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Lecture notes in computer science

How Many Quantum Circuit Identities Are Needed to Generate All Others?

Yuantian Ding, Nengkun Yu, Xiaokang Qiu

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Abstract Quantum circuit optimizers use rewrite rules from circuit equivalences, yet prior work has identified thousands of such identities, creating substantial challenges for their storage, management, and effective application. For many widely used unitary gate sets, including Clifford+T, this apparent complexity is largely redundant, raising a fundamental question: How many quantum circuit identities are actually needed to generate all others? In this work, we provide strong evidence that a small pruned set of identities suffices to generate all circuit equivalences of bounded depth. Surprisingly, for circuits on up to nine qubits in which each side of an equality has depth at most ten, fewer than twenty identities are sufficient to derive all others, and for circuits on up to five qubits with depth at most ten, only 17 rules–each involving at most three qubits–are enough. These results enable significantly more compact and efficient rewriting systems for quantum compiler optimization and reveal underlying algebraic structure in common gate sets, showing that the vast majority of known circuit identities are consequences of a small foundational basis.

DOI
10.1007/978-3-032-32537-2_14

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Abstract Quantum circuit optimizers use rewrite rules from circuit equivalences, yet prior work has identified thousands of such identities, creating substantial challenges for their storage, management, and effective application. For many widely used unitary gate sets, including Clifford+T, this apparent complexity is largely redundant, raising a fundamental question: How many quantum circuit identities are actually needed to generate all others? In this work, we provide strong evidence that a small pruned set of identities suffices to generate all circuit equivalences of bounded depth. Surprisingly, for circuits on up to nine qubits in which each side of an equality has depth at most ten, fewer than twenty identities are sufficient to derive all others, and for circuits on up to five qubits with depth at most ten, only 17 rules–each involving at most three qubits–are enough. These results enable significantly more compact and efficient rewriting systems for quantum compiler optimization and reveal underlying algebraic structure in common gate sets, showing that the vast majority of known circuit identities are consequences of a small foundational basis.
Abstract Existing quantum compilers focus on mapping a logical quantum circuit to a quantum device and its native quantum gates. Only simple circuit identities are used to optimize the quantum circuit during the compilation process. This approach misses more complex circuit identities, which could be used to optimize the quantum circuit further. We propose Quanto, the first quantum optimizer that automatically generates circuit identities. Quanto takes as input a gate set and generates provably correct circuit identities for the gate set. Quanto’s automatic generation of circuit identities includes single-qubit and two-qubit gates, which leads to a new database of circuit identities, some of which are novel to the best of our knowledge. In addition to the generation of new circuit identities, Quanto’s optimizer applies such circuit identities to quantum circuits and finds optimized quantum circuits that have not been discovered by other quantum compilers, including IBM Qiskit and Cambridge Quantum Computing Tket. Quanto’s database of circuit identities could be applied to improve existing quantum compilers and Quanto can be used to generate identity databases for new gate sets.
In this article, we discuss an algebraic approach for optimizing quantum circuits. This approach is based on the construction of a Gröbner basis (standard basis) on a set of equations describing the equivalences of gate operations. It enables us to derive additional equivalences up to a bound and to use generated equivalences as rewriting rules of quantum circuits. If such rules are generated sufficiently large, the ambiguity in the rewriting circuit, such as the occurrence of multiple different results, could be avoided.
Quantum circuit equivalence checking asks whether two circuits implement the same unitary. It guarantees compiler correctness and safe optimization, yet most existing approaches scale exponentially with the number of qubits or the circuit depth, or are restricted to specific circuit structures. In this work, we present an equivalence-checking method for circuits formed by arbitrary single-qubit layers interleaved with Clifford layers. This pattern is common in variational quantum algorithms and Hamiltonian simulation via Trotter decomposition. It can also represent any unitary with sufficient depth. We prove the existence of an efficient classical algorithm that determines whether a pair of circuits with shared single-qubit layers are equivalent for every possible choice of the shared single-qubit unitaries. The same algorithm can also certify their non-equivalence for fixed assignments of single-qubit unitaries. Our framework supports the validation of emerging quantum compilers and facilitate the discovery of novel circuit optimization passes.
Quantum computing faces challenges such as noise, short coherence time, and limited qubit connections. These challenges worsen as quantum circuits become more complex. One major issue is the increasing depth of quantum circuits. This research proposes an optimization framework targeting depth and gate count reduction in quantum circuits, specifically for Noisy Intermediate-Scale Quantum (NISQ) devices. The proposed approach combines unitary merging of single-qubit gates, CNOT cancellation, gate commuting, and rotation gate rewriting strategies. Consecutive gates acting on the same qubit, such asRx(θ1)⋅Rx(θ2)R_{x}\left(\theta_{1}\right) \cdot R_{x}\left(\theta_{2}\right)Rx​(θ1​)⋅Rx​(θ2​), are algebraically merged into a single rotation, while pairs of redundant CNOT gates are eliminated based on gate identity relations. The technique is implemented using Qiskit and validated across five diverse circuits including complex, random, and multi-qubit configurations. Experimental results show an average depth reduction of 33.33% and gate count reduction of 32.14%, with runtime improvement of up to 25%. For instance, an input circuit with a depth of 7 and gate count of 11 was reduced to a depth of 2 and 4 gates. All optimized circuits preserve functional correctness with a fidelityF≥0.99F \geq 0.99F≥0.99. High-resolution circuit diagrams are presented to visually demonstrate improvements before and after optimization. Additionally, global phase shifts such aseiπ/4e^{i \pi / 4}eiπ/4are preserved or analytically characterized where relevant. This work enhances the viability of quantum computations on near-term hardware and opens pathways for future AI-driven quantum optimizations.
Equational reasoning is central to quantum circuit optimisation and verification: one replaces subcircuits by provably equivalent ones using a fixed set of rewrite rules viewed as equations. A finite rule set is most informative when it separates the genuine algebra of a circuit fragment from the structural treatment of wires. This paper gives six near-Clifford fragments a common PROP treatment, where wire permutations are structural: qubit Clifford, real Clifford, Clifford+T (up to two qubits), Clifford+CS (up to three qubits), CNOT-dihedral, and qutrit Clifford. Starting from prior completeness theorems, we transfer completeness into this setting and remove redundant non-structural rules, then check minimality by separating interpretations tailored to individual axioms; the resulting presentations are minimal in all arities for qubit Clifford, real Clifford, and CNOT-dihedral, minimal in bounded ranges for the remaining fragments, and comparable by one transfer-and-separation pattern.
Diagonal unitary operators are well known to be key building blocks of many quantum algorithms and quantum computing procedures, and thus resource-efficient quantum circuit implementations are in demand. Here we propose a constructive algorithm that can generate a quantum circuit over the primitive gate set {cnot, RZ{R}_{Z}RZ​} for realizing any given diagonal unitary operator piece by piece. The theoretical analysis reveals that, for the general case, our generated circuit not only ensures the asymptotically optimal gate count, but also nearly halves the circuit depth compared with the previous Welch's method [New J. Phys. 16, 033040 (2014)]. Specifically, this substantial depth optimization originates from the use of a uniform circuit rewriting rule developed here. The performance of our circuit synthesis algorithm is further validated by numerical evaluations on two examples. First, we achieve a nearly 50% depth reduction over Welch's method for synthesizing random diagonal unitary operators with up to 16 qubits. Second, we achieve an average of 22.05% depth reduction for resynthesizing the diagonal part of the specific quantum approximate optimization algorithm (QAOA) circuits with up to 14 qubits.