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arXiv:2609.10141

The Sample Complexity of Quantum Entanglement Allocation

Nathan Roll

quant-phcs.LGSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
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Abstract

How many past requests are needed to decide which qubits should share entanglement? We show that the answer depends on the allocation choices created by the queries: a larger memory can require no more data. The memory stores a classical bit and answers requests through a fixed detector that preserves coherence within each measured sector. For independent commuting $X$- and $Z$-type Pauli queries, we characterize the full attainable prediction-contrast region and construct encodings that preserve the bit at every nonzero vertex. With sharp reports, a $d$-qubit path and groups of at most $k$ qubits have minimax excess error after $m$ requests proportional to $k^{-1}\min\{1,\sqrt{d\log(k+1)/m}\}$, uniformly for $2\leq k<d$. Connected biclique regions can grow without increasing sample demand when depth, region count and connections per region stay bounded. Preparation noise introduces a separate calibration requirement. We derive an exact tradeoff with extra fresh detector calls and transfer the learning law to structured transaction co-location. Population-risk experiments test the statistical predictions. We also compare encodings on a native 15-qubit device and learned partitions on public purchase baskets. The full chain wins on the device; frequency grouping outperforms basket search in the largest-capacity retail setting.

Comments
61 pages, 15 figures. Includes code and recorded data as ancillary files

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How many past requests are needed to decide which qubits should share entanglement? We show that the answer depends on the allocation choices created by the queries: a larger memory can require no more data. The memory stores a classical bit and answers requests through a fixed detector that preserves coherence within each measured sector. For independent commuting XXX- and ZZZ-type Pauli queries, we characterize the full attainable prediction-contrast region and construct encodings that preserve the bit at every nonzero vertex. With sharp reports, a ddd-qubit path and groups of at most kkk qubits have minimax excess error after mmm requests proportional to k−1min⁡{1,dlog⁡(k+1)/m}k^{-1}\min\{1,\sqrt{d\log(k+1)/m}\}k−1min{1,dlog(k+1)/m​}, uniformly for 2≤k<d2\leq k<d2≤k<d. Connected biclique regions can grow without increasing sample demand when depth, region count and connections per region stay bounded. Preparation noise introduces a separate calibration requirement. We derive an exact tradeoff with extra fresh detector calls and transfer the learning law to structured transaction co-location. Population-risk experiments test the statistical predictions. We also compare encodings on a native 15-qubit device and learned partitions on public purchase baskets. The full chain wins on the device; frequency grouping outperforms basket search in the largest-capacity retail setting.
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.
How complex can the responses of a quantum device become as it runs longer with a fixed internal memory? We quantify this complexity through sequential response capacity: how many adaptive testing stages, each using a fresh run, can continue to separate possible processes by a prescribed gap in response probabilities. For fixed system and memory sizes, we establish a tight law relating this capacity to run length and probability resolution. At fixed resolution, the capacity grows on the order of Klog⁡KK\log KKlogK, where KKK is the number of time steps in each run. Our construction attains this growth using time-dependent phase rotations on a single visible qubit with no additional internal memory; its tests give response probabilities exactly zero or one. Under the same tests, classical stochastic processes that measure in a fixed basis at every step have only linear capacity at fixed sizes and resolution. For phase sequences selected by a stored classical label, we then quantify how known independent Pauli noise changes this logarithmic enhancement. With ideal controls and weak residual phase noise after correction, we prove matching capacity bounds at a fixed small probability gap. These bounds identify the inverse residual phase-flip probability as the coherence timescale that limits the extra logarithmic growth.
Learning unknown noise processes in quantum systems reveals their physical origin and informs error suppression, mitigation, and correction. Characterizing a general quantum process requires exponentially many parameters inferred from noncommuting measurements. Because these measurements cannot be performed simultaneously, the sample complexity grows exponentially. For Pauli channels, quantum memory and entangling operations can transform this task into measurements of commuting observables, reducing complexity exponentially. However, noise in these resources increases the overhead, leaving open whether any advantage remains in realistic devices. Here, we introduce error-mitigated entanglement-enhanced learning, analyze it theoretically, and demonstrate it experimentally. We quantify the noise-induced overhead, perform hypothesis testing with up to 64 qubits, and learn intrinsic noise in parallel-gate layers using up to 16 qubits of a superconducting processor. We show that noisy quantum memory provides a learning advantage, with a current experimental overhead of 1.33 ± 0.05 per qubit, below the no-entanglement lower bound of 2. The authors show that entanglement with noisy quantum memory can significantly speed up learning of quantum processes. With error mitigation, their method characterizes quantum noise at scale and outperforms entanglement-free approaches.
How much quantum information must a sequential optical test retain, and how reliably must that information survive between source preparations? This paper gives preparation budgets and impossibility bounds that distinguish useful quantum storage from regimes where the classical sample order is unavoidable. Optical source and fully counted memory A fixed unknown normal state supplies independent, identical preparations on mmm bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least 1/321/321/32 from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most DDD, while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability aaa, or is erased with a classical flag. The quantity ka(D,m)k_a(D,m)ka​(D,m) is the smallest deterministic original-preparation cap achieving worst-case error at most 1/31/31/3. Failed preparations, discarded trials and reserved positions are all charged. A retained register of qqq qubits has D=2qD=2^qD=2q. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is Θ(m)\Theta(m)Θ(m). Along arbitrary growing-mode sequences, the precise criterion for improving that order is ka(D,m)=o(m)⟺am→∞ and aDm→∞. k_a(D,m)=o(m)\quad\Longleftrightarrow\quad am\to\infty\ \text{and}\ \frac{aD}{m}\to\infty. ka​(D,m)=o(m)⟺am→∞ and maD​→∞. The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when a=O(1/m)a=O(1/m)a=O(1/m), yet a survival probability tending to zero can still permit an advantage. Let A=max⁡{1,m2/D}A=\max\{1,m^2/D\}A=max{1,m2/D}. At the fixed energy and separation above, constants c,C>0c,C>0c,C>0, independent of m,D,am,D,am,D,a, give cmin⁡{m,A/a}≤ka(D,m)≤Cmin⁡{m,A/a+min⁡(a−2,m2/3)},0<a≤1. c\min\{m,A/a\}\le k_a(D,m)\le C\min\{m,A/a+\min(a^{-2},m^{2/3})\},\qquad 0<a\le1. cmin{m,A/a}≤ka​(D,m)≤Cmin{m,A/a+min(a−2,m2/3)},0<a≤1. At a=0a=0a=0, the sample cost is Θ(m)\Theta(m)Θ(m). For ideal memory the complete rate is k1(D,m)=Θ ⁣(min⁡{m,max⁡(1,m2/D)}). k_1(D,m)=\Theta\!\left(\min\{m,\max(1,m^2/D)\}\right). k1​(D,m)=Θ(min{m,max(1,m2/D)}). With loss, the bounds match whenever A≥min⁡{a−1,am2/3}A\ge\min\{a^{-1},am^{2/3}\}A≥min{a−1,am2/3}. For example, a=m−1/6a=m^{-1/6}a=m−1/6 and D=⌈m11/6⌉D=\lceil m^{11/6}\rceilD=⌈m11/6⌉ give Θ(m1/3)\Theta(m^{1/3})Θ(m1/3) preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for D≥m2D\ge m^2D≥m2, a=m−1/2a=m^{-1/2}a=m−1/2, the stated bounds are Ω(m1/2)\Omega(m^{1/2})Ω(m1/2) and O(m2/3)O(m^{2/3})O(m2/3). Why the construction and converse fit the same resources A physical completely positive contraction combines first- and second-moment information before the next wait. Although the temporary pair output has order m2m^2m2 labels, only a positive register of dimension m+1m+1m+1 is retained. The three-visit route requires D≥m+1D\ge m+1D≥m+1 and pays two actual survival events; a classical alternative handles the centering statistics. The converse covers arbitrary adaptive instruments and persistent noncommuting memory. Erasure removes the coherent prediction contribution while preserving information already learned in the full classical history. Unflagged noise and applicability For a fixed memoryless channel on the entire retained register with a physical reset decomposition N=aΨ+(1−a)Rτ\mathcal N=a\Psi+(1-a)\mathcal R_\tauN=aΨ+(1−a)Rτ​, where Ψ\PsiΨ is CPTP and Rτ(X)=Tr⁡(X)τ\mathcal R_\tau(X)=\operatorname{Tr}(X)\tauRτ​(X)=Tr(X)τ, an exact flagged-erasure simulation gives kN(D,m)≥ka(D,m)k_{\mathcal N}(D,m)\ge k_a(D,m)kN​(D,m)≥ka​(D,m). The established positive quantum Doeblin coefficient optimizes this lower obstruction. Whole-register depolarizing noise with unchanged fraction 1−q=O(1/m)1-q=O(1/m)1−q=O(1/m) therefore forces Θ(m)\Theta(m)Θ(m) samples for every memory dimension. General unflagged noise inherits a converse, not the flagged model's complete achievability guarantees. The results quantify asymptotic preparation and persistent-memory resources under ideal visit-local control. Temporary auxiliary systems and gate complexity have no imposed budget, and the explicit sufficient finite-sample constants are conservative. The task is pure Gaussianity testing, rather than membership in the full family of mixed Gaussian states. The single integrated PDF contains the short main text, full proofs, four figures and precise resource accounting.
How can a learned representation remain useful when the question it must answer is chosen only after encoding? Preserving information is essential, but does not by itself ensure access to the requested target. We address this problem by deriving "neural information causality (Neural-IC)," a representation law that connects causal access to the physically specified interface capacity. For a fresh database of independent bits, the aggregate information recoverable across later queries cannot exceed that capacity, even with pre-established quantum resources. A decomposition separates capacity slack, information organized jointly across targets, and losses in producing answers. Exhaustive evaluation of trained binary codes reveals why nearly full use of a memory's information budget can coexist with limited retrieval. In a nested Bell-correlation construction, quantum assistance improves average retrieval over the optimal classical one-bit benchmark within the same communicated budget. These results give representation learning a design principle that connects information capacity to the organization and extraction of target information. By locating the constraints on later answers, the framework can inform memory allocation, encoding objectives, and decoder outputs, providing a basis for seeking better retrieval within a fixed budget.
We investigate the operational limits of entanglement distribution through quantum channels reused without reinitialization. Building on an effective repeated-use channel description, we show that residual excitations left in the channel from previous uses act as a memory source that progressively suppresses coherent quantum transmission. To quantify this limitation, we derive an analytical lower bound on the quantum capacity and combine it with the Holevo-Werner semidefinite-programming upper bound. We also provide an analytical criterion to identify entanglement-breaking channels, with thresholds that depend jointly on chain length and reuse. For perfect-state-transfer chains, the resulting capacity window yields tighter estimates than previous bounds, with the relative improvement becoming substantially more pronounced closer to the entanglement-breaking threshold. Moreover, on the second use, our lower bound certifies a maximal relative timing-error tolerance across all chains with internal memory that lies just below 22.63% and is attained by the three-site chain. We further derive a fundamental relative timing-error threshold of 36.41%, above which we analytically guarantee zero unassisted quantum capacity for every chain with internal memory, independent of the number of uses. These results establish a clear benchmark model for repeated entanglement distribution, and identify timing precision, chain length, and memory accumulation as fundamental constraints on scalable quantum-network architectures.