Here it is shown that minimized expressions retain algebraic structure minimization cannot reach, arising from containment and complementary-polarity relationships among their terms, and that extracting it yields circuits cheaper to execute despite having more operations.
Abstract
Quantum algorithms promise advantages beyond classical reach, but running them on error-corrected hardware requires translating Boolean specifications into reversible circuits, and the resources that translation demands determine what is executable. Established methods minimize a Boolean expression and map it to a circuit, assuming the minimized form is best. Here we show that minimized expressions retain algebraic structure minimization cannot reach, arising from containment and complementary-polarity relationships among their terms, and that extracting it yields circuits cheaper to execute despite having more operations. The decisive quantity is not a circuit's operation count but the control count of its widest operation, a superlinear cost; extracting shared factors trades a few wide operations for many narrow ones and reduces qubit count. Across benchmarks and oracles from quantum search and factoring algorithms, at the representation level the transformation never increases either cost measure, a guarantee from its construction. Translation to an executable circuit returns part of that advantage, since auxiliary lines must be uncomputed, yet the factorized circuit still left a leading circuit-level optimizer reaching lower final counts, and faster, than unaided. The representation of a computation is therefore itself a resource, optimizable before compilation and distinct from both logic minimization and circuit-level optimization.
A measurement of what diagrammatic post-processing recovers from structural redundancy in the Solovay-Kitaev algorithm, which optimizes for numerical convergence rather than circuit economy, and its output carries structural redundancy that a gate-level compiler cannot see.
Dulari De Silva, A. Mahasinghe, Chon-Fai Kam et al.· 0 citations
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.
A Clifford+T quantum circuit construction that approximately implements any classically specified unitary to within error $\epsilon$ and achieves a worst-case $T$-count with leading exponential scaling of $2^{5n/4}$ whenever $\log(1/\epsilon)=\operatorname{poly}(n)$.
It is explained that the efficiently preparable states, device-generated distributions, variationally learned loading, and amortized preparation are required to get advantage from quantum machine learning and close with a checklist for evaluating input-dependent advantage claims.
A general framework to define quantum circuit architecture witnesses, which certify the incompatibility of a unitary transformation with a specified quantum circuit architecture, and exploit the stabiliser formalism to reduce the construction to linear programming.
A unified, highly scalable methodology for the optimal design of reversible circuits across both libraries is presented, establishing best-known upper bounds that significantly outperform heuristic literature benchmarks.
G. Papakonstantinou· Quantum Reports· 0 citations
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