A Localized Reality Appears To Underpin Quantum Circuits

James Saslow; Ken Wharton; Raylor Liu; Roderick Sutherland; Titus Amza

REVIEW 4 major objections 4 minor 1 cited by

A Localized Reality Appears To Underpin Quantum Circuits

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper argues that the local weak values of individual qubits in a quantum circuit remain localized and evolve by simple local rules even when the global state is entangled, providing evidence that a spacetime-localized, all-at-once…

desk verdict Clean, checkable algebra on dynamically local weak values (the oscillator equation is genuinely new) under a universal claim that outruns the evidence; the Section V model is an honest template, not a demonstration. read the letter →

arxiv 2412.05456 v1 pith:FKMVUCIB submitted 2024-12-06 quant-ph physics.hist-ph

Ken Wharton , Roderick Sutherland , Titus Amza , Raylor Liu , James Saslow This is my paper · ORCID

classification quant-phphysics.hist-ph PACS 03.65.Ta03.67.Lx

keywords weakvaluesquantumcircuitslocalhiddenvariablesall-at-oncemodelsretrocausalityexchangeinteractionsqrt(SWAP)gateBell'stheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the local weak values of individual qubits in any complete quantum circuit behave as localized, real properties, even when the overall quantum state is massively entangled. On circuit wires the weak values stay constant; through single-qubit gates they rotate exactly as the gate rotates a Bloch vector; and during an exchange interaction they evolve by a simple second-order differential equation of coupled oscillators. The paper takes these regularities as evidence that a spacetime-localized, all-at-once account could underpin any quantum circuit, with resources scaling linearly in the number of qubits rather than exponentially. It also presents a toy model using hidden complex 3-vectors whose solutions reproduce the weak values on average and whose counting reproduces outcome probabilities for a sqrt(SWAP) circuit.

What carries the argument

The main object is the local weak value vector $w$, a complex 3-vector assigned to each wire of a quantum circuit, defined by Eq. (1) with single-qubit Pauli operators and identity on the remaining qubits. The load-bearing mechanical results are: constancy of $w$ on free wires (Section III.B), the rotation rule through single-qubit gates (Section III.C), and the oscillator equation inside the exchange interaction, derived analytically in Appendix A. The authors also propose a template hidden-variable model (Section V) in which each qubit carries a complex 3-vector $s$ constrained by $s \cdot s = 1$ and, during exchange, by $s_a \cdot s_b = 1$ and first-order torque equations; these constraints imply the oscillator dynamics, and counting the solutions of a $\sqrt{\mathrm{SWAP}}$ circuit reproduces the Born-rule probabilities of the three allowed outcomes.

What would settle it

Perform a weak-measurement experiment on the $\sqrt{\mathrm{SWAP}}$ circuit of Figure 4, preparing $|x\rangle|y\rangle$ and post-selecting on $|10\rangle$, and compare the measured $\mathrm{Re}[w_a(\tau)]$ and $\mathrm{Re}[w_b(\tau)]$ with the analytic forms in Appendix B. If the trajectories deviate from the coupled-oscillator equation, or if the pre-interaction weak values are found not to depend on the choice of the final measurement basis while the preparation and post-selected outcome are held fixed, the central locality claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that the complex weak-value vector $w = (W[\sigma_x], W[\sigma_y], W[\sigma_z])$ for each qubit, computed from the usual post-selected weak-value formula, obeys dynamic locality in any circuit built from single-qubit rotations and the exchange interaction. If a qubit is acted on by no gate, its $w$ is constant even when other qubits evolve in the same entangled state; if it passes through a single-qubit gate, its $w$ rotates exactly as the corresponding Bloch vector rotates, independent of the rest of the state; and inside a $\mathrm{SWAP}_\alpha$ exchange gate its components obey $\frac{d^2 w_a}{d\tau^2} = \frac{\pi^2}{2}(w_b - w_a)$, a classical coupled-oscillator equation. The authors take these results as evidence that entangled quantum circuits can be underpinned by localized variables that are future-input-dependent: the hidden variables must be solved all-at-once, using both past preparation and future measurement constraints, rather than by a forward-causal dynamical law.

Load-bearing premise

The model requires that the hidden vectors of two qubits entering an exchange interaction already satisfy $s_a \cdot s_b = 1$ before the interaction begins, which means the past states are constrained by future measurement settings; if retrocausal or future-boundary constraints are disallowed, the model cannot reproduce entangled weak values or outcome statistics.

Editorial extensions

If this is right

If the central claim is right, no instantaneous nonlocal connection is needed to explain Bell-inequality violations in circuits; the locality is carried by weak values that only depend on past and future boundary conditions, not on distant simultaneous events.
A generic quantum circuit can be analyzed "all-at-once", replacing the exponentially large state vector with a set of $N$ local hidden vectors, provided the future measurement setting is allowed to constrain earlier hidden states.
Weak values can be generated without state vectors as an intermediate step, at least in the $\sqrt{\mathrm{SWAP}}$ example, because averaging the hidden $s$ solutions reproduces $\mathrm{Re}(w)$ throughout the gate.
The outcome probabilities of a partially entangled two-qubit circuit emerge from the number of allowed hidden-variable solutions, matching the Born-rule weights for the $|00\rangle$, $|10\rangle$, $|11\rangle$ outcomes, and vanishing for the impossible $|01\rangle$ outcome.
The same modular strategy applied to a universal gate set (single-qubit rotations plus $\sqrt{\mathrm{SWAP}}$) suggests an infinite family of entanglement geometries could be treated by local rules, one gate at a time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the weak-value locality carries over to larger circuits, an experimental test could track weak values through a controlled $\sqrt{\mathrm{SWAP}}$ interaction and check whether the measured trajectories satisfy the oscillator equation for each outcome; this would distinguish the all-at-once picture from forward-causal hidden-variable models.
The future-input dependence required by the model is a retrocausal element: the final measurement basis is a boundary condition on the past. A reader who rejects retrocausal constraints will see the exchange-interaction constraint $s_a \cdot s_b = 1$ as the point where a purely dynamical account breaks down.
The paper's approach connects naturally to the "second-order qubit" program mentioned in the text; expanding the per-qubit hidden space from 6 to 7 or 8 real variables would relieve the over-constraint problem and may yield a general local model for universal circuits.
If the solution-counting rule generalizes, it would provide a probabilistic interpretation in terms of hidden-state counting rather than wavefunction amplitudes, offering a route to quantum probabilities that does not invoke a collapse postulate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—

The thing worth knowing about this paper is that the algebraic core is real and the interpretive superstructure is not. Sections III and IV, plus Appendix A, show that local weak values in any quantum circuit are dynamically local: constant on noninteracting wires no matter what the rest of the entangled state is doing, rotating through one-qubit gates, and obeying a clean oscillator equation d²w_a/dτ² = (π²/2)(w_b − w_a) during an exchange interaction. That last result is new, and Appendix A derives it directly from the weak-value definition. I checked the logic; it is straightforward and sound. Anyone working with weak values or retrocausal models will want this.

The soft spot is the leap from those results to the claim that a localized reality underpins any quantum circuit. That leap is carried by the Section V model, and the model is explicitly a template. It is overconstrained: 14 real constraints on 12 real initial variables. The probability rule—counting solutions—does not generalize, and the authors admit it does not square with the single-qubit 1/Re[w]^2 rule. They recover Im(w) only by discarding half the solutions. The future-input dependence (Eq. 29 and the measurement constraints) is an assumption of the research program, not a derivation. So the universal claim is not supported yet. But the authors say this themselves in Section V.D. The paper is honest about its own gaps, which is more than many papers in this area manage.

The dynamic-locality results stand on their own. They are formal, reproducible algebra: no free parameters, no invented entities beyond the s(t) vectors in the template model. The citation pattern is appropriate—Wharton and Argaman's Colloquium, Sutherland's work, the weak-value experimental literature. Self-citation is not a problem here; the cited work is relevant.

Who should read this: quantum foundations people interested in weak values, retrocausality, and all-at-once accounts. It won't convert a skeptic, but it gives the program a concrete, checkable handle on circuit dynamics. As a referee, I would send it out. The algebraic results deserve scrutiny, and the Section V model deserves a referee's push on whether the constraints can be relaxed or the probability rule improved. The paper should survive as a contribution if the claims are scaled back to “this is a promising template, not a demonstration.”

Recommendation: accept for review. If I were the editor, I'd want a referee who can check the algebra and who is sympathetic but not credulous.

Referee Report

4 major / 4 minor

Summary. The paper studies local (single-qubit) weak values in complete quantum circuits. It proves (Sections III and IV, with Appendix A) that these weak values are dynamically local: constant on non-interacting wires, rotating through single-qubit gates, and obeying the coupled oscillator equation d²w/dτ² = (π²/2)(w_b − w_a) during an exchange interaction, regardless of entanglement elsewhere. On this basis the authors argue that any quantum circuit can plausibly be underpinned by a localized, all-at-once hidden-variable model. Section V presents a template model in which each qubit carries a complex 3-vector s(t), constrained by s·s = 1 and by preparation/measurement constraints, and shows that for a single √SWAP circuit the number of solutions matches the Born probabilities while the average Re[s] reproduces Re[w]. The authors themselves identify serious open problems with this model: it is overconstrained (14 real constraints on 12 real variables) and lacks a general probability rule.

Significance. If the program were completed, the result would be significant: it would suggest a spacetime-localized, linearly-scaling account of arbitrary quantum circuits, with retrodictive constraints replacing nonlocal causation. The algebraic core of the paper is strong and reproducible: the dynamic-locality results in Sections III–IV follow directly from Eq. (1), and Appendix A gives a clean derivation of Eq. (22). The authors are also commendably explicit about the limitations of the Section V template. However, the central claim of the paper—that a localized reality 'appears to underpin' any quantum circuit—is not established by the evidence presented. The dynamic-locality results concern weak values, not a complete hidden-variable model, and the only concrete model offered (Section V) is overconstrained, future-input-dependent, and fitted to a single example. The paper is therefore best read as a strong motivation and a preliminary template, not as a demonstration of the universal claim.

major comments (4)

[§V.D] The overconstraint problem is load-bearing for the universality claim. The authors count 14 real constraints on the 12 real initial variables of the two-qubit model (three complex equations (27)–(29) give six real constraints, and the four preparation/measurement constraints (36)–(39) give eight more). They then show that the √SWAP example happens to have solutions, but no argument is given that generic circuits, or even generic two-qubit states, admit solutions. Since the paper's conclusion that 'any quantum circuit can plausibly be underpinned by localized variables' depends on the Section V model being a template, the overconstraint means the central claim is currently unsupported beyond the single example.
[§II.C and §V.C] The model uses two incompatible probability rules. In the single-qubit sector, Eq. (11) assigns probability P(w) ∝ 1/Re(w)², whereas the √SWAP analysis in Section V.C assigns equal weight to each solution of the constraint equations, with the correct Born probabilities emerging from the number of solutions. The authors acknowledge in §V.D that no unified rule is known ('It is not clear what probability rule would work for both this example and for the single-qubit case'). Without a general probability rule that reproduces Born probabilities for arbitrary circuits, the Section V model cannot underpin quantum circuits, and the claim that weak values 'could generate weak values without using state vectors' remains confined to the worked example.
[§V.A, Eqs. (29), (38)–(39)] The model's constraints are future-input-dependent in a way that is not derived but imposed. Eq. (29), s_a·s_b = 1, requires the two qubits' hidden vectors to be correlated before they interact, and the measurement constraints (38)–(39) depend on the future outcome. The authors honestly label this 'counterintuitive' and 'evidently retrocausal'. However, the paper presents this future-input dependence as a feature that explains Bell violations, not as a hypothesis to be tested. Because no mechanism or independent motivation is given for why these particular constraints should hold, the empirical match in Section V.C is essentially a reverse-engineering of the constraints from the known weak values. This weakens the inference from 'the model fits one example' to 'localized reality underpins circuits.'
[§VI.A] The leap from dynamic locality of weak values to a localized underlying reality is not supported by the algebraic results alone. Sections III–IV show that weak values, as computed from Eq. (1), evolve locally; but weak values are averages over an ensemble (or best estimates), and the paper itself introduces s(t) in Section V precisely because w(t) does not satisfy the model constraints in all cases (e.g., the |10⟩ outcome). Thus the dynamic-locality results are necessary but not sufficient for the conclusion. The only bridge is the Section V template, whose limitations are acknowledged. To make the universal claim credible, the paper would need either a more complete model or a clearly weakened conclusion stating that the evidence supports only the possibility, not the plausibility, of such an underpinning.

minor comments (4)

[Eqs. (20)–(21)] The denominator in both equations is written as ⟨f|U_ex[α]|ψ⟩, but it should be ⟨f|U_ex[α]|i⟩, consistent with Eq. (1).
[§IV.C] In the expression for Im(dw_a/dτ), the second cross product is written as Im(w_b)×Im(w_a); from Eq. (23) it should be Im(w_b)×Re(w_a) (with the first term Re(w_b)×Im(w_a)).
[§V.C] The text refers to a 'BRST algorithm' for gradient descent; this is presumably a typo for a standard optimizer such as BFGS. Please clarify the algorithm used.
[Figure 3] The axis labels and legend are difficult to read in the reproduced figure; please increase the font size and ensure all curve labels are legible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dynamic-locality derivations follow directly from the standard weak-value definition and are algebraically self-contained; the Section V model is explicitly labeled a speculative template, not a derivation.

full rationale

The core derivation chain is self-contained. Section III's dynamic-locality results are direct consequences of Eq. (1): for a noninteracting wire, W[sigma⊗I] is constant because sigma⊗I commutes with I⊗U_{N-1}, and through a single-qubit gate the weak value rotates because U1 sigma_j U1^{-1} = sigma_f. These are algebraic corollaries, not assumptions smuggled in. Section IV's central result, Eq. (22), is proved in Appendix A directly from Eqs. (1) and (19), using only a Taylor expansion of the exchange unitary and the identities (A5)-(A7); no fitted parameter or imported conclusion is involved. The single-qubit relations (9)-(11) are also consequences of Eq. (1): Eq. (11) simply rewrites the Born probability P ∝ 1/Re(w)^2 for that two-outcome example, so it is a derived expression rather than a fitted rule. The only place where reverse-engineering could be suspected is Section V: the constraints (27)-(31) are chosen so that the first-order equations yield exactly the second-order dynamics (26), with Eq. (29) imposed to make the bracketed factors in (32)-(33) collapse to 2. But the paper explicitly calls this 'a template', says the constraints are 'guessed', and Section V.D admits the model is overconstrained (14 real constraints on 12 variables), lacks a general probability rule, and recovers Im(w) only after discarding half the solutions. Because the model is not presented as a prediction or a first-principles derivation, choosing constraints to match the already-established oscillator equation is a modelling choice, not a circular step. The self-citations, including [1], [15], [19], [23], [24], are contextual and are not load-bearing for the mathematical results in Sections III-IV. No step was found where an input is defined in terms of the claimed output, or where a fitted quantity is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central claim rests on standard weak-value formalism plus several domain assumptions (exchange interaction, retrocausality) and a set of ad hoc model constraints. The invented entity s has no independent evidence and is designed to match the weak values.

assumptions (5)

standard math The weak value definition (Eq. 1) gives the best estimate of a local observable given pre- and post-selection.
Invoked at Eq. (1) and throughout; a standard result in weak measurement theory.
domain assumption The exchange interaction Hamiltonian Hex = J(σ⊗σ) generates the unitary Uex[α] in Eq. (19).
Taken as the model for two-qubit interactions; this is a standard physical assumption.
domain assumption Future measurement settings can constrain earlier hidden variables (future-input-dependence / retrocausality).
Stated in Sec. I and used to justify the all-at-once model; without it the hidden-variable model cannot reproduce Bell correlations.
ad hoc to paper The constraint sa·sb = 1 (Eq. 29) is imposed on hidden vectors even before the two qubits interact.
Chosen to make the first-order equations (30)-(31) imply the second-order equation (26); no physical mechanism provided.
ad hoc to paper The probability of an outcome is proportional to the number of solutions of the constraint equations.
Used to match Born probabilities in Sec. V.C; authors acknowledge it is not generalizable.

invented entities (1)

s(t)
purpose: A complex 3-vector per qubit, proposed as the fundamental localized hidden variable; its real part averages to the weak value w.
Introduced in Sec. V.A as a template; there is no independent experimental or theoretical handle outside the paper, and its constraints are chosen to reproduce w.

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Cite this review

Pith. "Pith review of A Localized Reality Appears To Underpin Quantum Circuits." pith.science (2026). https://pith.science/paper/FKMVUCIB

@misc{pith2026241205456,
  author       = {Pith},
  title        = {Pith review of: A Localized Reality Appears To Underpin Quantum Circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FKMVUCIB}},
  note         = {Machine review of arXiv:2412.05456}
}

read the original abstract

Although entangled state vectors cannot be described in terms of classically realistic variables, localized in space and time, any given entanglement experiment can be built from basic quantum circuit components with well-defined locations. By analyzing the (local) weak values for any given run of a quantum circuit, we present evidence for a localized account of any circuit's behavior. Specifically, even if the state is massively entangled, the weak values are found to evolve only when they pass through a local circuit element. They otherwise remain constant and do not evolve when other qubits pass through their circuit elements. A further surprise is found when two qubits are brought together in an exchange interaction, as their weak values then evolve according to a simple classical equation. The weak values are subject to both past and future constraints, so they can only be determined by considering the entire circuit "all-at-once", as in action principles. In the context of a few basic quantum gates, we show how an all-at-once model of a complete circuit could generate weak values without using state vectors as an intermediate step. Since these gates comprise a universal quantum gate set, this lends support to the claim that any quantum circuit can plausibly be underpinned by localized variables, providing a realistic, lower-level account of generic quantum systems.

Figures

Figures reproduced from arXiv: 2412.05456 by the authors.

**Figure 2.** FIG. 2. The x-z plane of the Bloch sphere, showing the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. This graph shows a typical pair of local weak value [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. The local weak values [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: FIG. 6. The comprehensive set of solutions for the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Wharton and N

K. Wharton and N. Argaman, Colloquium: Bell’s theo- rem and locally mediated reformulations of quantum me- chanics, Reviews of Modern Physics 92, 021002 (2020)

work page 2020

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work page 1978

[3] [3]

Aharonov, D

Y. Aharonov, D. Z. Albert, and L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Physical review letters 60, 1351 (1988)

work page 1988

[4] [4]

R. I. Sutherland, Density formalism for quantum theory, Foundations of Physics 28, 1157 (1998)

work page 1998

[5] [5]

Campagne-Ibarcq, L

P. Campagne-Ibarcq, L. Bretheau, E. Flurin, A. Auff` eves, F. Mallet, and B. Huard, Observing interferences between past and future quantum states in resonance fluorescence, Physical review letters 112, 180402 (2014)

work page 2014

[6] [6]

Weber, A

S. Weber, A. Chantasri, J. Dressel, A. N. Jordan, K. W. Murch, and I. Siddiqi, Mapping the optimal route be- tween two quantum states, Nature 511, 570 (2014)

work page 2014

[7] [7]

D. Tan, S. Weber, I. Siddiqi, K. Mølmer, and K. Murch, Prediction and retrodiction for a continuously moni- tored superconducting qubit, Physical review letters114, 090403 (2015)

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[8] [8]

Rybarczyk, B

T. Rybarczyk, B. Peaudecerf, M. Penasa, S. Gerlich, B. Julsgaard, K. Mølmer, S. Gleyzes, M. Brune, J. Rai- mond, S. Haroche, et al. , Forward-backward analysis of the photon-number evolution in a cavity, Physical Review A 91, 062116 (2015)

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