We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works \cite{Hastings2017} and \cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+ T gate set we achieve an average non-Clifford gate count of 0.23 log 2 ( 1 / ε ) + 2.13 and T-count 0.56 log 2 ( 1 / ε ) + 5.3 with mixed fallback approximations for diamond norm accuracy ε .This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+ T ). We also provide detailed numerical results for Clifford+T and Clifford+ T gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.
We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works \cite{Hastings2017} and \cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+ T gate set we achieve an average non-Clifford gate count of 0.23 log 2 ( 1 / ε ) + 2.13 and T-count 0.56 log 2 ( 1 / ε ) + 5.3 with mixed fallback approximations for diamond norm accuracy ε .This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+ T ). We also provide detailed numerical results for Clifford+T and Clifford+ T gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.
We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using multi-qubit gate sets such as Real Clifford and CCZ. The key tools are repeat-until-success circuits, lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.
Resource-efficient and high-precision approximate synthesis of quantum circuits expressed in the Clifford+T gate set is vital for Fault-Tolerant quantum computing.Efficient optimal methods are known for single-qubit R Z unitaries, otherwise the problem is generally intractable.Search-based methods, like simulated annealing, empirically generate low resource cost approximate implementations of general multi-qubit unitaries so long as low precision (Hilbert-Schmidt distances of ε ≥ 10 -2 ) can be tolerated.These algorithms build up circuits that directly invert target unitaries.We instead leverage search-based methods to first approximately diagonalize a unitary, then perform the inversion analytically.This lets difficult continuous rotations be bypassed and handled in a postprocessing step.Our approach improves both the implementation precision and run time of synthesis algorithms by orders of magnitude when evaluated on unitaries from real quantum algorithms.On benchmarks previously synthesizable only with analytical techniques like the Quantum Shannon Decomposition, diagonalization uses an average of 95% fewer non-Clifford 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.
Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle θ, requiring O(log1/δ) T gates. We show that it is possible to do much better for small angles, reducing the T cost to O~(θ2/δ), and returning to existing O(log1/δ) results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new θ-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of θ, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources.
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
Matchgate unitaries are ubiquitous in quantum computation due to their relation to non-interacting fermions and because they can be used to benchmark quantum computers. Implementing such unitaries on fault-tolerant devices requires first compiling them into a discrete universal gate set, typically Clifford+T. Here, we propose a different approach for their synthesis: compile matchgate unitaries using only matchgate gates. To this end, we first show that the matchgate-Clifford group (the intersection of the matchgate and Clifford groups) plus the T gate (a T unitary up to a phase) is universal for the matchgate group. Our approach leverages the connection between n-qubit matchgate circuits and the standard representation of SO(2n), which reduces the compilation from 2n×2n unitaries to 2n×2n ones, thus reducing exponentially the size of the target matrix. Moreover, we rigorously show that this scheme is efficient, as an approximation error εSO(2n) incurred in this smaller-dimensional representation translates at most into an O(nεSO(2n)) error in the exponentially large unitary. In addition, we study the exact version of the matchgate synthesis problem, and we prove that all matchgate unitaries U such that U⊗U∗ has entries in the ring Z[1/2,i] can be exactly synthesized by a finite sequence of gates from the matchgate-Clifford+T set, without ancillas. We then use this insight to map optimal exact matchgate synthesis to Boolean satisfiability, and compile the circuits that diagonalize the free-fermionic XX Hamiltonian on n=4,8 qubits.