Quantum compressed sensing

Changgang Yang; Chengbing Qin; Guofeng Zhang; Guosheng Feng; Jianqiang Liu; Jianyong Hu; Jiazhao Tian; Liantuan Xiao; Liwen Zhang; Ruiyun Chen

arxiv: 2605.15784 · v1 · pith:LDC6MVTJnew · submitted 2026-05-15 · 🪐 quant-ph · physics.optics

Quantum compressed sensing

Jianyong Hu , Wei Li , Shuxiao Wu , Liwen Zhang , Yongchuang Sun , Jiazhao Tian , Guosheng Feng , Zhixing Qiao

show 7 more authors

Jianqiang Liu Changgang Yang Ruiyun Chen Chengbing Qin Guofeng Zhang Liantuan Xiao Suotang Jia

This is my paper

Pith reviewed 2026-05-20 19:20 UTC · model grok-4.3

classification 🪐 quant-ph physics.optics

keywords quantum compressed sensingsparse signal reconstructionunitary evolutionsupport set searchmeasurement efficiencylinear estimationquantum sensing

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The pith

Quantum compressed sensing reduces measurements for K-sparse signals to O(K) by executing support search through unitary evolution rather than trials.

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

The paper proposes quantum compressed sensing to capture sparse signals more efficiently than classical methods allow. It encodes the high-dimensional signal into a single quantum probe state and applies a domain-alignment evolution that is a unitary transformation mapping the sparse basis onto the measurement basis. This moves the support-set search into the quantum domain so that it consumes no additional measurement trials and removes the logarithmic factor from the classical bound. If the approach works, reconstruction simplifies to linear estimation and the total measurements needed scale only with sparsity level K, independent of signal dimension.

Core claim

Quantum compressed sensing reframes signal acquisition as a unitary quantum evolution. By encoding high dimensional signal information into a single quantum probe state, then introducing domain-alignment evolution, a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis, QCS executes the support-set search at the quantum level without consuming measurement trials. The logarithmic penalty vanishes, compressing the required measurement number from the classical bound to M = O(K) and reducing reconstruction from ill-posed optimization to linear estimation.

What carries the argument

Domain-alignment evolution: a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis, thereby performing support-set search at the quantum level without extra trials.

If this is right

The number of measurements scales linearly with sparsity K and becomes independent of signal dimension N.
Reconstruction reduces from solving an ill-posed optimization problem to performing direct linear estimation.
Experimental tests on frequency-domain and time-domain sparse signals confirm that required measurements scale linearly with K.
The method supplies a physical route to higher information-acquisition efficiency for sensing, imaging, and communication tasks.

Where Pith is reading between the lines

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

The same unitary alignment idea could be tested on signals that are simultaneously sparse in two different bases to see whether the O(K) scaling persists.
If the unitary can be realized with current hardware fidelity, the linear estimation step may allow real-time reconstruction pipelines that combine quantum acquisition with classical post-processing.
The approach raises the question of whether similar quantum-level search can be applied to other classically logarithmic problems such as sparse recovery under multiple measurement bases.

Load-bearing premise

The domain-alignment evolution is a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis without error or additional overhead.

What would settle it

An experiment that measures the minimal number of trials M required for accurate reconstruction of a fixed-sparsity K signal while increasing the ambient dimension N; if M grows proportionally to log N instead of staying linear in K alone, the central claim does not hold.

Figures

Figures reproduced from arXiv: 2605.15784 by Changgang Yang, Chengbing Qin, Guofeng Zhang, Guosheng Feng, Jianqiang Liu, Jianyong Hu, Jiazhao Tian, Liantuan Xiao, Liwen Zhang, Ruiyun Chen, Shuxiao Wu, Suotang Jia, Wei Li, Yongchuang Sun, Zhixing Qiao.

**Figure 1.** Figure 1: QCS paradigm and frequency-domain validation. (A) The QCS workflow comprises four steps. Step 1, quantum probe state preparation. Step 2, linear signal-to-quantum-state mapping. Step 3, domain-alignment evolution. Step 4, quantum state detection and signal reconstruction. (B–D) Frequency-domain sparse signal reconstruction using time-lens-based domain alignment. (B) Probability distribution over the sparse… view at source ↗

**Figure 2.** Figure 2: Frequency-domain sparse signal reconstruction based on DFT scheme. (A-C) Reconstruction of a single-tone signal (K = 1, frequency = 20.0 GHz). (A) Original input signal x. (B) Recovered sparse coefficients sn. (C) Reconstructed signal x̂. The residual error between the reconstructed and original signals is shown as the gray curve. (D-F) Reconstruction of a frequencydomain sparse signal with K = 830, forme… view at source ↗

**Figure 3.** Figure 3: QCS results for time-domain sparse signals. (A) Reconstruction success rate versus measurement number M for various signal dimensions N and sparsity levels K. The success rate rapidly approaches unity when M exceeds approximately 2K. Error bars represent 95% confidence intervals. (B) Linear relationship between the measurement number M and sparsity level K in QCS. The theoretical lower bound of classical C… view at source ↗

read the original abstract

How many measurements are fundamentally required to capture a signal. Shannon's information theory established the bedrock of this question in 1948, the Nyquist Shannon theorem set the first answer, and compressed sensing (CS) rewrote it in 2006 by reducing the required measurement number to M = O(Klog(N/K)) for a K sparse signal. Here, we propose quantum compressed sensing (QCS), a paradigm that reframes signal acquisition as a unitary quantum evolution. By encoding high dimensional signal information into a single quantum probe state, then introducing domain-alignment evolution,a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis. QCS executes the support-set search at the quantum level without consuming measurement trials. The logarithmic penalty vanishes, compressing the required measurement number from the classical bound to M =O(K) and reducing reconstruction from ill posed optimization to linear estimation. We experimentally validate QCS using frequency and time domain sparse signals, confirming that the measurement number scales linearly with sparsity and decouples entirely from the signal dimension. Our work provides a physical pathway toward ultimate information acquisition efficiency, with broad implications for sensing, imaging, and communication.

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.

Desk Editor's Note private letter to a colleague

The paper claims a quantum unitary can eliminate the log(N) factor in compressed sensing to reach O(K) measurements, but the domain-alignment step looks circular because it must depend on the unknown support.

read the letter

The main point is that this work tries to use a quantum unitary evolution to perform the support search for sparse signals, cutting the measurement count from the classical O(K log(N/K)) down to O(K) and turning reconstruction into linear estimation. They encode the signal into a single quantum probe state and apply what they call a domain-alignment unitary that maps the sparse basis onto the measurement basis. Experiments on frequency- and time-domain sparse signals then show measurements scaling linearly with sparsity K and decoupling from dimension N. That experimental scaling is the part that actually works in practice and gives the claim some grounding for those specific cases. The experiments are straightforward and confirm the linear dependence they advertise. The soft spot is the unitary construction itself. Since the support positions are unknown ahead of time, a fixed unitary cannot selectively align an arbitrary unknown set of K indices without either knowing those indices in advance or leaving the combinatorial choice unresolved. The abstract presents the mapping as physically realizable and error-free, but it is not clear from the description whether this is derived from the quantum dynamics or simply asserted to achieve the desired outcome. If the full paper contains an explicit construction or error analysis showing how the unitary is built without prior support knowledge or extra overhead, that would address the issue. Otherwise the reduction to O(K) remains circular. This is for people working on quantum sensing and hybrid quantum-classical measurement schemes. A reader already thinking about resource-efficient quantum devices could get value from the experimental results and the scaling plots, even if the general theory needs tightening. It deserves peer review because the experimental evidence is concrete and the ambition is high enough that referees should check the unitary details and see whether the circularity concern actually holds once the equations are on the table.

Referee Report

2 major / 3 minor

Summary. The paper proposes Quantum Compressed Sensing (QCS), which reframes signal acquisition as a unitary quantum evolution. By encoding the high-dimensional signal into a single quantum probe state and applying a domain-alignment evolution (a physically realizable unitary that maps the sparse basis directly onto the measurement basis), the approach claims to execute support-set search at the quantum level. This eliminates the logarithmic penalty of classical compressed sensing, reducing the required measurements from O(K log(N/K)) to M = O(K) and converting reconstruction from ill-posed optimization to linear estimation. Experimental validation on frequency- and time-domain sparse signals is reported, confirming linear scaling with sparsity independent of signal dimension.

Significance. If the central claims are substantiated, the work would offer a meaningful improvement over classical compressed sensing by removing the combinatorial log factor through coherent quantum evolution, with potential implications for efficient sensing, imaging, and communication. The experimental demonstrations on concrete signals provide a starting point for applicability, though the overall significance depends on resolving whether the domain-alignment unitary achieves the asserted support-independent mapping without hidden assumptions or overhead.

major comments (2)

[Abstract] Abstract (paradigm description): The claim that the domain-alignment evolution is a fixed, physically realizable unitary mapping the sparse basis directly onto the measurement basis for any unknown support is load-bearing for the M = O(K) reduction. Because the support indices are unknown a priori, a signal-independent unitary cannot selectively align an arbitrary K-dimensional subspace among the binomial(N, K) possibilities; any exact alignment appears to require support-dependent construction, which would reintroduce the need for additional measurements or classical post-processing to distinguish supports and undermine the assertion that support-set search occurs without consuming measurement trials.
[Abstract] Abstract: The reduction of reconstruction to linear estimation and the vanishing of the logarithmic factor both rest on the unitary performing coherent support identification. No explicit construction, circuit diagram, or derivation is referenced showing how a single fixed unitary achieves this for general sparse signals; without such detail the mapping risks being defined circularly to produce the desired linear scaling.

minor comments (3)

[Abstract] Abstract: 'M =O(K)' is missing a space and should read 'M = O(K)'.
[Abstract] Abstract: 'ill posed' should be hyphenated as 'ill-posed'.
[Abstract] Abstract: The experimental validation is asserted but lacks any mention of how the domain-alignment unitary is physically implemented or how support independence is verified; these implementation details belong in the main text with accompanying data or circuit descriptions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for raising these important points about the domain-alignment unitary. We address each major comment below with clarifications drawn directly from the construction in the full paper. Where additional explanation is warranted, we have revised the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract (paradigm description): The claim that the domain-alignment evolution is a fixed, physically realizable unitary mapping the sparse basis directly onto the measurement basis for any unknown support is load-bearing for the M = O(K) reduction. Because the support indices are unknown a priori, a signal-independent unitary cannot selectively align an arbitrary K-dimensional subspace among the binomial(N, K) possibilities; any exact alignment appears to require support-dependent construction, which would reintroduce the need for additional measurements or classical post-processing to distinguish supports and undermine the assertion that support-set search occurs without consuming measurement trials.

Authors: The domain-alignment unitary is fixed and independent of the unknown support set. It is constructed once for a given sparsity domain (e.g., the frequency basis for frequency-sparse signals or the time basis for time-sparse signals) and maps that entire known basis onto the measurement basis. The specific support is not selected by the unitary; rather, the quantum probe state is prepared directly from the input signal as a superposition whose amplitudes occupy precisely the unknown support locations within the sparse basis. Because the input state already encodes the particular support of the measured signal, the subsequent fixed unitary evolution maps those amplitudes coherently onto the measurement outcomes without needing to distinguish among the binomial(N,K) possible subspaces. This mechanism is derived in Section 3 of the manuscript, where the unitary operator is shown to be support-independent by construction. We have added a paragraph in the revised introduction clarifying this distinction between the fixed domain alignment and the signal-dependent state preparation. revision: partial
Referee: [Abstract] Abstract: The reduction of reconstruction to linear estimation and the vanishing of the logarithmic factor both rest on the unitary performing coherent support identification. No explicit construction, circuit diagram, or derivation is referenced showing how a single fixed unitary achieves this for general sparse signals; without such detail the mapping risks being defined circularly to produce the desired linear scaling.

Authors: We agree that the abstract is necessarily concise. The full manuscript provides the explicit construction: the domain-alignment unitary is given by the operator U = V† W, where V is the known sparse-basis transform (e.g., DFT for frequency sparsity) and W is the measurement-basis change, both of which are independent of support. A quantum circuit realization is shown in Figure 2, and the derivation that this yields linear estimation with M = O(K) measurements appears in Section 3.2. To address the concern, we have revised the abstract to include a short reference to this construction and added a brief derivation summary in the main text. revision: yes

Circularity Check

1 steps flagged

Domain-alignment unitary defined to map unknown sparse basis, forcing M=O(K) and linear estimation by construction

specific steps

self definitional [Abstract]
"then introducing domain-alignment evolution,a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis. QCS executes the support-set search at the quantum level without consuming measurement trials. The logarithmic penalty vanishes, compressing the required measurement number from the classical bound to M =O(K) and reducing reconstruction from ill posed optimization to linear estimation."

The unitary is defined precisely as the transformation that aligns the unknown sparse basis with the measurement basis. This definition presupposes the support information needed to build the unitary, so the 'quantum-level support-set search' and consequent elimination of the log(N/K) factor are true by the definition of the introduced evolution rather than derived from it.

full rationale

The paper's central reduction from O(K log(N/K)) measurements and combinatorial optimization to M=O(K) linear estimation rests entirely on the introduction of a 'domain-alignment evolution' unitary. This unitary is presented as mapping the sparse basis (whose support is unknown by definition) directly onto the measurement basis, thereby executing support-set search at the quantum level. Because any such mapping requires the unitary to be constructed from the specific unknown support indices, the claimed independence from support knowledge and the resulting linear scaling hold only by the definition of the unitary itself rather than by an independent derivation or physical mechanism shown to work for arbitrary unknown supports. No equations or explicit construction are supplied in the provided text to demonstrate how a fixed, signal-independent unitary could achieve this for general sparse signals.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the physical realizability of a domain-alignment unitary that perfectly aligns bases and on the assumption that quantum evolution can perform the support search without measurement cost. No free parameters or invented particles are stated in the abstract.

axioms (1)

domain assumption A physically realizable unitary transformation exists that maps any chosen sparse basis directly onto the measurement basis.
Invoked in the description of domain-alignment evolution as the mechanism that removes the logarithmic factor.

invented entities (1)

Domain-alignment evolution no independent evidence
purpose: Unitary transformation that encodes the sparse basis into the measurement basis so support search occurs at the quantum level.
Newly introduced construct whose independent experimental verification is not supplied in the abstract.

pith-pipeline@v0.9.0 · 5779 in / 1332 out tokens · 45959 ms · 2026-05-20T19:20:09.264401+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

domain-alignment evolution—a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis... QCS executes the support-set search at the quantum level without consuming measurement trials... M ~ K
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

reconstruction from ill-posed optimization to linear estimation

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

E. J. Candès , J. Romberg, T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52, 489-509 (2006)

work page 2006
[2]

Rudelson, R

M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements. Comm. Pure Appl. Math. 61, 1025-1045 (2008)

work page 2008
[3]

Schirhagl, K

R. Schirhagl, K. Chang, M. Loretz, C. L. Degen, Nitrogen- vacancy centers in diamond: Nanoscale sensors for physics and biology. Annu. Rev. Phys. Chem. 65, 83-105 (2014)

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[4]

H. T. Zhou, C. B. Qin, R. Y . Chen, Y . M. Liu, W. J. Zhou, G. F. Zhang, Y . Gao, L. T. Xiao, S. T. Jia, Quantum coherent modulation-enhanced single-molecule imaging microscopy. J. Phys. Chem. Lett. 10, 223-228 (2019)

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Y . S. Fan, J. Y . Hu, S. X. Wu, Z. X. Qiao, G. S. Feng, C. G. Yang, J. Q. Liu, R. Y . Chen, C. B. Qin, G. F. Zhang, L. T. Xiao, S. T. Jia, Quantum Compressed Sensing Enables Image Classification with a Single Photon. arXiv:2604.25480 [quant-ph] (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[6]

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, D. Gorinevsky, An interior-point method for large-scale l1-regularized least squares. IEEE J. Sel. Top. Signal Process. 1, 606-617 (2007)

work page 2007

[1] [1]

E. J. Candès , J. Romberg, T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52, 489-509 (2006)

work page 2006

[2] [2]

Rudelson, R

M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements. Comm. Pure Appl. Math. 61, 1025-1045 (2008)

work page 2008

[3] [3]

Schirhagl, K

R. Schirhagl, K. Chang, M. Loretz, C. L. Degen, Nitrogen- vacancy centers in diamond: Nanoscale sensors for physics and biology. Annu. Rev. Phys. Chem. 65, 83-105 (2014)

work page 2014

[4] [4]

H. T. Zhou, C. B. Qin, R. Y . Chen, Y . M. Liu, W. J. Zhou, G. F. Zhang, Y . Gao, L. T. Xiao, S. T. Jia, Quantum coherent modulation-enhanced single-molecule imaging microscopy. J. Phys. Chem. Lett. 10, 223-228 (2019)

work page 2019

[5] [5]

Y . S. Fan, J. Y . Hu, S. X. Wu, Z. X. Qiao, G. S. Feng, C. G. Yang, J. Q. Liu, R. Y . Chen, C. B. Qin, G. F. Zhang, L. T. Xiao, S. T. Jia, Quantum Compressed Sensing Enables Image Classification with a Single Photon. arXiv:2604.25480 [quant-ph] (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[6] [6]

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, D. Gorinevsky, An interior-point method for large-scale l1-regularized least squares. IEEE J. Sel. Top. Signal Process. 1, 606-617 (2007)

work page 2007