Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.23195

Localized charges, reset noise, and boundary memory in matrix-product-conserving quantum chains

Ron Rubin

quant-phcond-mat.stat-mechmath-phSubmitted Sep 19, 2026• Announced Sep 22, 2026• 0 citations
arXivHTMLPDF

Abstract

We construct boundary observables that retain memory for a quantified time window in quantum chains whose basis configurations carry a conserved ordered product of matrix labels. Under strong-irreducibility and proximality hypotheses on the label matrices, projective contraction of random matrix products localizes a conserved charge in the root-mean-square over uniform basis configurations. A finite-time inequality then converts readout overlap and reset leakage into a correlation bound: for a conserved reference observable $H$, a readout $F$, accumulated squared leakage $\mathcal B$, and the normalized Hilbert-Schmidt inner product, $\langle F,Φ(F)\rangle\ge 2\langle H,F\rangle^2/(\|H\|_2^2+\mathcal B)-\|F\|_2^2$. Here $Φ$ is a sequence of conserving channels and partial resets. The bound holds for each such circuit, without averaging its gates. It yields an exponential memory window in the distance from the noisy region, with quantitative corrections for spatially distributed noise and imperfect conservation. For a thirteen-state elementary-matrix model, exact polynomial certificates give root-mean-square localization error $e^{-r/1400}$ between the charge and its restriction to the last $r$ sites, and limiting variance at least $1/19$ for a rational charge. A related normalized-Gram charge gives an eight-site certificate retaining more than half the specified readout correlation through two boundary-reset rounds. Uniform depolarization imposes an inverse-noise-rate lifetime ceiling. A two-qubit IBM experiment illustrates the general reset inequality; it does not realize the matrix-product chain. We provide proofs, exact certificate programs, and an independent recount of the archived experimental outcomes.

Comments
43 pages, 3 figures. Ancillary file: exact certificate programs, primary experimental data and offline recount code

Related papers

Exact Localized Information Capacity of a Local Observable in a Hilbert-Space-Shattered System

Hikaru Wakaura, Taiki Tanimae

Abstract How many bits of information a local observable can retain forever is a sharp, operational measure of a system's memory, but it has been quantified only for many-body localization, where it rests on approximate integrals of motion, and for generic systems, where it is obtained numerically. Systems with strong Hilbert-space fragmentation possess an \emph{exact and complete} commutant algebra, yet their memory has been characterized only through autocorrelation Mazur bounds or by counting protected qubits---never as an information capacity in bits. Here we define the localized information capacity of a local observable as the Holevo/mutual information its infinite-time-retained state carries about the initial state, and show that in a fragmented system it equals, in \emph{closed form}, the projection of the observable onto the commutant, $\Cuniv=\sum_\alpha[\Tr(A P_\alpha)]^2/(D\,\Tr P_\alpha)$. This value is Hamiltonian-independent and is a rigorous lower bound on the true infinite-time autocorrelation; the excess is non-universal dynamical memory, which we show grows under disorder. For the spin-1 dipole-conserving chain we derive the sector count in closed form from a finite transfer structure on sector boundary types---Pell-number growth, $K_L=2P_{L+1}-1\sim(1+\sqrt2)^L$, the silver ratio; the capacity density provably converges, with the numerical value $\chi_\infty=1.09(1)$ bits per site, and fragmentation nearly doubles the autocorrelation floor guaranteed by all polynomial multipole charges. For the $t$--$J_z$ chain the capacity closes fully: exactly $2/3$ bit per site, the particle density times one retained spin bit. We further show why the complete labelling of the retained bits resists strict locality: the height dynamics is a partially-commutative (trace) monoid whose natural complete invariant---computable by a strictly local sweep---is provably incomplete for the gated physical moves, the missing labels being frozen orderings that are precisely statistically localized data. Our results turn the memory of a shattered Hilbert space into an exact bit count and explain, from first principles, why its complete integrals of motion are statistically rather than sharply localized.

•Jul 16, 2026•0 citations

Finite-time quantum equilibration from local correlations

Piotr Ćwikliński

An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. The two mean-square quantities are linearly comparable. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length, and the result extends to nonnegative observation weights with nonnegative Fourier transform. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.

quant-pharXiv:2610.16351•Oct 13, 2026•0 citations

Complexity of tensor network simulation for noisy quantum circuits

Yuguo Shao, Zishuo Zhao, Song Cheng +1 more

We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with $\mathrm{poly}(n)$ bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\order{1}$ depth, while relative error demands $\order{\log n}$ depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of $\mathrm{poly}(n)$ bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient $c<1/3$ yield an $\order{1}$ OEE plateau with probability $1-Te^{-Ω(n)}$, while arbitrary gates with $c<1/48$ give $\order{\log n}$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth $\mathrm{poly}(n)$ average boundary-bond dimensions for projected entangled pair operators~(PEPO) across every cut -- under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.

cond-mat.str-elCross-listedarXiv:2606.00474•Jun 2, 2026•0 citations

Quantum Cantor–Bendixson Machine: Exact and Simultaneous Boundary Repair

Hutan Ashrafian

Quantum programmes described through successive approximations need a consistent way to retain their limiting behaviour when individual approximating settings are discarded. I propose a Quantum Cantor–Bendixson Machine that certifies and repairs quantum operations at boundaries of a classical control space. For continuous matrix-valued observables over a compact zero-dimensional metric space and a finite retained boundary, the smallest completely bounded repair cost is exactly twice the largest boundary leakage, and a continuous unital completely positive map attains it. I extend this result to simultaneous finite boundary constraints, identifying when their costs combine without an additional penalty. For independent operations on any finite product of convergent-sequence spaces, a product repair preserves every Cantor–Bendixson derivative, including the infinite intermediate boundaries, at the exact cost already forced by the final boundary. A coherent qubit example establishes why the corresponding cost for a quantum subspace can instead scale as the square root of leakage. Computable operations can have noncomputable optimal repair costs: their possible leakages are exactly the lower semicomputable reals in the unit interval. An explicit tail certificate permits finite bounds and a computable repair arbitrarily close to the optimum. These results specify when one operation can respect several topological reductions, what this consistency costs, and which additional information makes its certification effective.

•Oct 29, 2026•0 citations

Sigma-Time Quantum Dynamics III: Measurement Events, Reset Ledgers, and Benchmark Certificates

Ben F.T. Tibola

This paper develops the measurement and reset layer of sigma-time quantum dynamics. The first paper treats unitary BV-clock evolutions with gates and pulses; the second treats bounded open-system channel products with CPTP atom maps and random-clock averages. Here we add finite-outcome measurement and reset events. A measurement event is modeled as a quantum instrument placed at a locally finite atom of a BV sigma clock. Selective outcomes are completely positive trace-nonincreasing branch maps; their traces are joint outcome probabilities. Positive-probability branches normalize to density matrices, while the nonselective sum over all outcomes is a completely positive trace-preserving channel. Reset atoms and measurement-thenreset chains are included by finite Kraus certificates. We also give a finite adaptive scaffolding theorem: when the next event instrument is chosen from a finite recorded outcome history, the same branch and nonselective closure follow by induction on the event tree. The paper includes event ledger schemas, examples, refusal walls, and a reproducible qubit benchmark for projective measurement, weak POVM measurement, reset, measurement-reset, and a two-stage adaptive event tree. Zeno accumulation, continuous outcomes, unbounded measurement domains, and hardware fidelity are explicitly outside the Scope of the results.

•May 28, 2026•0 citations

Qubit recycling and the path-counting problem

Zijian Song, Isaac H. Kim

Recently, it was shown that the qudits used in circuits with a convolutional form (e.g., matrix product states and the multiscale entanglement renormalization Ansatz) can be reset unitarily [Anikeeva et al., Phys. Rev. A 103, 042613 (2021)], even without measurement. We analyze the fidelity of this protocol for a family of quantum circuits that interpolates between such circuits and local quantum circuits, averaged over Haar-random gates. We establish a connection between this problem and counting directed paths on a graph, which is determined by the shape of the quantum circuit. This connection leads to an exact expression for the fidelity of the protocol for the entire family that interpolates between a convolutional circuit and a random quantum circuit. For convolutional circuits with a constant window size, the rate of convergence to unit fidelity is shown to be $\frac{{q}^{2}}{{q}^{2}+1}$, independent of the window size, where $q$ is the local qudit dimension. Since most applications of convolutional circuits use constant-size windows, our result suggests that the unitary reset protocol will likely work well in such a regime. We also derive two extra results in the convolutional limit, which may be of independent interest. First, we derive exact expressions for the correlations between reset qudits and show that they decays exponentially in the distance. Second, we derive an expression for the fidelity in the presence of noise, expressed in terms of the quantities that define the property of the channel, such as the entanglement fidelity.

arXiv:2301.03725•Physical Review A•Oct 18, 2023•1 citation
We construct boundary observables that retain memory for a quantified time window in quantum chains whose basis configurations carry a conserved ordered product of matrix labels. Under strong-irreducibility and proximality hypotheses on the label matrices, projective contraction of random matrix products localizes a conserved charge in the root-mean-square over uniform basis configurations. A finite-time inequality then converts readout overlap and reset leakage into a correlation bound: for a conserved reference observable HHH, a readout FFF, accumulated squared leakage B\mathcal BB, and the normalized Hilbert-Schmidt inner product, ⟨F,Φ(F)⟩≥2⟨H,F⟩2/(∥H∥22+B)−∥F∥22\langle F,Φ(F)\rangle\ge 2\langle H,F\rangle^2/(\|H\|_2^2+\mathcal B)-\|F\|_2^2⟨F,Φ(F)⟩≥2⟨H,F⟩2/(∥H∥22​+B)−∥F∥22​. Here ΦΦΦ is a sequence of conserving channels and partial resets. The bound holds for each such circuit, without averaging its gates. It yields an exponential memory window in the distance from the noisy region, with quantitative corrections for spatially distributed noise and imperfect conservation. For a thirteen-state elementary-matrix model, exact polynomial certificates give root-mean-square localization error e−r/1400e^{-r/1400}e−r/1400 between the charge and its restriction to the last rrr sites, and limiting variance at least 1/191/191/19 for a rational charge. A related normalized-Gram charge gives an eight-site certificate retaining more than half the specified readout correlation through two boundary-reset rounds. Uniform depolarization imposes an inverse-noise-rate lifetime ceiling. A two-qubit IBM experiment illustrates the general reset inequality; it does not realize the matrix-product chain. We provide proofs, exact certificate programs, and an independent recount of the archived experimental outcomes.
Abstract How many bits of information a local observable can retain forever is a sharp, operational measure of a system's memory, but it has been quantified only for many-body localization, where it rests on approximate integrals of motion, and for generic systems, where it is obtained numerically. Systems with strong Hilbert-space fragmentation possess an \emph{exact and complete} commutant algebra, yet their memory has been characterized only through autocorrelation Mazur bounds or by counting protected qubits---never as an information capacity in bits. Here we define the localized information capacity of a local observable as the Holevo/mutual information its infinite-time-retained state carries about the initial state, and show that in a fragmented system it equals, in \emph{closed form}, the projection of the observable onto the commutant, $\Cuniv=\sum_\alpha[\Tr(A P_\alpha)]^2/(D\,\Tr P_\alpha)$. This value is Hamiltonian-independent and is a rigorous lower bound on the true infinite-time autocorrelation; the excess is non-universal dynamical memory, which we show grows under disorder. For the spin-1 dipole-conserving chain we derive the sector count in closed form from a finite transfer structure on sector boundary types---Pell-number growth, KL=2PL+1−1∼(1+2)LK_L=2P_{L+1}-1\sim(1+\sqrt2)^LKL​=2PL+1​−1∼(1+2​)L, the silver ratio; the capacity density provably converges, with the numerical value χ∞=1.09(1)\chi_\infty=1.09(1)χ∞​=1.09(1) bits per site, and fragmentation nearly doubles the autocorrelation floor guaranteed by all polynomial multipole charges. For the ttt--JzJ_zJz​ chain the capacity closes fully: exactly 2/32/32/3 bit per site, the particle density times one retained spin bit. We further show why the complete labelling of the retained bits resists strict locality: the height dynamics is a partially-commutative (trace) monoid whose natural complete invariant---computable by a strictly local sweep---is provably incomplete for the gated physical moves, the missing labels being frozen orderings that are precisely statistically localized data. Our results turn the memory of a shattered Hilbert space into an exact bit count and explain, from first principles, why its complete integrals of motion are statistically rather than sharply localized.
An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. The two mean-square quantities are linearly comparable. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length, and the result extends to nonnegative observation weights with nonnegative Fourier transform. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.
We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with poly(n)\mathrm{poly}(n)poly(n) bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\order{1}$ depth, while relative error demands $\order{\log n}$ depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of poly(n)\mathrm{poly}(n)poly(n) bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient c<1/3c<1/3c<1/3 yield an $\order{1}$ OEE plateau with probability 1−Te−Ω(n)1-Te^{-Ω(n)}1−Te−Ω(n), while arbitrary gates with c<1/48c<1/48c<1/48 give $\order{\log n}$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth poly(n)\mathrm{poly}(n)poly(n) average boundary-bond dimensions for projected entangled pair operators~(PEPO) across every cut -- under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.
Quantum programmes described through successive approximations need a consistent way to retain their limiting behaviour when individual approximating settings are discarded. I propose a Quantum Cantor–Bendixson Machine that certifies and repairs quantum operations at boundaries of a classical control space. For continuous matrix-valued observables over a compact zero-dimensional metric space and a finite retained boundary, the smallest completely bounded repair cost is exactly twice the largest boundary leakage, and a continuous unital completely positive map attains it. I extend this result to simultaneous finite boundary constraints, identifying when their costs combine without an additional penalty. For independent operations on any finite product of convergent-sequence spaces, a product repair preserves every Cantor–Bendixson derivative, including the infinite intermediate boundaries, at the exact cost already forced by the final boundary. A coherent qubit example establishes why the corresponding cost for a quantum subspace can instead scale as the square root of leakage. Computable operations can have noncomputable optimal repair costs: their possible leakages are exactly the lower semicomputable reals in the unit interval. An explicit tail certificate permits finite bounds and a computable repair arbitrarily close to the optimum. These results specify when one operation can respect several topological reductions, what this consistency costs, and which additional information makes its certification effective.
This paper develops the measurement and reset layer of sigma-time quantum dynamics. The first paper treats unitary BV-clock evolutions with gates and pulses; the second treats bounded open-system channel products with CPTP atom maps and random-clock averages. Here we add finite-outcome measurement and reset events. A measurement event is modeled as a quantum instrument placed at a locally finite atom of a BV sigma clock. Selective outcomes are completely positive trace-nonincreasing branch maps; their traces are joint outcome probabilities. Positive-probability branches normalize to density matrices, while the nonselective sum over all outcomes is a completely positive trace-preserving channel. Reset atoms and measurement-thenreset chains are included by finite Kraus certificates. We also give a finite adaptive scaffolding theorem: when the next event instrument is chosen from a finite recorded outcome history, the same branch and nonselective closure follow by induction on the event tree. The paper includes event ledger schemas, examples, refusal walls, and a reproducible qubit benchmark for projective measurement, weak POVM measurement, reset, measurement-reset, and a two-stage adaptive event tree. Zeno accumulation, continuous outcomes, unbounded measurement domains, and hardware fidelity are explicitly outside the Scope of the results.
Recently, it was shown that the qudits used in circuits with a convolutional form (e.g., matrix product states and the multiscale entanglement renormalization Ansatz) can be reset unitarily [Anikeeva et al., Phys. Rev. A 103, 042613 (2021)], even without measurement. We analyze the fidelity of this protocol for a family of quantum circuits that interpolates between such circuits and local quantum circuits, averaged over Haar-random gates. We establish a connection between this problem and counting directed paths on a graph, which is determined by the shape of the quantum circuit. This connection leads to an exact expression for the fidelity of the protocol for the entire family that interpolates between a convolutional circuit and a random quantum circuit. For convolutional circuits with a constant window size, the rate of convergence to unit fidelity is shown to be q2q2+1\frac{{q}^{2}}{{q}^{2}+1}q2+1q2​, independent of the window size, where qqq is the local qudit dimension. Since most applications of convolutional circuits use constant-size windows, our result suggests that the unitary reset protocol will likely work well in such a regime. We also derive two extra results in the convolutional limit, which may be of independent interest. First, we derive exact expressions for the correlations between reset qudits and show that they decays exponentially in the distance. Second, we derive an expression for the fidelity in the presence of noise, expressed in terms of the quantities that define the property of the channel, such as the entanglement fidelity.