Reducing Circuit Depth in Quantum State Preparation for Quantum Simulation Using Measurements and Feedforward

Ha Eum Kim; Hyeonjun Yeo; Ilkwon Sohn; Kabgyun Jeong

REVIEW 5 major objections 6 minor 2 cited by

Reducing Circuit Depth in Quantum State Preparation for Quantum Simulation Using Measurements and Feedforward

T0 review · 5 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that adaptive circuits with unary encoding prepare sparse states, sums of Slater determinants, and Bethe wavefunctions in logarithmic or constant depth, at the price of more qubits.

desk verdict Clever depth-width tradeoffs for state preparation, but the headline Bethe width claim doesn't survive a close check of its own appendix. read the letter →

arxiv 2501.02929 v2 pith:7DMCS44Z submitted 2025-01-06 quant-ph

Hyeonjun Yeo , Ha Eum Kim , IlKwon Sohn , Kabgyun Jeong This is my paper · ORCID

classification quant-ph MSC 81P68 PACS 03.67.Lx

keywords quantumstatepreparationcircuitdepthreductionmeasurementsandfeedforwardunaryencodingBethewavefunctionSlaterdeterminantssparsesimulation

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

The paper argues that mid-circuit measurement and feedforward, which allow constant-depth implementations of quantum fan-out and related logic gates, can be converted into a depth-width trade-off for preparing quantum states that commonly seed quantum simulations. By using unary encoding as a reversible bridge between two superpositions over ordered integer tuples, it constructs an $\mathcal{O}(\log d)$-depth, $\mathcal{O}(d n \log n)$-width circuit for arbitrary $d$-sparse $n$-qubit states, and extends the same machinery to sums of Slater determinants in first quantization at depth $\mathcal{O}(d)$ or $\mathcal{O}(\log d)$. Its central result is Theorem 4: an $L$-site $M$-particle Bethe wavefunction with $M \le L/2$ can be prepared with a constant-depth adaptive circuit of width $\widetilde{\mathcal{O}}(L^2 \log L)$ and success probability $\mathcal{O}(1/M!)$. If correct, initial-state preparation for these state families would no longer be the depth bottleneck in quantum simulation, at the cost of substantially wider circuits and probabilistic success.

What carries the argument

The load-bearing objects are (i) the Clifford-ladder circuit, which realizes the quantum fan-out gate in constant depth through Bell pairs, mid-circuit measurements, and feedforward; (ii) the Uncompress and Compress operations, which reversibly map a binary integer to its unary (one-hot) representation and back in constant depth, so that a superposition over integer tuples can be re-routed through an addressable unary register; and (iii) the parallelization of a product of controlled pairwise-commuting gates by simultaneously diagonalizing them and using a unary-encoded control register. These routines are what let the paper trade width for depth: each previously sequential step is replaced by parallel fan-out-mediated operations acting on many replicated registers. The width bounds in Theorems 1 through 4 are dominated by these replications.

What would settle it

Compile the Compress subroutine for small sizes, e.g. $\eta = 2$ and $\log N = 2$, apply it to a superposition with known amplitudes, and check whether the binary register returns exactly to $|00\ldots0\rangle$ with the original amplitudes intact; any residual phase or amplitude error falsifies Lemma 3. For the Bethe claim, run Theorem 4's circuit for $L=4, M=2$ with the exact Dicke option, measure the frequency with which the postselection register reads $|00\ldots0\rangle$, and compare the output state's fidelity to $|\psi(\theta,\vec{k})\rangle$ against $1 - \mathcal{O}(1/2^L)$; a success probability far from $1/M!$ or a fidelity below that bound would falsify Theorem 4.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that two constant-depth primitives, the binary-unary-binary conversion and the parallelization of controlled pairwise-commuting phase gates, are enough to restructure the whole state-preparation pipeline for structured states. The first primitive reduces the problem of transforming a superposition over one ordered integer set into another to constant depth with width $\widetilde{\mathcal{O}}(\eta \log N)$, yielding the sparse-state and Slater-determinant results. The second primitive attaches the scattering phases $A_\sigma \exp(i \sum_\ell k_{\sigma(\ell)} x_\ell)$ of the Bethe wavefunction to a Dicke-state backbone in one parallel layer, after the inverse permutation has been encoded in unary; combining these with a probabilistic postselection gives the constant-depth Bethe preparation. The authors state these results as Theorems 1, 3, and 4, and give width bounds for each.

Load-bearing premise

The central bet is that every auxiliary register can be uncomputed again in constant depth without disturbing the target superposition, and that the Bethe postselection succeeds with the cited $\mathcal{O}(1/M!)$ probability, so the final state is left clean at the claimed depth.

Editorial extensions

If this is right

Any $d$-sparse $n$-qubit state can be initialized in depth $\mathcal{O}(\log d)$ with $\mathcal{O}(d n \log n)$ qubits, matching the best previous depth bound at a polynomial width cost.
Sums of Slater determinants in first quantization can be prepared in depth $\mathcal{O}(\log d)$ with width $\widetilde{\mathcal{O}}(\eta^2 d^2 \log N)$, or in depth $\mathcal{O}(d)$ with width $\widetilde{\mathcal{O}}(\eta^2(\eta \log \eta + \log N) + \log d)$.
Bethe wavefunctions with up to half-filling ($M \le L/2$) can be prepared in constant depth with width $\widetilde{\mathcal{O}}(L^2 \log L)$ and success probability $\mathcal{O}(1/M!)$, either exactly under $M = \mathcal{O}(\sqrt{L})$ or with infidelity $\mathcal{O}(1/2^L)$ in the general case.
Because integer-tuple superpositions can be rewired in constant depth, the unary bridge is a reusable subroutine that can be attached to any amplitude-loading circuit that produces a superposition over ordered indices.

Reading between the lines

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

The same binary-unary-binary bridge should apply to other index-symmetric state families in first quantization, such as permanents or fermionic antisymmetrized geminals, whenever the amplitudes factor over ordered tuples; the paper notes permanents as a byproduct but does not develop the consequences.
A direct computation of the Bethe postselection probability from the authors' own inverse-phase circuit, rather than a citation, would sharpen the runtime estimate; the paper's constant-depth claim is otherwise insensitive to the multiplicative success overhead.
The width bounds suggest a practical crossover: on devices where idle qubits are cheap and coherence is short, these adaptive circuits should outperform alternatives, while on qubit-starved hardware the $\widetilde{\mathcal{O}}(L^2 \log L)$ width for Bethe states will dominate the resource budget.
Extending the phase-attachment lemma to complex scattering matrices would require encoding more than pairwise ordering information per permutation; the paper mentions this as a future direction but leaves it open.

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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

Quick take: this paper has a genuinely useful core—the unary-encoding bridge plus constant-depth fan-out tools gives new depth-width tradeoffs for sparse states and Slater determinant sums. The sparse-state result (Thm 1) looks solid, and the two SOS circuits in Thm 3 are plausible and worth taking seriously. But the Bethe wavefunction claim (Thm 4) has a width-accounting problem that the authors' own appendix exposes.

What's new: the parallelized constant-depth unary transformation (Lemmas 2,3) is a real contribution, even though the bridge itself was in Ref [47]. Combining it with the controlled-pairwise-commuting-gate trick from Ref [18] to get O(log d) sparse prep and O(log d) or O(d) SOS prep is a decent step forward. The paper is clearly written and properly credits the primitives it imports. No fitted parameters, no hand-waving about resources of external tools—the constructions are explicit enough to check.

Soft spots: the Bethe width. Lemma 7's proof (Appendix D) states the first summand costs O(M^3 log M). For M=O(√L) that's fine, but the theorem claims O~(L^2 log L) without the constraint (using the approximate Dicke route of Ref [13]). For M≈L/2, M^3 log M = Θ(L^3 log L), which dominates the stated width. So the claimed width bound in Theorem 4 is not derived; the construction as written gives O(M^3 log M + L^2 log L). That needs a fix, either by restricting M or by changing the bound. Second, the uncomputation steps in Appendix C and the postselection probability in Appendix H are asserted, not proven. Those are load-bearing for the depth claims. Minor typos and notation slips don't affect the math, but the paper would benefit from a careful pass.

The reader's CONDITIONAL verdict is on the mark. The core ideas are sound and new enough to deserve referee time, but the Bethe width inconsistency must be addressed before this can be accepted.

Recommendation: send it to peer review, but with a clear request for the authors to supply a corrected width analysis for Theorem 4 and explicit circuit decompositions for the uncomputation steps. My guess is the sparse and SOS results survive; the Bethe result may need to be scaled back.

Referee Report

5 major / 6 minor

Summary. The paper proposes constant-depth quantum circuits with mid-circuit measurements and feedforward for preparing several quantum states relevant to quantum simulation: sparse quantum states (Theorem 1), single (anti)symmetric states (Theorem 2), sums of Slater determinants in first quantization (Theorem 3), and Bethe wavefunctions (Theorem 4). The main technical tool is a constant-depth transformation between binary and unary encodings (Uncompress and Compress, Lemmas 2 and 3), combined with known constant-depth primitives such as the quantum fan-out gate, Equal, GreaterThan, and Dicke-state preparation. For Bethe wavefunctions, the authors combine a prepared Dicke state with permutation registers and controlled pairwise commuting phase gates to encode the phases A_σ(θ) and exp(i k·x), claiming constant depth, width O~(L^2 log L), and success probability O(1/M!).

Significance. If the constructions are correct, the paper would demonstrate a significant depth reduction for state preparation tasks that previously required depth O(log d) or O(L), at the cost of increased width. The unary-encoding transformation between integer sets is clearly presented and makes use of established constant-depth feedforward primitives in a coherent way, which is a genuine strength. The claimed constant-depth Bethe wavefunction preparation, together with explicit resource counts, is a falsifiable and potentially useful contribution to early fault-tolerant quantum simulation. However, the significance is conditional on resolving several load-bearing proof gaps, most notably a width accounting inconsistency in the Bethe theorem that currently invalidates the claimed O~(L^2 log L) bound in the worst case.

major comments (5)

[Appendix D / Lemma 7 / Theorem 4] The width bound in Theorem 4 is not derived from the provided construction. In Appendix D, the proof of Lemma 7 states that 'the first part of the desired quantum state can be prepared using a constant-depth quantum circuit of width O(M^3 log M)', which follows from Lemma 5 with η = M and ζ = M^2. For the allowed range M ≤ L/2, taking M ≈ L/2 gives Θ(L^3 log L), which asymptotically dominates the claimed O~(L^2 log L) width of Theorem 4. The later Cleaning step, also quoted as O~(L^2 log L), cannot reduce the cost of the first summand. The theorem's stated width is therefore not supported by the proof; as written, the construction supports only O(M^3 log M + L^2 log L).
[Appendix C (both versions of Theorem 3)] Both SOS constructions end with an asserted uncomputation: the first version concludes 'Finally, applying the inverse operations of Compress and Uncompress ensures that only the information of r0 remains', and the second version ends with 'we obtain the desired state after uncomputing all redundant states'. The uncomputation of the coefficient register and of the intermediate index registers is not shown. If these inverse operations cannot be implemented in constant depth (or within the stated width), the O(d) and O(log d) depth claims of Theorem 3 fail. The proof should either provide an explicit circuit for the final uncomputation or invoke a lemma that guarantees the required disentangling within the claimed resources.
[Appendix H / Theorem 4] The success probability in Theorem 4 is stated as O(1/M!), but the proof does not derive this for the authors' own circuit. Step 3 asserts that after applying the inverse phase-encoding gates, the probability of measuring |00...0⟩ is 'known to be O(1/M!) [40]'. Reference [40] derives this for a different Bethe-state preparation scheme, not for the circuit constructed here, which includes additional registers and phase gates (Lemmas 8–10 and the Cleaning step). Since the success probability is a central part of the theorem's claim, the authors should either compute the postselection amplitude for their specific circuit or prove that the cited bound carries over to their construction.
[Appendix E / Lemma 8] The proof of Lemma 8 contains a circular reference: it states that the phase-attaching gates can be parallelized 'Using Theorem 4', but Lemma 8 is itself used in the proof of Theorem 4. This is not a valid proof step. The intended meaning is presumably that the parallelization follows from the quantum fan-out technique of Refs. [17,18], but the proof of Lemma 8 must be self-contained or explicitly reference the appropriate prior lemma rather than the theorem being proved.
[Theorem 4 statement] The first sentence of Theorem 4 claims that the Bethe wavefunction for M ≤ L/2 can be prepared in constant depth with width O~(L^2 log L). The subsequent sentence, however, splits the result into two cases: exact preparation restricted to M = O(√L), or approximate preparation with infidelity O(1/2^{LL}) and no restriction on M. The theorem statement should reflect this dichotomy explicitly, since as written it suggests exact constant-depth preparation for all M ≤ L/2, which is not established.

minor comments (6)

[Theorem 1 proof] The resource count transitions from O(d log^2 n log log^2 n) to O(d n log n); the equality is not justified, and the two expressions are not asymptotically equal (the first is smaller for large n). The bound is true in the sense of being an upper bound, but the presentation is misleading.
[Eq. (13)] In the definition of the Bethe wavefunction, the sum over σ is written as S_η, but the number of particles is denoted M; it should be S_M. The same inconsistency appears in the surrounding discussion.
[Lemma 3] Equation (4) omits the summation on the right-hand side; it should read Σ_i α_i |0⟩_{log N} |e_i⟩_η.
[Appendix D] The 'Cleaning operation' from Ref. [17] is invoked without definition; its action and width scaling should be specified so that the reader can verify the claimed O~(L^2 log L) contribution.
[Throughout] There are several typographical issues, including 'Greatherthan' in Table II and Appendix A, 'Preparaing' in the heading of Section IV.B, 'spririt' in Appendix H, and inconsistent capitalization of 'Uncompress'/'uncompress'.
[Figure 1] The labels in Figure 1b are partially garbled (e.g., '𝑟𝑖log𝑁𝑟 0𝜂 0log𝑁𝑝'), making the figure difficult to interpret.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constructions compose external constant-depth primitives and explicit input data; the flagged width issue is a correctness concern, not a circular reduction.

full rationale

No circularity found. The paper's derivation chain is modular and relies on externally established constant-depth adaptive primitives: Clifford-ladder/fan-out/Equal/GreaterThan/OR from Refs. [17,18], Dicke-state preparation from Refs. [17,13], and the O(1/M!) success probability from Ref. [40]. None of these references are authored by the present authors, and none are instantiated with parameters fitted to the paper's own target states. The claimed reductions are explicit compositions (Uncompress/Compress, controlled-pairwise-commuting-gate parallelization, parity via GreaterThan, inverse-permutation encoding via fan-out) whose inputs are classical data (r_i, q_i, theta, k) and external state-preparation subroutines, not the target outputs. The circuit constructions are given step by step in the appendices. Two concerns that might look like circularity are actually not: (i) the Bethe success probability O(1/M!) is cited to prior independent work rather than derived, but this is an imported external result, not an assumption of the theorem; (ii) the width bound in Lemma 7/Appendix D appears internally inconsistent (first-summand width O(M^3 log M) versus the final O~(L^2 log L) when M is roughly L/2), but this is a proof-consistency or correctness issue, not a circular reduction from the theorem to its own inputs. There is no fitted-input-called-prediction, no load-bearing self-citation, and no renaming of a known result as a new one. The central claims therefore have independent content and should be evaluated on correctness, not circularity.

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

The construction introduces no fitted parameters and no new physical entities. It imports constant-depth primitives (fan-out, Equal, GreaterThan, Dicke preparation) from prior work. The main hidden inputs are the known classical data defining each target state and the external Dicke-preparation results, plus the assumptions about real Bethe parameters.

assumptions (5)

domain assumption The Clifford-ladder circuit and quantum fan-out gate can be implemented in constant depth with O(n) width using measurements and feedforward (Lemma 1, from Ref [17]).
Invoked in Lemmas 2 and 3, Theorem 1, and throughout the paper. If this external primitive fails, all depth claims collapse.
domain assumption Mid-circuit measurement and feedforward, plus classical computation of depth O(log n), are available and are treated as ideal devices.
Section II A and Section III A define the hybrid circuit model the paper assumes throughout.
domain assumption The scattering matrix theta and quasimomenta k are real, are obtained by solving Bethe equations, and satisfy theta_{i,j} = -theta_{j,i}.
Section V A and Lemma 10 rely on these properties to encode the Bethe phases using GreaterThan gates.
domain assumption Exact Dicke state preparation from Ref [17] succeeds deterministically under M <= sqrt(L), and the approximate Dicke circuit from Ref [13] can reach very small infidelity within the allowed width.
Lemma 7 and Theorem 4 inherit these external constructions and their constraints.
standard math A set of pairwise commuting unitaries can be simultaneously diagonalized, and this diagonalization can be realized by a unitary T with the stated depth.
Lemma 4 is a direct restatement of Theorem 3.2 in Ref [18], which the paper uses to parallelize controlled phase gates.

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Pith. "Pith review of Reducing Circuit Depth in Quantum State Preparation for Quantum Simulation Using Measurements and Feedforward." pith.science (2026). https://pith.science/paper/7DMCS44Z

@misc{pith2026250102929,
  author       = {Pith},
  title        = {Pith review of: Reducing Circuit Depth in Quantum State Preparation for Quantum Simulation Using Measurements and Feedforward},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DMCS44Z}},
  note         = {Machine review of arXiv:2501.02929}
}

read the original abstract

Reducing circuit depth and identifying an optimal trade-off between circuit depth and width is crucial for successful quantum computation. In this context, midcircuit measurement and feedforward have been shown to significantly reduce the depth of quantum circuits, particularly in implementing logical gates. By leveraging these techniques, we propose several parallelization strategies that reduce quantum circuit depth at the expense of increasing width in preparing various quantum states relevant to quantum simulation. With measurements and feedforward, we demonstrate that utilizing unary encoding as a bridge between two quantum states substantially reduces the circuit depth required for preparing quantum states, such as sparse quantum states and sums of Slater determinants within the first quantization framework, while maintaining an efficient circuit width. Additionally, we show that a Bethe wave function, characterized by its high degree of freedom in its phase, can be probabilistically prepared in a constant-depth quantum circuit using measurements and feedforward. We anticipate that our study will contribute to the reduction of circuit depth in initial state preparation, particularly for quantum simulation, which is a critical step toward achieving quantum advantage.

Figures

Figures reproduced from arXiv: 2501.02929 by the authors.

**Figure 2.** FIG. 2. (a) A series of multi-controlled pairwise commuting [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Quantum circuit for encoding inverse of permutation [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

119 extracted references · 47 canonical work pages · cited by 2 Pith papers

[47]

G. Wang, D. S. Fran¸ ca, R. Zhang, S. Zhu, and P. D. Johnson, Quantum algorithm for ground state energy es- timation using circuit depth with exponentially improved dependence on precision, Quantum7, 1167 (2023)

2023
[18]

|σ(M−1)⟩|σ−1(0)⟩

Using Lemma 9 to encode the inverse of permuta- tion into a quantum state X σ∈SM |σ(0)⟩. . .|σ(M−1)⟩|σ−1(0)⟩. . .|σ−1(M−1)⟩, which can be implemented in a constant-depth quantum circuit of width ˜O(M 2 logM) using mea- surements and feedforward. 20
[13]

For simplicity, let us focus on the (j+ 1)-th pair of quantum states|e i⟩⊗η|σ(j)⟩ ⊗ηd from Eq. (C1). 16 Now, we represent this state as follows, |e0 i ⟩⊗η|σ(j)⟩ ⊗η . . .|ed−1 i ⟩⊗η|σ(j)⟩ ⊗η,(C2) where the state|e l i⟩refers to the (l+ 1)-th qubit of |ei⟩, which is equal to|1 i=l⟩. For the (l+ 1)-th pair of quantum states|e l i⟩⊗η|σ(j)⟩ ⊗η in Eq. (C2), we ...
[40]

G. H. Low, V. Kliuchnikov, and L. Schaeffer, Trading T gates for dirty qubits in state preparation and unitary synthesis, Quantum8, 1375 (2024)

2024
[1]

This can be im- plemented using a constant-depth quantum circuit of widthO(η 2 logζlog logζ)

For each|k σ(i)⟩, apply theUncompressoperation using prior knowledge of{k i}η−1 i=0 . This can be im- plemented using a constant-depth quantum circuit of widthO(η 2 logζlog logζ). The resulting state is X σ∈Sη |kσ(0)⟩logζ |eσ(0)⟩η . . .|kσ(η−1) ⟩logζ |eσ(η−1) ⟩η
[2]

This step is achieved via a constant-depth quantum circuit of width O(η2 logζ), which results as X σ∈Sη |0⟩logζ |eσ(0)⟩η

Remove the information of{k i}η−1 i=0 using the Compressoperation. This step is achieved via a constant-depth quantum circuit of width O(η2 logζ), which results as X σ∈Sη |0⟩logζ |eσ(0)⟩η . . .|0⟩logζ |eσ(η−1) ⟩η
[3]

[25], followed by antisymmetriza- tion

The circuit width of this approach is ˜O(ηdlogNlogη) and ours is ˜O(η2d2 logN).For the case whered= poly(N), a sparse quantum state can be prepared using the algorithm in Ref. [25], followed by antisymmetriza- tion. Although the result cannot be directly compared due to differences in measurement metrics, we emphasize that our Theorem 3 provides significa...

1931
[4]

This step re- quires a constant-depth quantum circuit of width O(η2 logNlog logN)

Using the prior knowledge of{r i}η−1 i=0 , ap- ply the inverse ofCompress. This step re- quires a constant-depth quantum circuit of width O(η2 logNlog logN). The resulting state is X σ∈Sη |rσ(0)⟩logN |eσ(0)⟩η . . .|rσ(η−1) ⟩logN |eσ(η−1) ⟩η

Show all 119 references

[5]

The resulting state is X σ∈Sη |rσ(0)⟩logN |0⟩η

Apply the inverse ofUncompressto erase informa- tion in the unary registers, by a constant-depth quantum circuit of widthO(η 2 logNlog logN). The resulting state is X σ∈Sη |rσ(0)⟩logN |0⟩η . . .|rσ(η−1) ⟩logN |0⟩η. Given the conditionζ≥η 2, we can setζ=η 2 without loss of gene...
[6]

Furthermore, we will refer to the qubits|i⟩in the state Pd−1 i=0 ψi|i⟩logd as the coefficient register

A quantum circuit of width logarithmic ind Although the following process will be iterated fori= 0 tod−1, this proof focuses on the case ofi= 0, as the procedure for the other terms is analogous. Furthermore, we will refer to the qubits|i⟩in the state Pd−1 i=0 ψi|i⟩logd as the...
[7]

|σ(η− 1)⟩in order to encode⃗ r0,

Our objective is to entangle each|i⟩ logd in the coef- ficient register Pd−1 i=0 ψi|i⟩logd , with|σ(0)⟩. . .|σ(η− 1)⟩in order to encode⃗ r0, . . . , ⃗ rd−1. To achieve this, we first apply theEqual 0 gate, where the coeffi- cient register serves as the control, to generate the...
[8]

Apply theUncompressoperation to each quantum state|1 i=0⟩1|σ(j)⟩for 0≤j≤η−1. Instead of us- ing the originalEqual η gate as described in Lemma 2, we employ theEqual η+m gate to solely modify the states associated with|i= 0⟩ logd in the su- perposition Pd−1 i=0 ψi|i⟩logd . Cons...
[9]

To do so, we first apply a quantum fan-out gate to uncompute|1 i=0⟩⊗η back to|1 i=0⟩

Next, we need to delete the information encoded in |i= 0⟩ logd and|1 i=0⟩, leaving only the antisymmet- ric state. To do so, we first apply a quantum fan-out gate to uncompute|1 i=0⟩⊗η back to|1 i=0⟩. Subse- quently, we apply quantum fan-out gates, where the controlled qubit i...
[10]

To encode⃗ r0 using the quantum state|e σ(j) ⟩η, we applyCompressop- eration to each state|σ(j)⟩for 0≤j≤η−1

Henceforth, we dismiss the qubit|0⟩ 1 that was pre- viously used to encode|1 i=0⟩. To encode⃗ r0 using the quantum state|e σ(j) ⟩η, we applyCompressop- eration to each state|σ(j)⟩for 0≤j≤η−1. This transforms the state as follows, X σ∈Sη h ψ0|0⟩logd |0⟩logη |eσ(0)⟩η . . .|0⟩log...
[11]

To achieve this, we transform the state Pd−1 i=0 ψi|i⟩logd as Pd−1 i=0 ψi|ei⟩d, which requires ˜O(dlogd) additional qubits

With the quantum circuit width polynomial ind If we aim to utilize a quantum circuit width ofO(d), it becomes possible to access the unary encoding ofPd−1 i=0 |ei⟩d to reduce the circuit depth. To achieve this, we transform the state Pd−1 i=0 ψi|i⟩logd as Pd−1 i=0 ψi|ei⟩d, whi...
[12]

|σ(η−1)⟩

Similar to the previous case, we need to en- tangle the coefficient register with the state |σ(0)⟩. . .|σ(η−1)⟩. To avoid a circuit depth of O(d), we first copy the coefficient registerη 2 times, transforming the state as d−1X i=0 ψi|ei⟩⊗η d |σ(0)⟩logη . . .|ei⟩⊗η d |σ(η−1)⟩ l...
[14]

Specifically, we operateORgates, where the con- trolled qubits are the qubits in thep-th unary reg- ister, and the target qubit is thep-th qubit of the coefficient register

To disentangle the coefficient register from the other states, we utilize the location of|e σ(j) ⟩η. Specifically, we operateORgates, where the con- trolled qubits are the qubits in thep-th unary reg- ister, and the target qubit is thep-th qubit of the coefficient register. Th...
[15]

Since differentψ i share the same state|σ(j)⟩, we canCompress|σ(j)⟩collectively. To achieve this, we attach additionalηqubits|0⟩ ⊗η 1 and apply ORgates, where the controlled qubits are theq- th qubits of each unary registers, and the target qubit is theq-th qubit of the newly ...
[16]

, ⃗ rd−1

Now, we need to encode the information of ⃗ r0, . . . , ⃗ rd−1. To do so, we operate the inverse of Compressoperations in parallel. First, we attach |0⟩logN to all unary registers and apply Hadamard 17 gates to these newly added qubits, resulting as ψ0 η−1O j=0 N−1X l0 j ,...,...
[17]

This can be achieved by extracting the information of the permutationσ from the binary registers, as all binary registers share the same permutation

Now, we need to disentangle the unary registers from the binary registers. This can be achieved by extracting the information of the permutationσ from the binary registers, as all binary registers share the same permutation. Specifically, we can applyUncompressoperation 1) fro...
[19]

By using quantum fan-out gates, we copy|σ −1(i)⟩for alli to obtain X σ∈SM |σ−1(0)⟩⊗M−1

From now on, we neglect the state|σ(j)⟩, as it is not used in the remainder of our proof. By using quantum fan-out gates, we copy|σ −1(i)⟩for alli to obtain X σ∈SM |σ−1(0)⟩⊗M−1 . . .|σ−1(M−1)⟩ ⊗M−1 , where the quantum circuit width ofO(M 2 logM) is required for this process
[20]

The resulting state is repre- sented as X σ∈SM |σ−1(0)⟩⊗M−1 |1σ−1(0)<σ−1(1)⟩

ApplyGreatherthangates to all possible pairs of |σ−1(i)⟩and|σ −1(j)⟩. The resulting state is repre- sented as X σ∈SM |σ−1(0)⟩⊗M−1 |1σ−1(0)<σ−1(1)⟩. . .|1σ−1(0)<σ−1(M−1) ⟩ ⊗ |σ−1(1)⟩⊗M−1 |1σ−1(1)<σ−1(2)⟩. . .|1σ−1(1)<σ−1(M−1) ⟩ ⊗. . . ⊗ |σ−1(M−2)⟩ ⊗M−1 |1σ−1(M−2)<σ −1(M−1) ⟩ ⊗ ...
[21]

This is implemented simultaneously for all 0≤ i < j≤M−1

Let us define a phase gateP θ(i, j) as Pθ(i, j) = exp(iθj,i/2) 0 0 exp(iθ i,j/2) ,(G1) By operating phase gates on the targets of the Greatherthangates, which are|1 σ−1(i)<σ−1(j)⟩, we attach the appropriate phase exp( i 2 θi,j) or exp( i 2 θj,i), irrespective of the specific p...
[22]

|jσ(M−1) ⟩|σ(0)⟩

By using Lemma 7, we obtain the state X ⃗j∈ΞM 2 M X σ∈SM |jσ(0)⟩. . .|jσ(M−1) ⟩|σ(0)⟩. . .|σ(M−1)⟩| M−1M i=0 eji ⟩ ⊗ X ⃗ x∈ΞL M |x0⟩. . .|xM−1 ⟩| M−1M k=0 exk ⟩ by a constant-depth quantum circuit of width ˜O(L2 logL). While preparing this state, let us as- sume that we use th...
[23]

|xM−1 ⟩using a quantum circuit of width ˜O(M LlogL), we attach the phases Aσ exp(i PM−1 i=0 kσ(i)xi) by Lemmas 8 and 10

After performingUncompressoperation on |x0⟩. . .|xM−1 ⟩using a quantum circuit of width ˜O(M LlogL), we attach the phases Aσ exp(i PM−1 i=0 kσ(i)xi) by Lemmas 8 and 10. This phase attachment can be implemented by a quantum circuit of width ˜O(M 2(L+ log 2 M)). By applying the ...
[24]

|xM−1 ⟩, 21 only retaining the complete Bethe wavefunction

Finally, we need to remove the information about the permutationσand the state|x 0⟩. . .|xM−1 ⟩, 21 only retaining the complete Bethe wavefunction. The latter can be removed by performingClean- ingoperation described in [17], which requires a quantum circuit of width ˜O(L2 log...
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Reviewed August 10, 2026 · model on record in the stance chip above.

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|σ(M−1)⟩|σ−1(0)⟩

Using Lemma 9 to encode the inverse of permuta- tion into a quantum state X σ∈SM |σ(0)⟩. . .|σ(M−1)⟩|σ−1(0)⟩. . .|σ−1(M−1)⟩, which can be implemented in a constant-depth quantum circuit of width ˜O(M 2 logM) using mea- surements and feedforward. 20

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For simplicity, let us focus on the (j+ 1)-th pair of quantum states|e i⟩⊗η|σ(j)⟩ ⊗ηd from Eq. (C1). 16 Now, we represent this state as follows, |e0 i ⟩⊗η|σ(j)⟩ ⊗η . . .|ed−1 i ⟩⊗η|σ(j)⟩ ⊗η,(C2) where the state|e l i⟩refers to the (l+ 1)-th qubit of |ei⟩, which is equal to|1 i=l⟩. For the (l+ 1)-th pair of quantum states|e l i⟩⊗η|σ(j)⟩ ⊗η in Eq. (C2), we ...

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This can be im- plemented using a constant-depth quantum circuit of widthO(η 2 logζlog logζ)

For each|k σ(i)⟩, apply theUncompressoperation using prior knowledge of{k i}η−1 i=0 . This can be im- plemented using a constant-depth quantum circuit of widthO(η 2 logζlog logζ). The resulting state is X σ∈Sη |kσ(0)⟩logζ |eσ(0)⟩η . . .|kσ(η−1) ⟩logζ |eσ(η−1) ⟩η

[6] [2]

This step is achieved via a constant-depth quantum circuit of width O(η2 logζ), which results as X σ∈Sη |0⟩logζ |eσ(0)⟩η

Remove the information of{k i}η−1 i=0 using the Compressoperation. This step is achieved via a constant-depth quantum circuit of width O(η2 logζ), which results as X σ∈Sη |0⟩logζ |eσ(0)⟩η . . .|0⟩logζ |eσ(η−1) ⟩η

[7] [3]

[25], followed by antisymmetriza- tion

The circuit width of this approach is ˜O(ηdlogNlogη) and ours is ˜O(η2d2 logN).For the case whered= poly(N), a sparse quantum state can be prepared using the algorithm in Ref. [25], followed by antisymmetriza- tion. Although the result cannot be directly compared due to differences in measurement metrics, we emphasize that our Theorem 3 provides significa...

1931

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This step re- quires a constant-depth quantum circuit of width O(η2 logNlog logN)

Using the prior knowledge of{r i}η−1 i=0 , ap- ply the inverse ofCompress. This step re- quires a constant-depth quantum circuit of width O(η2 logNlog logN). The resulting state is X σ∈Sη |rσ(0)⟩logN |eσ(0)⟩η . . .|rσ(η−1) ⟩logN |eσ(η−1) ⟩η