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arxiv: 2605.14822 · v1 · pith:DBDVZZEAnew · submitted 2026-05-14 · 🪐 quant-ph

Nonlinear Hamiltonians and Boolean satisfiability

Michael R. Geller , Victoria S. Ordonez , Yohannes Abate This is my paper

Pith reviewed 2026-06-30 20:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords nonlinear quantum computationBoolean satisfiabilityUNIQUE SAT3SAT#SATnonlinear Hamiltoniansancilla qubitsmean-field dynamics
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The pith

Nonlinear Hamiltonians on an ancilla qubit can determine the number of satisfying assignments to a Boolean formula and thereby solve UNIQUE SAT, 3SAT, and #SAT.

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

The paper proposes coupling a standard fault-tolerant quantum computer to ancilla qubits that evolve under a nonlinear Schrödinger equation of mean-field type. A quantum circuit first prepares the ancilla in a state that encodes the integer s, the number of satisfying assignments to an n-bit formula in conjunctive normal form. Different nonlinear terms are then applied to extract information about s: one distinguishes whether s equals zero or one, another distinguishes whether s equals zero or is positive, and a third extracts the exact value of s. These operations would solve the UNIQUE SAT, 3SAT, and #SAT problems respectively. A sympathetic reader cares because the construction shows how a specific form of nonlinearity could extract global properties of a solution set that remain hard to access with linear quantum evolution alone.

Core claim

Given an efficient quantum circuit that prepares an ancilla qubit whose state encodes the number s of satisfying assignments (0 ≤ s ≤ 2^n) to a Boolean formula in conjunctive normal form, three nonlinear Hamiltonians acting on the ancilla perform state discrimination. The term ⟨σ^z⟩ σ^z distinguishes s = 0 from s = 1 and thereby solves UNIQUE SAT under the promise of at most one solution. The term ⟨σ^x⟩ σ^y − ⟨σ^y⟩ σ^x distinguishes s = 0 from s > 0 and thereby solves 3SAT. The term ⟨σ^y⟩ ⟨σ^z⟩ σ^x − ⟨σ^x⟩ ⟨σ^z⟩ σ^y extracts the exact integer s and thereby solves #SAT. The nonlinear models are of mean-field type.

What carries the argument

Nonlinear state-discrimination gates on the ancilla qubit generated by mean-field Hamiltonians whose coefficients depend on expectation values taken in the current state.

If this is right

  • UNIQUE SAT is solved in polynomial time by the ⟨σ^z⟩ σ^z nonlinearity under the promise 0 ≤ s ≤ 1.
  • 3SAT is solved by the ⟨σ^x⟩ σ^y − ⟨σ^y⟩ σ^x nonlinearity that distinguishes s = 0 from s > 0.
  • The exact integer s is obtained by the double-expectation nonlinearity, solving #SAT.
  • All three solvers require a scalable fault-tolerant quantum computer to prepare the initial encoding of s.

Where Pith is reading between the lines

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

  • Physical realization of the required mean-field nonlinearities with ultracold atoms would provide a concrete experimental test of the proposed solvers.
  • The same encoding-plus-discrimination pattern could be applied to other counting or decision problems whose solution multiplicity can be prepared in superposition.

Load-bearing premise

The ancilla qubits evolve according to a nonlinear Schrödinger equation of mean-field type.

What would settle it

A laboratory demonstration or numerical simulation in which the proposed nonlinear evolution applied to the two ancilla states encoding s = 0 and s = 1 fails to produce distinguishable final states.

Figures

Figures reproduced from arXiv: 2605.14822 by Michael R. Geller, Victoria S. Ordonez, Yohannes Abate.

Figure 1
Figure 1. Figure 1: Efficient preparation of an ancilla qubit encoding the number of solutions [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Nonlinear Hamiltonians and their resulting velocity fields. (a) Torsion model. (b) Morse [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

We consider an extended model of quantum computation where a scalable fault-tolerant quantum computer is coupled to one or more ancilla qubits that evolve according to a nonlinear Schr\"odinger equation. Following the approach of Abrams and Lloyd, an efficient quantum circuit evaluating an $n$-bit Boolean function in conjunctive normal form is used to prepare an ancilla encoding its number $s$ of satisfying assignments ($0 \le s \le 2^n$). This is followed by a nonlinear quantum state discrimination gate on the ancilla qubit that is used to learn properties of $s$. Here we consider three types of state discriminators generated by different nonlinear Hamiltonians. First, given a restricted Boolean satisfiability problem with the promise of at most one satisfying assignment ($ 0 \le s \le 1$), we show that a qubit with $\langle \sigma^z \rangle \sigma^z$ nonlinearity can be used to efficiently determine whether $s = 0$ or $s = 1$, solving the UNIQUE SAT problem. Here $\langle A \rangle := \langle \psi | A |\psi \rangle $ denotes expectation in the current state. UNIQUE SAT is NP-hard under a randomized polynomial-time reduction (of course any discussion of complexity assumes a scalable, fault-tolerant implementation). Second, for unrestricted satisfiability problems with $ 0 \le s \le 2^n$, a Hamiltonian with $ \langle \sigma^x \rangle \sigma^y - \langle \sigma^y \rangle \sigma^x$ nonlinearity can be used to efficiently determine whether $s=0$ or $s>0$, thereby solving 3SAT, which is NP-complete. Finally, we show that $ \langle \sigma^y \rangle \langle \sigma^z \rangle \sigma^x - \langle \sigma^x \rangle \langle \sigma^z \rangle \sigma^y $ nonlinearity can be used to efficiently measure $s$ and solve #SAT, which is #P-complete. The nonlinear models are of mean field type and might be simulated with ultracold atoms.

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

1 major / 0 minor

Summary. The paper proposes extending quantum computation by coupling a fault-tolerant linear quantum computer to ancilla qubits governed by nonlinear Schrödinger equations of mean-field type. A standard quantum circuit prepares an ancilla state encoding the number s of satisfying assignments for a Boolean formula in CNF; three specific nonlinear Hamiltonians are then applied to the ancilla to discriminate properties of s, yielding efficient solutions to UNIQUE-SAT (via ⟨σ^z⟩σ^z), 3SAT (via ⟨σ^x⟩σ^y − ⟨σ^y⟩σ^x), and #SAT (via ⟨σ^y⟩⟨σ^z⟩σ^x − ⟨σ^x⟩⟨σ^z⟩σ^y). The nonlinear models are suggested to be simulable with ultracold atoms.

Significance. If the nonlinear dynamics can be consistently defined on entangled states and physically realized, the construction would place NP-complete and #P-complete problems in polynomial time, with direct implications for complexity theory. The explicit gate constructions for the three cases and the reference to Abrams-Lloyd provide a concrete starting point; the mean-field form is a clear modeling choice.

major comments (1)
  1. [Abstract and model statement] Abstract and model statement: the central claim requires that the nonlinear Hamiltonian (e.g., H = ⟨σ^z⟩ σ^z) acts on the ancilla after a linear circuit has entangled it with the main register. No definition is given for lifting the expectation-value-dependent term to the joint Hilbert space, nor for evaluating ⟨A⟩ on the reduced ancilla state without an auxiliary measurement. This definition is load-bearing for the claimed deterministic discrimination between s=0 and s=1 (and the analogous claims for 3SAT and #SAT).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for an explicit definition of the nonlinear Hamiltonian on the joint system. We address the major comment below and will incorporate the requested clarification in a revised version.

read point-by-point responses

  1. Referee: [Abstract and model statement] Abstract and model statement: the central claim requires that the nonlinear Hamiltonian (e.g., H = ⟨σ^z⟩ σ^z) acts on the ancilla after a linear circuit has entangled it with the main register. No definition is given for lifting the expectation-value-dependent term to the joint Hilbert space, nor for evaluating ⟨A⟩ on the reduced ancilla state without an auxiliary measurement. This definition is load-bearing for the claimed deterministic discrimination between s=0 and s=1 (and the analogous claims for 3SAT and #SAT).

    Authors: We agree that an explicit definition is required. In the revised manuscript we will state that the nonlinear Hamiltonian acts on the joint space as the identity on the main register tensored with an ancilla Hamiltonian whose coefficients are determined by expectation values computed on the reduced density operator of the ancilla (obtained by partial trace over the main register). The time-dependent Schrödinger equation is then integrated deterministically using these reduced-state expectations at each instant; no auxiliary measurement is performed. This construction follows the mean-field nonlinear models referenced via Abrams and Lloyd and is sufficient to realize the claimed state discrimination for UNIQUE-SAT, 3SAT, and #SAT. revision: yes

Circularity Check

0 steps flagged

No circularity: independent nonlinear model proposed without reduction to inputs

full rationale

The paper introduces a new mean-field nonlinear Schrödinger model on ancilla qubits (with terms such as ⟨σ^z⟩σ^z) to discriminate the number s of satisfying assignments prepared by a linear circuit, thereby claiming solutions to UNIQUE-SAT, 3SAT and #SAT. No step reduces a claimed result to a fitted parameter, self-defined quantity, or self-citation chain; the nonlinear Hamiltonians are posited as external assumptions rather than derived from the target complexity statements. The derivation therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Central claim depends on physical existence of mean-field nonlinear evolution for ancilla qubits and its compatibility with fault-tolerant quantum computation; no free parameters are fitted in the abstract.

axioms (2)
  • domain assumption Ancilla qubits evolve under a nonlinear Schrödinger equation
    Core modeling choice stated in the abstract opening.
  • domain assumption Mean-field nonlinear models can be simulated with ultracold atoms
    Physical realization claim in the final sentence.
invented entities (1)
  • Nonlinear Hamiltonians (⟨σ^z⟩σ^z, ⟨σ^x⟩σ^y − ⟨σ^y⟩σ^x, ⟨σ^y⟩⟨σ^z⟩σ^x − ⟨σ^x⟩⟨σ^z⟩σ^y) no independent evidence
    purpose: Generate state discrimination gates to extract properties of s
    Introduced as generators of the three discriminators

pith-pipeline@v0.9.1-grok · 5925 in / 1396 out tokens · 40679 ms · 2026-06-30T20:11:58.157105+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    Constructs quantum filtered oracle for Valiant-Vazirani theorem reducing SAT to UNIQUE SAT, enabling polynomial-time NP solution via torsion nonlinearity in noise-free limit but not #P.

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