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arXiv:2603.25101

Gradient-Based Optimization Enables Scalable Fault-Tolerant Quantum Gate Synthesis

Marc Grau Davis, Ed Younis, Mathias Weiden, Hyeongrak Choi, Dirk Englund

quant-phcs.ETSubmitted Mar 26, 2026• Announced Mar 27, 2026• 0 citations• Revised Oct 18, 2026 (v3)
v1 · Mar 26, 2026v2 · May 15, 2026v3 · Oct 18, 2026
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Abstract

We develop a numerical approach to optimizing a quantum circuit over a discrete gate set by using gradient descent to tune $R_Z$ angles to a solution where many $R_Z$ gates can be rounded to a Clifford or $T$ gate without introducing further synthesis error. Through benchmarking, we show that our approach is able to match state-of-the-art techniques for multi-qubit Clifford + $T$ circuit synthesis, and is able to find circuits for unitaries that previous approaches fail to find solutions for. Our approach can adapt to any discrete gate set, which we demonstrate using the Clifford + $T$ + $\sqrt T$ gate set.

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10 pages 3 figures and tables

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We develop a numerical approach to optimizing a quantum circuit over a discrete gate set by using gradient descent to tune RZR_ZRZ​ angles to a solution where many RZR_ZRZ​ gates can be rounded to a Clifford or TTT gate without introducing further synthesis error. Through benchmarking, we show that our approach is able to match state-of-the-art techniques for multi-qubit Clifford + TTT circuit synthesis, and is able to find circuits for unitaries that previous approaches fail to find solutions for. Our approach can adapt to any discrete gate set, which we demonstrate using the Clifford + TTT + T\sqrt TT​ gate set.
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.
Abstract We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS = diag(1, 1, 1, i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operator U and outputs a Clifford+CS circuit for U, which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worst-case lower bound of 5log2(1ϵ)+O(1)5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)5log2​(ϵ1​)+O(1) 5 log 2 ( 1 ϵ ) + O ( 1 ) on the number of CS gates required to ϵ-approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of fault-tolerant quantum circuits.
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 θθθ, requiring O(log⁡1/δ)O(\log 1/δ)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)\tilde{O}(θ^2/δ)O~(θ2/δ), and returning to existing O(log⁡1/δ)O(\log1/δ)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 θθθ-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 θθθ, 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.
We propose a method for exact circuit synthesis using a discrete gate set, as required for fault-tolerant quantum computing. Our approach translates the problem of synthesizing a gate specified by its unitary matrix into a boolean satisfiability (SAT) instance. It leverages the algebraic properties of the coefficients of the matrices that constitute the gate set, enabling the transformation of the constraint of equality between complex matrices into a boolean expression to satisfy. For a specified number of gates, the SAT solver finds a circuit implementing the target or proves that none exist. Optimality of the number of gates is proven by iterating on the number of gates. We also propose some extensions of the method, for example, handling ancillary qubits, post-selection, or classical feedback. The time-to-solution scales double-exponentially with the number of qubits, making it impractical for large circuits. However, since many quantum algorithms rely on small, frequently used subcircuits, and because of the intrinsic value of having a provably optimal circuit synthesis, we believe that our tool are valuable for quantum circuit design.
Fault-tolerant quantum computing requires translating application-level quantum circuits into the Clifford+TTT gate set, where the TTT gate is the dominant resource cost. Phase kickback is an ancilla-based technique that can dramatically reduce TTT-count for rotations with dyadic angles, but has previously been limited to highly structured circuit families. We present Dyadic Phase Fixing (DPF), a general multi-qubit synthesis tool that extends phase kickback to general quantum circuits. DPF uses numerical unitary synthesis to greedily extract dyadic angle rotations from any input circuit. Combined with a decision matrix to automatically size the final phase gradient register, our end-to-end workflow achieves up to 70% reduction in TTT-count compared to \texttt{gridsynth} and up to 60% compared to Repeat-Until-Success synthesis on a diverse set of benchmarks. We map these compiled circuits to a surface-code architecture to evaluate space-time volume, demonstrating up to a 60\% reduction in this metric as well. However, for some circuits and mapping strategies the two metrics diverge significantly, demonstrating that TTT-count alone is a useful but incomplete proxy for fault-tolerant program costs.
Fault-tolerant gate sets whose generators belong to the Clifford hierarchy form the basis of many protocols for scalable quantum computing architectures. At the beginning of the decade, number-theoretic techniques were employed to analyze circuits over these gate sets on single qubits, providing the basis for a number of state-of-the-art quantum compiling algorithms. In this dissertation, I further this program by employing number-theoretic techniques for higher-dimensional gate sets on both qudit and multi-qubit circuits. First, I introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. I show that these canonical forms are T-optimal and describe an algorithm which takes as input a Clifford+T circuit and outputs the canonical form for that operator. The algorithm runs in time linear in the number of gates of the circuit. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits. Finally, we show that a similar extension of these normal forms to higher dimensions exists, but do not establish uniqueness. Moving to multi-qubit circuits, I provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/√2, i]. We focus on the subrings Z[1/2], Z[1/√2], Z[1/√−2], and Z[1/2, i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X, CX, CCX} with an analogue of the Hadamard gate and an optional phase gate. I then establish the existence and uniqueness of a normal form for one of these gate sets, the two-qubit gate set of Clifford+Controlled Phase gate CS. This normal form is optimal in the number of CS gates, making it the first normal form that is non-Clifford optimal for a fault tolerant universal multi-qubit gate set. We provide a synthesis algorithm that runs in a time linear in the gate count and outputs the equivalent normal form. In proving the existence and uniqueness of the normal form, we likewise establish the generators and relations for the two-qubit Clifford+CS group. Finally, we demonstrate that a lower bound of 5 log2 (1/ε) + O(1) CS gates are required to ε-approximate any 4 × 4 unitary matrix. Lastly, using the characterization of circuits over the Clifford+CS gate set and the existence of an optimal normal form, I provide an ancilla-free inexact synthesis algorithm for two-qubit unitaries using the Clifford+SC gate set for Pauli-rotations. These operators require 6 log2 (1/ε) + O(1) CS gates to synthesize in the typical case and 8 log2 (1/ε) + O(1) in the worst case.