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.
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.
In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Gröbner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Gröbner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gröbner bases over F 2.
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.
Near-term quantum computers are expected to work in an environment where each operation is noisy, with no error correction. Therefore, quantum-circuit optimizers are applied to minimize the number of noisy operations. Today, physicists are constantly experimenting with novel devices and architectures. For every new physical substrate and for every modification of a quantum computer, we need to modify or rewrite major pieces of the optimizer to run successful experiments. In this paper, we present QUESO, an efficient approach for automatically synthesizing a quantum-circuit optimizer for a given quantum device. For instance, in 1.2 minutes, QUESO can synthesize an optimizer with high-probability correctness guarantees for IBM computers that significantly outperforms leading compilers, such as IBM's Qiskit and TKET, on the majority (85%) of the circuits in a diverse benchmark suite. A number of theoretical and algorithmic insights underlie QUESO: (1) An algebraic approach for representing rewrite rules and their semantics. This facilitates reasoning about complex symbolic rewrite rules that are beyond the scope of existing techniques. (2) A fast approach for probabilistically verifying equivalence of quantum circuits by reducing the problem to a special form of polynomial identity testing . (3) A novel probabilistic data structure, called a polynomial identity filter (PIF), for efficiently synthesizing rewrite rules. (4) A beam-search-based algorithm that efficiently applies the synthesized symbolic rewrite rules to optimize quantum circuits.
Optimizing quantum circuits is critical: the number of quantum operations needs to be minimized for a successful evaluation of a circuit on a quantum processor. In this paper we unify two disparate ideas for optimizing quantum circuits, rewrite rules, which are fast standard optimizer passes, and unitary synthesis, which is slow, requiring a search through the space of circuits. We present a clean, unifying framework for thinking of rewriting and resynthesis as abstract circuit transformations. We then present a radically simple algorithm, guoq, for optimizing quantum circuits that exploits the synergies of rewriting and resynthesis. Our extensive evaluation demonstrates the ability of guoq to strongly outperform existing optimizers on a wide range of benchmarks.
Today’s quantum devices are noisy, so reducing circuit size is critical for reliable execution. Existing rule-based optimizers often rely on large rule sets that are difficult to manage and still miss long-distance transformations. We present QSymb, a framework for synthesizing compact and expressive quantum-circuit rewrite rules with formal guarantees. We formalize symbolic rewrite rules in which a symbolic gate represents infinitely many subcircuits. We then define canonical symbolic rules of the form L ; S = S ; R and prove that they constitute a compact generative core from which general symbolic rules can be derived. On top of this formal foundation, given a gate set, QSymb synthesizes (1) a small, non-derivable concrete rule set that is complete up to chosen size and qubit bounds, and (2) a small but expressive canonical symbolic rule set that captures transformations beyond finite or monomial-only patterns. We further present rule anchoring to derive optimization-effective rules from canonical symbolic rules. Together, these results provide both expressiveness and guarantees: soundness of synthesized rules via validation, non-derivability, and bounded completeness. On the IBM-Eagle gate set, QSymb strictly outperforms state-of-the-art rewrite-based optimizers (Qiskit, Guoq, Quartz, TKET, and Queso) in two-qubit-gate reduction on 90%, 67%, 82%, 85%, and 83% of standard quantum algorithm benchmarks, respectively; on Nam gate set, the corresponding rates are 88%, 74%, 81%, 86%, and 82.9%. It achieves final average two-qubit-gate reductions of 27.44% and 29.95%, respectively.
Quantum computing is an emerging technology in which quantum mechanical properties are suitably utilized to perform certain compute-intensive operations faster than classical computers. Quantum algorithms are designed as a combination of quantum circuits that each require a large number of quantum gates, which is a challenge considering the limited number of qubit resources available in quantum computing systems. Our work proposes a technique to optimize quantum arithmetic algorithms by reducing the hardware resources and the number of qubits based on ZX calculus. We have utilised ZX calculus rewrite rules for the optimization of fault-tolerant quantum multiplier circuits where we are able to achieve a significant reduction in the number of ancilla bits and T-gates as compared to the originally required numbers to achieve fault-tolerance. Our work is the first step in the series of arithmetic circuit optimization using graphical rewrite tools and it paves the way for advancing the optimization of various complex quantum circuits and establishing the potential for new applications of the same.