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 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 describe a new efficient algorithm to approximate z-rotations by ancilla-free Clifford+V circuits, up to a given precision ϵ. 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/ϵ))), 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.
Approximate synthesis of general single-qubit unitaries over the Clifford+T gate set
For the standard Clifford+T gate set, deterministic, ancilla-free synthesis now attains the minimal T-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T gate rotates by half the angle of T, 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 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 to 10−8, the cost of Clifford+T circuits scales as 2.4log2(1/ε) compared to 3.0log2(1/ε) for T-count-optimal Clifford+T circuits. Once a one-time catalyst state is amortized, the Clifford+T circuits are never costlier than their Clifford+T 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)).
Building generic gates from a restricted gate set is a difficult but important problem, especially in the noisy regime where only a limited set of noise-resistant gates are available, e.g., fault-tolerant Clifford gates (generated by Hadamard, phase, and gates) and fault-tolerant Toffoli gates. The Toffoli is also often used as a building block of many algorithms and will need to be constructed if not directly available. This makes Clifford + Toffoli an attractive gate set for building generic gates. In this Letter we give a simple and efficient algorithm for building an approximate single-qubit rotation using only Clifford + Toffoli , which in turn enables any generic single-qubit unitary. An important difference compared to earlier attempts is that the use of the Toffoli allows us to use simple rounding as opposed to a complicated approximation algorithm. The resulting gate array does not rely on repeatedly applying a fixed rotation, but immediately applies a rotation R θ * that is ε -close to the desired rotation R θ , with a success probability strictly greater than 1 / 2 . It can be rerun upon failure, giving an expected number of repetitions strictly less than 2, an expected Toffoli count strictly less than 4 ⌈ log 1 ε ⌉ , an expected gate depth strictly less than 4 ⌈ log 1 ε ⌉ + 6 , and uses 2 ⌈ log 1 ε ⌉ ancillas. The small circuit depth of our construction enables low-noise gates on existing quantum computational devices, and allows for arbitrary precision using a very modest number of ancillas. Published by the American Physical Society 2024
We present pygridsynth, an open-source Python library for ancilla-free approximate Clifford+T synthesis that runs in O(log(1/ε)) for precision ε. For n=1,2 qubits, the library builds upon established efficient and high-precision synthesis routines, such as nearly optimal Z-rotation synthesis and magnitude approximation. For n≥3 qubits, we introduce a partial-decomposition technique that generalizes the magnitude approximation, reducing constant factors in the T-count as (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+T circuits, for which we empirically find that the synthesis error is reduced from ε to ε2/(2n). Taken together, these features make pygridsynth a Python-native platform for high-precision Clifford+T synthesis and for benchmarking unitary and mixed synthesis strategies on multi-qubit instances.
We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves ε-approximations using circuits of length O(log(1/ε)), which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Clifford+π/12 [arXiv:1409.3552], running on average in time polynomial in O(log(1/ε)) (conditional on a number-theoretic conjecture). Ours is the first such algorithm that works for a wide range of gate sets and provides insight into what should constitute a "good" gate set for a fault-tolerant quantum computer. Joint work with Alex Bocharov, Martin Roetteler, Jon Yard: http://arxiv.org/abs/1510.03888