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:2610.36972

Quantum and Classical Memory for Storage and Retrieval of Quantum Measurements

Filip Turoboś, Ludmiła Marcinkowska, Zbigniew Puchała

quant-phSubmitted Oct 27, 2026• Announced Oct 28, 2026• 0 citations
arXivHTMLPDF

Abstract

We study the persistent quantum and classical memory required for exact heralded storage and retrieval of quantum measurements. Given only black-box access to the measurement channel, we construct an exact compression of the port-based program into an outcome-type record and a quantum register, with total dimension polynomial in the query count. For every universal retrieval processor we prove the bounds $Q+C\ge\log_2 n$ and $2Q+C\ge\log_2\bigl((n-1)d^2+1\bigr)$ on the numbers $Q$ of qubits and $C$ of bits of the program, for $n$-outcome POVMs on $\mathbb{C}^d$ with $d\ge2$. For $n\ge2$, no finite classical memory removes the need for quantum memory: any such processor requires a quantum program of dimension at least $d$, at that dimension its classical alphabet must have at least $n$ symbols, and both minima are attained by a single-query protocol with success $1/d^2$. Combining direct-program compression with a converse from approximate measurement programming gives logarithmic total-memory scaling in $1/(1-p)$ as the uniform success probability $p$ approaches one, at fixed $d,n\ge2$. At fixed input dimension and target success below one, the optimal total-memory scaling is logarithmic in $n$. We also construct self-contained programs under branch-decomposition access and compare the quantum and classical costs of all constructions, including where the outcome-type record is kept: for POVMs with $n$ nonzero rank-one effects the direct program admits a purely quantum allocation whose register is never larger than that of the minimal-Naimark unitary program and which succeeds more often whenever $n>d$.

Comments
32 pages, 1 figure

Related papers

Optimal Probabilistic Storage and Retrieval of Unitary Channels

Michal Sedlák, Alessandro Bisio, Mário Ziman

We address the question of quantum memory storage for quantum dynamics. In particular, we design an optimal protocol for N→1 probabilistic storage and retrieval of unitary channels on d-dimensional quantum systems. If we access the unknown unitary gate only N times, the optimal success probability of perfect single-use retrieval is N/(N-1+d^{2}). The derived size of the memory system exponentially improves the known upper bound on the size of the program register needed for probabilistic programmable quantum processors. Our results are closely related to probabilistic perfect alignment of reference frames and probabilistic port-based teleportation.

arXiv:1809.04552•Physical Review Letters•May 3, 2019•60 citations

Size and reliability limits of quantum memory for heterodyne uncertainty

Zixuan He

Can a larger quantum memory compensate for a less reliable one? This paper answers that question for optical heterodyne quantum conditional entropy under a total photon budget and a strict physical memory dimension. It separates an uncertainty loss that additional storage can reduce from an erasure floor that no increase in dimension can remove. Preparation and noise model. The optical system has \(m\) modes and total mean photon number at most \(E\). Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most \(d\). Independently, the entire register survives with probability \(1-p\) or is erased into an orthogonal readable failure state. The physical output therefore has \(d+1\) levels; a total \(q\)-qubit register permits \(d\le2^q-1\). Every event is retained, with no postselection. For heterodyne measure \(d^{2m}z/\pi^m\) and natural logarithms, the exact all-preparation reduction is \[V_{\rm er}(m,d,E,p)=\min_{\operatorname{rank}\rho\le d,\;\operatorname{Tr}(N\rho)\le E}\{W(\rho)-(1-p)S(\rho)\}.\] Here \(W\) is Wehrl entropy, \(S\) is von Neumann entropy and \(N\) is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define \(g(x)=(1+x)\ln(1+x)-x\ln x\), \(S_*=m g(E/m)\), the ideal unlimited-memory value \(B_m(E)=m-E\ln(1+m/E)\), and \(a_E=1-[(1+E/m)\ln(1+m/E)]^{-1}\). For any target excess satisfying \(0<\varepsilon<a_ES_*\), \[p_{\rm crit}=\frac{\varepsilon}{m g(E/m)}.\] A finite register reaches \(V_{\rm er}\le B_m(E)+\varepsilon\) if and only if \(p<p_{\rm crit}\). At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let \(S_d\) be the largest entropy of any rank-at-most-\(d\) optical state within the same photon budget, and \(\Delta_d=S_*-S_d\). For \(0\le p<a_E\), \[pS_*+(a_E-p)\Delta_d\le V_{\rm er}-B_m(E)\le pS_*+(1-p)\Delta_d.\] The first contribution survives at unlimited dimension, whereas \(\Delta_d\) tends to zero as storage grows. An exact finite Gibbs program computes \(S_d\), including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At \(0<E\le10^{-4}\), \(2\le d\le m+1\) and \(p<1/8\), an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.

•Oct 26, 2026•0 citations

Quantum memory advantage for quantum process tomography

Carlos Bravo-Prieto, Weiyuan Gong, Antonio Anna Mele

Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be $Θ(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)$, where $d_{\mathrm{in}}$ and $d_{\mathrm{out}}$ are the channel input and output dimensions, respectively, and $\varepsilon$ is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires $Ω(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)$ queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound $O(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)$. Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case $d_{\mathrm{in}}=1$. By contrast, coherent protocols with quantum memory achieve query complexity $Θ(d_{\mathrm{in}}^2 d_{\mathrm{out}}^2/\varepsilon^2)$. Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.

quant-pharXiv:2607.13476•Jul 16, 2026•0 citations

Quantum Advantage in Storage and Retrieval of Isometry Channels

Satoshi Yoshida, Jisho Miyazaki, Mio Murao

Storage and retrieval refer to the task of encoding an unknown quantum channel Λ into a quantum state, known as the program state, such that the channel can later be retrieved. There are two strategies for this task: classical and quantum strategies. The classical strategy uses multiple queries to Λ to estimate Λ and retrieves the channel based on the estimate represented in classical bits. The classical strategy turns out to offer the optimal performance for the storage and retrieval of unitary channels. In this work, we analyze the asymptotic performance of the classical and quantum strategies for the storage and retrieval of isometry channels. We show that the optimal fidelity for isometry estimation is given by F=1-[d(D-d)/n]+O(n^{-2}), where d and D denote the input and output dimensions of the isometry, and n is the number of queries. This result indicates that, unlike in the case of unitary channels, the classical strategy is suboptimal for the storage and retrieval of isometry channels, which requires n=Θ(ε^{-1}) to achieve the diamond-norm error ε. We propose a more efficient quantum strategy based on port-based teleportation, which stores the isometry channel in a program state using only n=Θ(1/sqrt[ε]) queries, achieving a quadratic improvement over the classical strategy. As an application, we extend our approach to general quantum channels, achieving improved program cost compared to prior results by Gschwendtner, Bluhm, and Winter [Quantum 5, 488 (2021)2521-327X10.22331/q-2021-06-29-488].

arXiv:2507.10784•Physical Review Letters•Mar 27, 2026•1 citation

Exact Virtual Channel Programming with Vanishing Excess Overhead

Mingrui Jing, Mengbo Guo, Hongshun Yao +1 more

A finite-dimensional physical processor cannot exactly program a continuous family of distinct unitary channels. We show that this obstruction becomes quantitative when the target channel is stored in a normalized Choi state and its output observables are reconstructed by sampling physical channels and classically post-processing their measurement outcomes. For arbitrary $d$-dimensional channels, we construct a target-independent exact reconstruction protocol and prove the optimal one-copy sampling overhead, which grows quadratically with system dimension. We further prove the sharp fixed-$d$ law that the excess overhead vanishes inversely with the number of identical Choi programs. The upper bound combines deterministic port-based teleportation with a quasi-decomposition that corrects its depolarizing distortion. The converse maps any low-overhead reconstruction protocol to a physical learner of unknown unitaries and uses local quantum estimation to recover the same leading coefficient. These results recast the universal no-programming obstruction as a quantitative trade-off between quantum program memory and classical sampling, with a leading cost that reflects the locally learnable unitary degrees of freedom.

quant-pharXiv:2609.01419•Sep 2, 2026•0 citations

Quantum Source Completions and Memory Environment Tradeoffs

Jeremy Rodgers

How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.

•Oct 28, 2026•0 citations
We study the persistent quantum and classical memory required for exact heralded storage and retrieval of quantum measurements. Given only black-box access to the measurement channel, we construct an exact compression of the port-based program into an outcome-type record and a quantum register, with total dimension polynomial in the query count. For every universal retrieval processor we prove the bounds Q+C≥log⁡2nQ+C\ge\log_2 nQ+C≥log2​n and 2Q+C≥log⁡2((n−1)d2+1)2Q+C\ge\log_2\bigl((n-1)d^2+1\bigr)2Q+C≥log2​((n−1)d2+1) on the numbers QQQ of qubits and CCC of bits of the program, for nnn-outcome POVMs on Cd\mathbb{C}^dCd with d≥2d\ge2d≥2. For n≥2n\ge2n≥2, no finite classical memory removes the need for quantum memory: any such processor requires a quantum program of dimension at least ddd, at that dimension its classical alphabet must have at least nnn symbols, and both minima are attained by a single-query protocol with success 1/d21/d^21/d2. Combining direct-program compression with a converse from approximate measurement programming gives logarithmic total-memory scaling in 1/(1−p)1/(1-p)1/(1−p) as the uniform success probability ppp approaches one, at fixed d,n≥2d,n\ge2d,n≥2. At fixed input dimension and target success below one, the optimal total-memory scaling is logarithmic in nnn. We also construct self-contained programs under branch-decomposition access and compare the quantum and classical costs of all constructions, including where the outcome-type record is kept: for POVMs with nnn nonzero rank-one effects the direct program admits a purely quantum allocation whose register is never larger than that of the minimal-Naimark unitary program and which succeeds more often whenever n>dn>dn>d.
We address the question of quantum memory storage for quantum dynamics. In particular, we design an optimal protocol for N→1 probabilistic storage and retrieval of unitary channels on d-dimensional quantum systems. If we access the unknown unitary gate only N times, the optimal success probability of perfect single-use retrieval is N/(N-1+d^{2}). The derived size of the memory system exponentially improves the known upper bound on the size of the program register needed for probabilistic programmable quantum processors. Our results are closely related to probabilistic perfect alignment of reference frames and probabilistic port-based teleportation.
Can a larger quantum memory compensate for a less reliable one? This paper answers that question for optical heterodyne quantum conditional entropy under a total photon budget and a strict physical memory dimension. It separates an uncertainty loss that additional storage can reduce from an erasure floor that no increase in dimension can remove. Preparation and noise model. The optical system has mmm modes and total mean photon number at most EEE. Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most ddd. Independently, the entire register survives with probability 1−p1-p1−p or is erased into an orthogonal readable failure state. The physical output therefore has d+1d+1d+1 levels; a total qqq-qubit register permits d≤2q−1d\le2^q-1d≤2q−1. Every event is retained, with no postselection. For heterodyne measure d2mz/πmd^{2m}z/\pi^md2mz/πm and natural logarithms, the exact all-preparation reduction is Ver(m,d,E,p)=min⁡rank⁡ρ≤d,  Tr⁡(Nρ)≤E{W(ρ)−(1−p)S(ρ)}.V_{\rm er}(m,d,E,p)=\min_{\operatorname{rank}\rho\le d,\;\operatorname{Tr}(N\rho)\le E}\{W(\rho)-(1-p)S(\rho)\}.Ver​(m,d,E,p)=minrankρ≤d,Tr(Nρ)≤E​{W(ρ)−(1−p)S(ρ)}. Here WWW is Wehrl entropy, SSS is von Neumann entropy and NNN is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define g(x)=(1+x)ln⁡(1+x)−xln⁡xg(x)=(1+x)\ln(1+x)-x\ln xg(x)=(1+x)ln(1+x)−xlnx, S∗=mg(E/m)S_*=m g(E/m)S∗​=mg(E/m), the ideal unlimited-memory value Bm(E)=m−Eln⁡(1+m/E)B_m(E)=m-E\ln(1+m/E)Bm​(E)=m−Eln(1+m/E), and aE=1−[(1+E/m)ln⁡(1+m/E)]−1a_E=1-[(1+E/m)\ln(1+m/E)]^{-1}aE​=1−[(1+E/m)ln(1+m/E)]−1. For any target excess satisfying 0<ε<aES∗0<\varepsilon<a_ES_*0<ε<aE​S∗​, pcrit=εmg(E/m).p_{\rm crit}=\frac{\varepsilon}{m g(E/m)}.pcrit​=mg(E/m)ε​. A finite register reaches Ver≤Bm(E)+εV_{\rm er}\le B_m(E)+\varepsilonVer​≤Bm​(E)+ε if and only if p<pcritp<p_{\rm crit}p<pcrit​. At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let SdS_dSd​ be the largest entropy of any rank-at-most-ddd optical state within the same photon budget, and Δd=S∗−Sd\Delta_d=S_*-S_dΔd​=S∗​−Sd​. For 0≤p<aE0\le p<a_E0≤p<aE​, pS∗+(aE−p)Δd≤Ver−Bm(E)≤pS∗+(1−p)Δd.pS_*+(a_E-p)\Delta_d\le V_{\rm er}-B_m(E)\le pS_*+(1-p)\Delta_d.pS∗​+(aE​−p)Δd​≤Ver​−Bm​(E)≤pS∗​+(1−p)Δd​. The first contribution survives at unlimited dimension, whereas Δd\Delta_dΔd​ tends to zero as storage grows. An exact finite Gibbs program computes SdS_dSd​, including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At 0<E≤10−40<E\le10^{-4}0<E≤10−4, 2≤d≤m+12\le d\le m+12≤d≤m+1 and p<1/8p<1/8p<1/8, an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.
Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2)Θ(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Θ(din3​dout3​/ε2), where dind_{\mathrm{in}}din​ and doutd_{\mathrm{out}}dout​ are the channel input and output dimensions, respectively, and ε\varepsilonε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2)Ω(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Ω(din3​dout3​/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2)O(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)O(din3​dout3​/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1d_{\mathrm{in}}=1din​=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2)Θ(d_{\mathrm{in}}^2 d_{\mathrm{out}}^2/\varepsilon^2)Θ(din2​dout2​/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.
Storage and retrieval refer to the task of encoding an unknown quantum channel Λ into a quantum state, known as the program state, such that the channel can later be retrieved. There are two strategies for this task: classical and quantum strategies. The classical strategy uses multiple queries to Λ to estimate Λ and retrieves the channel based on the estimate represented in classical bits. The classical strategy turns out to offer the optimal performance for the storage and retrieval of unitary channels. In this work, we analyze the asymptotic performance of the classical and quantum strategies for the storage and retrieval of isometry channels. We show that the optimal fidelity for isometry estimation is given by F=1-[d(D-d)/n]+O(n^{-2}), where d and D denote the input and output dimensions of the isometry, and n is the number of queries. This result indicates that, unlike in the case of unitary channels, the classical strategy is suboptimal for the storage and retrieval of isometry channels, which requires n=Θ(ε^{-1}) to achieve the diamond-norm error ε. We propose a more efficient quantum strategy based on port-based teleportation, which stores the isometry channel in a program state using only n=Θ(1/sqrt[ε]) queries, achieving a quadratic improvement over the classical strategy. As an application, we extend our approach to general quantum channels, achieving improved program cost compared to prior results by Gschwendtner, Bluhm, and Winter [Quantum 5, 488 (2021)2521-327X10.22331/q-2021-06-29-488].
A finite-dimensional physical processor cannot exactly program a continuous family of distinct unitary channels. We show that this obstruction becomes quantitative when the target channel is stored in a normalized Choi state and its output observables are reconstructed by sampling physical channels and classically post-processing their measurement outcomes. For arbitrary ddd-dimensional channels, we construct a target-independent exact reconstruction protocol and prove the optimal one-copy sampling overhead, which grows quadratically with system dimension. We further prove the sharp fixed-ddd law that the excess overhead vanishes inversely with the number of identical Choi programs. The upper bound combines deterministic port-based teleportation with a quasi-decomposition that corrects its depolarizing distortion. The converse maps any low-overhead reconstruction protocol to a physical learner of unknown unitaries and uses local quantum estimation to recover the same leading coefficient. These results recast the universal no-programming obstruction as a quantitative trade-off between quantum program memory and classical sampling, with a leading cost that reflects the locally learnable unitary degrees of freedom.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.