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arxiv: 2409.03917 · v1 · pith:YISN3R63new · submitted 2024-09-05 · 🪐 quant-ph · cs.ET

qSAT: Design of an Efficient Quantum Satisfiability Solver for Hardware Equivalence Checking

Abhoy Kole , Mohammed E. Djeridane , Lennart Weingarten , Kamalika Datta , Rolf Drechsler This is my paper

Pith reviewed 2026-05-23 21:13 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET
keywords quantum SATGrover algorithmequivalence checkingBoolean circuitsCNF generationExclusive-Sum-of-Productquantum circuit depth
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The pith

An Exclusive-Sum-of-Product encoding for CNF clauses reduces qubits, gates, and depth in Grover-based quantum SAT solvers for Boolean circuit equivalence checking.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a quantum SAT solver that applies Grover's algorithm to check equivalence between Boolean circuits. It replaces standard CNF generation with an Exclusive-Sum-of-Product representation of the clauses. This change is claimed to lower the qubit count and shorten both gate count and circuit depth in the resulting quantum circuit. The approach is illustrated with reference circuits that influence the required Grover iterations, and results are reported from Qiskit simulations and runs on IBM quantum hardware. The goal is to mitigate the exponential runtime that classical SAT solvers encounter on certain verification instances.

Core claim

The Exclusive-Sum-of-Product based generation of the Conjunctive Normal Form equivalent clauses demand less qubits and minimizes the gates and depth of quantum circuit interpretation for equivalence checking of Boolean circuits employing Grover's algorithm.

What carries the argument

Exclusive-Sum-of-Product (ESP) representation used to generate the CNF clauses that are then encoded into the oracle of a Grover search circuit.

If this is right

  • Equivalence checking instances that were previously qubit-limited become feasible on near-term devices.
  • Shallower circuits reduce the number of Grover iterations needed for a target success probability.
  • Reference-circuit properties can be used to tune iteration count and resource allocation before execution.
  • The method extends directly to other SAT-encoded verification tasks that rely on Grover oracles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The shallower circuits may tolerate higher noise rates, potentially allowing larger instances before error correction becomes necessary.
  • Combining the ESP encoding with amplitude amplification variants could further lower the total gate count.
  • The same encoding technique might be tested on hybrid classical-quantum SAT solvers to measure crossover points.

Load-bearing premise

That the claimed reductions in qubits, gates, and depth from the ESP-based CNF generation will produce a net practical advantage once Grover iteration count and hardware noise are taken into account.

What would settle it

A side-by-side resource count on the same equivalence instance showing that a conventional CNF encoding requires no more qubits, gates, or depth than the ESP version while achieving at least equal success probability under Grover search.

Figures

Figures reproduced from arXiv: 2409.03917 by Abhoy Kole, Kamalika Datta, Lennart Weingarten, Mohammed E. Djeridane, Rolf Drechsler.

Figure 1
Figure 1. Figure 1: Time and memory requirements of SAT based verification of multipliers. scale quantum (NISQ) computers [5] enables one to formulate problems taking advantage of such systems. In this respect, the verification problem is designed as a quantum operator and a quantum version of SAT solver (qSAT) is formulated using Grover’s search algorithm [6] to find solutions with a quadratic speed-up over classical searchi… view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit interpretation of a Boolean function, F(a, b) = a ⇒ b. the proposed ESOP-based clause generation, miter formation, and the qSAT solver architecture, including a case study on multiplier and 1-bit full-adder operations. In Section IV, we demonstrate the efficiency of the proposed qSAT solver in terms of resource overhead, solution accuracy probability, and operational reliability. Finally, c… view at source ↗
Figure 4
Figure 4. Figure 4: An illustration of a quantum miter circuit realization; (a) classical miter circuit F for doing equivalence checking of Boolean circuits GI and GR; (b) quantum circuit interpretation of F additionally realizing operations inferred on auxiliary variables a1, a2 and a3; (c) uncomputing the inferences on a1, a2 and a3 applying respective operations in reverse order. p ⇔ (q ∨ r) |p⟩ • |q⟩ W ≡ |r⟩ |y⟩ • ≡ |0⟩ X… view at source ↗
Figure 6
Figure 6. Figure 6: Outline of the qSAT Algorithm. 1 binary values. In order to exclude m such CEX from the miter outcome, F one can consider the following augmented ESOP form: Fb = F ⊕Mm i=1 CEXi (14) where CEXi indicates the i-th counter-example. The quantum circuit interpretation of such augmented miter, Fb can be obtained by realizing the miter, F in either of VF or UF form followed by placing multiple-controlled-X operat… view at source ↗
Figure 7
Figure 7. Figure 7: (a) A reference multiplexer clause network realization using 4 auxiliary variables; (b) corresponding realization based on single auxiliary variable. environment, the additional miter (VF ) execution in the qSAT algorithm eliminates the need of classical verification of the measured CEX. A prior knowledge about the number of CEXs, (say, m) can reduce the number of executions of Grover’s iteration (GI), i.e… view at source ↗
Figure 9
Figure 9. Figure 9: qSAT accuracy in providing SAT for miters, GF ⊕ GR and GF ⊕ GbR. C. Evaluation of qSAT Outcome For evaluation of qSAT networks we consider Qiskit Aer simulator as an ideal environment. The probability of SAT outcome for each miter, M ∈ {GF ⊕GR, GF ⊕GbR}, is computed in the following way: PM(SAT) = 1 #run Xn i=1 P(CEXi) (24) where #run indicates the number of solver executions for M and P(CEXi) denotes the … view at source ↗
Figure 10
Figure 10. Figure 10: Fidelity of reference models, GR and GbR on a IBM quantum computer. the clauses are generated using ESOP based method, then the miter circuit is proposed. The proposed method results in circuit generation with fewer qubits, which is also linear in terms of number of gates present in the circuit. Finally, the qSAT architecture is proposed which exploits the benefit of Grovers search algorithm. Experimental… view at source ↗
read the original abstract

The use of Boolean Satisfiability (SAT) solver for hardware verification incurs exponential run-time in several instances. In this work we have proposed an efficient quantum SAT (qSAT) solver for equivalence checking of Boolean circuits employing Grover's algorithm. The Exclusive-Sum-of-Product based generation of the Conjunctive Normal Form equivalent clauses demand less qubits and minimizes the gates and depth of quantum circuit interpretation. The consideration of reference circuits for verification affecting Grover's iterations and quantum resources are also presented as a case study. Experimental results are presented assessing the benefits of the proposed verification approach using open-source Qiskit platform and IBM quantum computer.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a quantum SAT (qSAT) solver for equivalence checking of Boolean circuits that employs Grover's algorithm. It introduces an Exclusive-Sum-of-Product (ESP) method for generating the equivalent CNF clauses, claiming this yields fewer qubits and lower gate count and circuit depth than conventional encodings. Case studies on reference circuits are used to illustrate effects on Grover iterations and resources, with experimental demonstrations on the Qiskit simulator and IBM quantum hardware.

Significance. If the per-oracle qubit/gate reductions from the ESP-based CNF generation survive multiplication by the full Grover iteration count (~2^{n/2}) and remain advantageous under realistic noise, the work would provide a concrete route to more resource-efficient quantum oracles for SAT instances arising in hardware verification. The approach builds on standard Grover analysis without introducing new free parameters, but the lack of aggregate accounting prevents a clear assessment of net practical gain.

major comments (2)
  1. [Experimental Results] Experimental Results section: the abstract states that experimental results on Qiskit and IBM hardware assess the benefits of the proposed approach, yet no quantitative comparisons to a conventional CNF encoding, no error bars, no total gate counts for the complete Grover operator (oracle repeated sqrt(N) times), and no estimated fidelity under the hardware noise model are reported. Without these aggregate metrics the local per-oracle savings cannot be shown to deliver a net advantage.
  2. [Case Study] Case Study section: the claim that ESP-based CNF generation minimizes qubits, gates and depth is presented for selected reference circuits, but the manuscript does not compute or tabulate the total resource cost of the full Grover search (including diffusion operator and iteration count) nor compare it head-to-head with a standard CNF oracle under the same noise model. This leaves the central efficiency claim load-bearing but unverified at the algorithm level.
minor comments (1)
  1. [Abstract] The abstract and introduction use the phrase 'quantum circuit interpretation' without defining the term; a brief clarification of whether this refers to circuit depth, gate count, or transpilation would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. Below we respond point-by-point to the major comments and indicate planned revisions.

read point-by-point responses

  1. Referee: [Experimental Results] Experimental Results section: the abstract states that experimental results on Qiskit and IBM hardware assess the benefits of the proposed approach, yet no quantitative comparisons to a conventional CNF encoding, no error bars, no total gate counts for the complete Grover operator (oracle repeated sqrt(N) times), and no estimated fidelity under the hardware noise model are reported. Without these aggregate metrics the local per-oracle savings cannot be shown to deliver a net advantage.

    Authors: We agree that aggregate metrics would allow a clearer evaluation of net advantage. Full execution of the Grover operator (approximately 2^{n/2} iterations) is infeasible to simulate or run on current hardware for the reference circuits considered, which is why only per-oracle implementations were demonstrated experimentally. The per-oracle qubit, gate, and depth savings scale linearly with iteration count, preserving the relative advantage. In revision we will add estimated total gate counts (per-oracle cost multiplied by iteration count, including the diffusion operator) and a theoretical comparison table against conventional CNF encoding. Error bars and fidelity estimates for the complete operator cannot be provided for the same scalability reason and will be noted explicitly. revision: partial

  2. Referee: [Case Study] Case Study section: the claim that ESP-based CNF generation minimizes qubits, gates and depth is presented for selected reference circuits, but the manuscript does not compute or tabulate the total resource cost of the full Grover search (including diffusion operator and iteration count) nor compare it head-to-head with a standard CNF oracle under the same noise model. This leaves the central efficiency claim load-bearing but unverified at the algorithm level.

    Authors: The case study illustrates the effect of the ESP encoding on iteration count and per-oracle resources for reference circuits. We acknowledge that tabulating full Grover resources would strengthen the presentation. We will revise the section to include explicit calculations of total estimated resources (oracle plus diffusion, scaled by iteration count) and add a head-to-head theoretical resource comparison with standard CNF encoding. Direct experimental comparison under a noise model remains limited by the inability to execute the full search; this limitation and the linear scaling argument will be stated clearly. revision: yes

Circularity Check

0 steps flagged

No circularity in the qSAT derivation chain

full rationale

The paper presents a constructive design for an ESP-based CNF encoding to reduce qubits/gates/depth in a Grover-oracle formulation of circuit equivalence checking. No equations, fitted parameters, or self-citation chains are shown that would make any claimed resource saving equivalent to its own inputs by construction. The method is a standard application of Grover search to a classically generated oracle; the experimental Qiskit/IBM runs serve as direct validation rather than a self-referential prediction. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proposal rests on the standard quadratic speedup of Grover's algorithm and on the assumption that CNF clauses can be directly mapped to oracle circuits; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Grover's algorithm yields a quadratic speedup for unstructured search problems
    Invoked to justify the use of Grover search for SAT.
  • domain assumption CNF clauses generated from an ESP representation can be realized with fewer qubits than standard encodings
    Central design choice stated in the abstract.

pith-pipeline@v0.9.0 · 5650 in / 1311 out tokens · 25922 ms · 2026-05-23T21:13:22.880760+00:00 · methodology

discussion (0)

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Reference graph

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