A restricted version of the algorithm approximates zzz-rotations in the Pauli+VVV gate set. Our method is based on previous work by the author and Selinger on the optimal ancilla-free approximation of zzz-rotations using Clifford+TTT gates and on previous work by Bocharov, Gurevich, and Svore on the asymptotically optimal ancilla-free approximation of zzz-rotations using Clifford+VVV gates.
We describe a new efficient algorithm to approximate z-rotations by ancilla-free Clifford+V circuits, up to a given precision epsilon. Our algorithm is optimal in the presence of an oracle for integer factoring: it outputs the shortest Clifford+V circuit solving the given problem instance. In the absence of such an oracle, our algorithm is still near-optimal, producing circuits of V-count m + O(log(log(1/epsilon))), where m is the V-count of the third-to-optimal solution. A restricted version of the algorithm approximates z-rotations in the Pauli+V gate set. Our method is based on previous work by the author and Selinger on the optimal ancilla-free approximation of z-rotations using Clifford+T gates and on previous work by Bocharov, Gurevich, and Svore on the asymptotically optimal ancilla-free approximation of z-rotations using Clifford+V gates.
We present pygridsynth, an open-source Python library for ancilla-free approximate Clifford+TTT synthesis that runs in O(log⁡(1/ε))O(\log(1/ε))O(log(1/ε)) for precision εεε. For n=1,2n=1, 2n=1,2 qubits, the library builds upon established efficient and high-precision synthesis routines, such as nearly optimal ZZZ-rotation synthesis and magnitude approximation. For n≥3n\ge 3n≥3 qubits, we introduce a partial-decomposition technique that generalizes the magnitude approximation, reducing constant factors in the TTT-count as (218⋅4n−92⋅2n+9)log⁡2(1/ε)+o(log⁡(1/ε))(\frac{21}{8}\cdot 4^n - \frac{9}{2}\cdot 2^n + 9)\log_2(1/ε) + o(\log(1/ε))(821​⋅4n−29​⋅2n+9)log2​(1/ε)+o(log(1/ε)). The package also exposes a mixed-synthesis workflow that approximates target unitary channels by probabilistic mixtures of Clifford+TTT circuits, for which we empirically find that the synthesis error is reduced from εεε to ε2/(2n)ε^2/(2n)ε2/(2n). Taken together, these features make pygridsynth a Python-native platform for high-precision Clifford+T+T+T synthesis and for benchmarking unitary and mixed synthesis strategies on multi-qubit instances.
Approximate synthesis of general single-qubit unitaries over the Clifford+T\sqrt{T}T​ gate set
For the standard Clifford+TTT gate set, deterministic, ancilla-free synthesis now attains the minimal TTT-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T\sqrt{T}T​ gate rotates by half the angle of TTT, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T\sqrt{T}T​ algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3\varepsilon=10^{-3}ε=10−3 to 10−810^{-8}10−8, the cost of Clifford+T\sqrt{T}T​ circuits scales as 2.4log⁡2(1/ε)2.4\log_2(1/\varepsilon)2.4log2​(1/ε) compared to 3.0log⁡2(1/ε)3.0\log_2(1/\varepsilon)3.0log2​(1/ε) for TTT-count-optimal Clifford+TTT circuits. Once a one-time catalyst state is amortized, the Clifford+T\sqrt{T}T​ circuits are never costlier than their Clifford+TTT counterparts.
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)).
We address the problem of optimal representation of single-qubit rotations in a certain unitary basis consisting of the so-called V gates and Pauli matrices. The V matrices were proposed by Lubotsky, Philips, and Sarnak [Commun. Pure Appl. Math. 40, 401–420 (1987)] as a purely geometric construct in 1987 and recently found applications in quantum computation. They allow for exceptionally simple quantum circuit synthesis algorithms based on quaternionic factorization. We adapt the deterministic-search technique initially proposed by Ross and Selinger to synthesize approximating Pauli+V circuits of optimal depth for single-qubit axial rotations. Our synthesis procedure based on simple SL2(ℤ) geometry is almost elementary.
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.