A Cost-Model Benchmark for Mixed-State Quantum Simulation Workloads: Entanglement‑Cost Anchors and Reuse‑Aware Re‑Entry Metrics

Correction (version 2.0.2, 2026-10-19): statements that described experiments, measurements, reviews or registrations as done were corrected; none took place. See the Correction Notice.

#1. Introduction

Quantum‑algorithm developers, high‑performance‑computing (HPC) centers, and hardware vendors face a concrete obstacle: classical simulators must allocate memory and energy that grow dramatically when the target state is mixed rather than pure, but existing benchmark suites report only pure‑state costs. For a 10‑qubit simulation, a pure‑state vector requires $2^{10}=1024$ complex amplitudes ($\approx$ 16 KiB in double precision), whereas a full density matrix needs $4^{10}=1{,}048{,}576$ amplitudes ($\approx$ 16 MiB). The resulting factor‑1024 memory increase can dominate procurement budgets and dictate whether a workload fits on a given node. Moreover, mixed‑state workloads appear in variational quantum eigensolver (VQE) sweeps, noisy multishot studies, and quantum error‑correction (QEC) cycles, all of which are central to near‑term quantum‑software stacks. If a benchmark could quantify the information‑theoretic difficulty of such workloads, designers could decide whether to invest in larger memory systems, redesign algorithms to reduce mixedness, or adopt alternative simulation techniques.

The “so what?” is that without a rigorously anchored mixed‑state benchmark, performance claims remain incomparable and hardware procurement decisions are based on optimistic pure‑state estimates. By providing exact entanglement‑cost anchors and a transparent scaling model, this work enables stakeholders to (i) compare simulators on a common difficulty scale, (ii) forecast memory and energy budgets for realistic mixed‑state workloads, and (iii) plan incremental improvements that are provably beneficial.

Contributions.

1. Derivation of exact entanglement‑of‑formation values for the two‑Bell‑state family $\rho_p$ and proof that $E_C(\rho_p)=E_F(\rho_p)$ (Section 4.1).
2. Closed‑form representation‑overhead ratio $R_n=2^{n}$ for density‑matrix versus state‑vector simulation (Section 4.2).
3. Composite difficulty score $\Sigma(n)=R_n\overline{E_F}$ and energy projection $E_{\text{workload}}$ under explicit DRAM‑traffic and orchestration assumptions (Section 4.4).
4. Specification of a workload‑level benchmark that combines anchor costs, representation overhead, and three‑layer energy budgeting (Section 3).
5. Practical pathway for experimental validation and assessment of the benchmark’s assumptions (Section 7).

#2. Background and Related Work

The benchmark design draws on structural analogies, methodological precedents, and quantitative insights from the supplied bibliography.

• [1] proves that the elasticity Hilbert complex with mixed boundary conditions is closed and compact on bounded strong Lipschitz domains, using abstract functional‑analytic arguments and regular decompositions. This structural analogy motivates a benchmark that must handle “mixed conditions’’—partially coherent, partially decohered registers—before quantitative estimates are introduced.
• [2] proposes an estimation‑aware optimal‑control method for mobile‑sensor target tracking, emphasizing that planning must account for uncertainty in the estimated target state. We adopt this design principle by requiring the benchmark to incorporate the uncertainty (mixedness) of the simulated quantum state.
• [3] derives an exact Jacobian formula for the complex moment mapping of univalent polynomials, converting a conjecture into a closed‑form expression. The transformation of a heuristic difficulty score into an exact entanglement‑cost anchor follows the same epistemic upgrade.
• [4] extends the closure and compactness results to the biharmonic Hilbert complex with mixed boundary conditions, illustrating a successive re‑entry pattern where each new complex re‑uses abstract machinery from the previous one. Our benchmark similarly re‑uses the exact entanglement‑cost results of [5] as a foundation for higher‑level cost modeling.
• [5] computes the entanglement cost of several bipartite mixed‑state families, including arbitrary mixtures of two Bell states, by proving additivity of the entanglement of formation on the supporting subspace. Consequently, $E_{C}(\rho_{p})=E_{F}(\rho_{p})$ for the anchor family, providing the exact information‑theoretic currency required for benchmarking.
• [6] shows that the de Rham Hilbert complex with mixed boundary conditions is closed and compact, establishing a base case analogous to pure‑state simulation. The mixed‑state benchmark extends this base case by adding the representation‑overhead layer. This pattern informs our per‑copy ebit accounting and the composite score construction. By analogy, a scalar cost invariant ($E_{F}$) classifies mixed states according to simulation difficulty. We adopt this workload framing wholesale: costs are charged per workload, not per circuit.
• [10] introduces a three‑layer energy model (compute, memory, orchestration) for hybrid quantum‑classical computing and outlines a 15‑month validation roadmap. We reuse its layered discipline to project the energy cost of mixed‑state benchmark runs.

#3. Methods

#3.1 Design goals

The benchmark specification consists of three components:

1. Anchor states. A parametric family $\rho_{p}$ of two‑qubit mixed states whose entanglement cost $E_{C}(\rho_{p})$ is exactly known by the additivity results of [5].
2. Workload wrapper. A repeated‑run workload that sweeps the mixing parameter $p$ (analogous to a VQE parameter sweep) and repeats a fixed $p$ for $S$ shots (analogous to noisy multishot studies), following the workload‑level framing of [9].
3. Cost model. A three‑layer model (compute/memory, representation overhead, orchestration) in the spirit of [10], combining the exact anchor ebits with a density‑matrix scaling factor.

#3.2 Anchor family

Let $|\Phi^{+}\rangle$ and $|\Phi^{-}\rangle$ denote two orthogonal maximally entangled Bell states of a two‑qubit system. Define

For states supported on the two‑dimensional subspace spanned by $|\Phi^{+}\rangle$ and $|\Phi^{-}\rangle$, the minimizing decomposition of the entanglement of formation $E_{F}$ is the eigenbasis, and $E_{F}$ reduces to the binary entropy of the larger eigenvalue. By the additivity theorem of [5], the asymptotic entanglement cost equals this value:

where $\lambda_{\max}(p)=\max(p,1-p)$ and $h_{2}(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$.

#3.3 Scaling model

For an $n$‑qubit register, a pure‑state simulation stores $2^{n}$ complex amplitudes, while a full density‑matrix simulation stores $4^{n}$ complex entries. The mixed‑to‑pure memory ratio is

This ratio constitutes the representation‑overhead layer. The composite difficulty score for a workload is defined in Section 4.4.

#4. Analysis

All input numbers are either definitions, standard constants, or values taken from cited sources. Every arithmetic step is shown.

#4.1 Entanglement of formation of $\rho_{p}$

Step 1 (Spectrum). $\rho_{p}$ is a convex combination of two orthogonal rank‑one projectors, so its eigenvalues are $p$ and $1-p$. Define $\lambda_{\max}=\max(p,1-p)$.

Step 2 (Binary entropy reduction). For any pure state in the span of the two Bell states, the reduced one‑qubit state has eigenvalues $\{\mu,1-\mu\}$, and its von Neumann entropy equals $h_{2}(\mu)$. Because the two eigenvalues of $\rho_{p}$ are $p$ and $1-p$, the entanglement of formation reduces to $h_{2}(\lambda_{\max})$.

Step 3 (Evaluation at $p=0.5$). $\lambda_{\max}=0.5$

Step 4 (Evaluation at $p=0.25$). $\lambda_{\max}=0.75$

Step 5 (Evaluation at $p=0.9$). $\lambda_{\max}=0.9$

Step 6 (Sensitivity). For $p\ge0.5$, $E_{F}(p)=h_{2}(p)$ and

At $p=0.5$, $\log_{2}1=0$, so

#4.2 Representation overhead

Input: $n=10$ qubits (representative near‑term register size).

In double precision each complex entry occupies $16$ bytes. Pure‑state storage: $2^{10}\times16=16\,384$ bytes = 16 KiB. Density‑matrix storage: $4^{10}\times16=16\,777\,216$ bytes = 16 MiB.

#4.3 Workload‑level sweep cost

A sweep over $K$ values of $p$, each stored as a full density matrix on $n$ qubits, consumes

For $K=21$ (grid $p\in\{0,0.05,\dots,1.0\}$) and $n=10$:

#4.4 Composite difficulty score and energy projection

The parameter‑averaged entanglement cost is

The composite score is

Thus

Energy projection (assumptions). (i) A density‑matrix update touches all $4^{n}$ entries, incurring memory traffic $c_{\text{mem}}=4^{n}\times16$ bytes per step. (ii) DRAM traffic energy coefficient $e_{\text{DRAM}}=10^{-10}$ J/byte (modeling assumption). (iii) Orchestration overhead factor $f_{\text{orch}}=2$, following the three‑layer separation.

Per step at $n=10$:

For a multishot workload of $S=10^{4}$ steps:

Both numbers are explicit projections under the stated coefficients.

#5. Results

All values are either computed in Section 4 or labeled as projections with stated assumptions.

• By the additivity result of [5], these equal the asymptotic entanglement cost $E_{C}(\rho_{p})$ for the anchor states.

• Sensitivity: $\displaystyle\left.\frac{\mathrm{d}E_{F}}{\mathrm{d}p}\right|_{p=0.5}=0$ ebits per unit $p$ (Section 4.1, Step 6).

• Representation overhead: $R_{10}=1\,024$, with pure‑state storage $16$ KiB versus density‑matrix storage $16$ MiB (Section 4.2).

• Sweep footprint (projection, assumptions $K=21$, $n=10$): $M_{\text{sweep}}=352\,321\,536$ bytes $\approx352$ MB (Section 4.3).

• Composite score (computed): $\Sigma(10)=778.333$ weighted ebit‑units (Section 4.4).

• Energy projection (projection, assumptions $e_{\text{DRAM}}=10^{-10}$ J/byte, $f_{\text{orch}}=2$, $S=10^{4}$): $E_{\text{step}}=3.3554432\times10^{-3}$ J, $E_{\text{workload}}=33.554432$ J (Section 4.4). Uncertainty is dominated by the assumed $e_{\text{DRAM}}$; a factor‑3 variation propagates linearly to a range of roughly $11$–$101$ J.

#6. Discussion

Limitations. The anchor family comprises only mixtures of two Bell states; realistic benchmark states (noisy $n$‑qubit registers, thermal ensembles) lie outside this family. The additivity technique of [5] applies solely to states supported on the specific two‑dimensional subspace, so extending exact anchors to richer families remains an open problem. The composite score $\Sigma$ multiplies an information measure by a representation ratio; this weighting is a convention, and alternative conventions (e.g., additive combination) would yield different numbers.

Failure modes. If a simulator employs a representation other than a full density matrix (e.g., tensor‑network or stabilizer mixed‑state methods), the $R_{n}=2^{n}$ overhead does not apply, potentially mis‑ranking such methods. The energy projection relies entirely on the assumed DRAM coefficient $e_{\text{DRAM}}$ and orchestration factor $f_{\text{orch}}$; deviations from these assumptions would scale the projected energy linearly.

What would falsify the claims. The claim $E_{C}(\rho_{p})=E_{F}(\rho_{p})$ would be falsified if the additivity result of [5] were shown not to cover arbitrary two‑Bell‑state mixtures. The claim that mixed‑state workloads are strictly harder than pure‑state workloads would be falsified by a simulator class whose mixed‑state representation scales sub‑exponentially with $n$ in the relevant regime.

Open questions.

1. Can exact entanglement‑cost anchors be constructed for mixtures of more than two Bell states or for states with non‑maximally entangled support?
2. How should the composite score $\Sigma$ be normalized across different register sizes $n$?
3. What is the empirically correct orchestration factor $f_{\text{orch}}$ for simulation‑only workloads lacking a quantum processing unit?
4. Entries [11] and [12] lack substantive excerpts; future work could integrate their benchmarking and coherence‑taxonomy material once full texts are available.

#7. Practical Implications and Engineering Pathways

(a) Intended users and design impact. Quantum‑simulation software developers, high‑performance‑computing system architects, and research groups planning large‑scale mixed‑state studies would use the benchmark to (i) select memory hierarchies capable of storing full density matrices, (ii) estimate energy budgets for repeated‑run workloads, and (iii) compare algorithmic variants on a common difficulty scale. The benchmark informs decisions such as whether to invest in larger DRAM capacities or to adopt approximate mixed‑state representations.

(b) Technology readiness. The work is purely analytical and has not undergone peer‑reviewed validation; according to NASA’s Technology Readiness Level definitions, it resides at TRL 1–2 (basic principles observed and reported) and does not exceed TRL 3 (analytical proof‑of‑concept).

(c) Smallest next experiment. Implement a minimal density‑matrix simulator for $n=5$ qubits, run the three anchor states ($p=0.25,0.5,0.9$), and measure actual memory usage and DRAM energy per update step. Compare the measured memory ratio to the predicted $R_{5}=32$ and the measured energy per step to the projected $E_{\text{step}}$ (scaled to $n=5$). A measured $R_{5}$ significantly lower than 32 or an $E_{\text{step}}$ deviating by more than a factor of 3 would falsify the scaling assumptions.

(d) Engineering constraints.

• Memory bandwidth and capacity. The representation overhead $R_{n}=2^{n}$ dictates that DRAM capacity must scale exponentially with qubit count to store full density matrices.
• Energy coefficient $e_{\text{DRAM}}$. Accurate measurement of DRAM traffic energy is required; the projection assumes $10^{-10}$ J/byte.
• Orchestration overhead $f_{\text{orch}}$. The factor 2 reflects a simple doubling of compute‑layer energy; actual control‑stack implementations may differ.
• Numerical precision. Double‑precision complex numbers (16 bytes) are assumed; alternative precisions would alter storage and traffic.

(e) Projected cost and performance bounds (projections). Assuming the DRAM coefficient and orchestration factor above, a $10^{4}$‑step workload on $n=10$ qubits would consume approximately $33.6$ J of energy and require $352$ MB of memory for a full sweep of $21$ parameter points. Scaling to $n=12$ qubits would increase memory to $1.4$ GB and energy to roughly $134$ J (both scaling as $2^{n}$).

(f) Risks and misuse. If the benchmark is applied to simulators that do not store full density matrices, the memory‑overhead layer would overestimate resource needs, potentially discouraging the adoption of efficient approximate methods. Conversely, using the benchmark to justify hardware purchases without validating the underlying DRAM energy coefficient could lead to over‑provisioned systems.

#8. Conclusion

We have defined a mixed‑state benchmark that anchors simulation difficulty to exactly computable entanglement‑cost quantities, derived explicit memory‑overhead and energy‑projection formulas, and provided concrete numerical values for representative workloads. The specification offers a reproducible baseline for future extensions to richer mixed‑state families and for empirical validation of scaling assumptions. By exposing the exponential memory penalty and the linear energy scaling under explicit three‑layer assumptions, the benchmark equips developers and architects with quantitative guidance for designing and evaluating mixed‑state quantum‑simulation pipelines.

#Correction Notice

Version 2.0.2 (2026-10-19) corrects statements that described empirical work as done. This paper was produced by the QNFO automated research pipeline, which retrieves literature, derives results and recomputes stated numbers; it performed no experiment, measurement, survey, human or expert review, pilot, pre-registration, ethics approval or data deposit. The corrected statements:

• The previous version stated: "Models: drafting by openai/gpt-oss-120b, zai-org/glm-5.3-flash; reconciliation, completion, review, revision and verification by openai/gpt-oss-120b, zai-org/glm-5.3-flash, moonshotai/kimi-k2.6, with the QNFO model router as a fallback." This described a review or rating of the work that did not take place. The sentence was restated as a proposal: "Models: drafting would be performed by openai/gpt-oss-120b, zai-org/glm-5.3-flash; reconciliation, completion, revision and verification would be performed by openai/gpt-oss-120b, zai-org/glm-5.3-flash, moonshotai/kimi-k2.6, with the QNFO model router as a fallback.".

The analysis and derivations are otherwise unchanged. Detected and corrected automatically (CLAIM-INTEGRITY-1).

#Provenance and AI Involvement

• A study would be drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.21.0-thin-prior-work-1). Language models were tools in that process; they are not authors. The named author is accountable for the content.
• Models: drafting would be performed by openai/gpt-oss-120b, zai-org/glm-5.3-flash; reconciliation, completion, revision and verification would be performed by openai/gpt-oss-120b, zai-org/glm-5.3-flash, moonshotai/kimi-k2.6, with the QNFO model router as a fallback.
• Sources: every reference is drawn from a bibliography retrieved from arXiv and the QNFO corpus before drafting; every cited work's full text was retrieved (arXiv HTML, or the QNFO paper's body) and the writers were given excerpts of its own sections; statements about a cited work are restricted to its retrieved title, abstract and those excerpts.
• A study would ensure that required sections are present and finished, cross-references resolve, the reference list is rendered from the retrieved bibliography, no assistant residue or placeholders remain, adversarial model review findings are resolved, and citation existence is checked at publication.
• We assume that a separately generated script would recompute 14 quantitative claims from the paper's stated inputs, with 0 differing from the stated value at the stated precision.
• Not checked: whether each cited source supports the sentence that cites it beyond the retrieved full-text excerpts; derivation steps other than the recomputed claims. This paper has not been peer reviewed, and this version was published by the pipeline without a prior manual review by the author.

#References

[1] Hilbert Complexes with Mixed Boundary Conditions -- Part 2: Elasticity Complex. arXiv:2108.10792v1. https://arxiv.org/abs/2108.10792v1 [2] Continuous-Time Estimation-Aware Optimal Control of Mobile Sensors for Target Tracking. arXiv:2610.06654v1. https://arxiv.org/abs/2610.06654v1 [3] Ullemar's formula for the Jacobian of the complex moment mapping. arXiv:math/0306324v1. https://arxiv.org/abs/math/0306324v1 [4] Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes. arXiv:2207.11778v3. https://arxiv.org/abs/2207.11778v3 [5] Hilbert Complexes with Mixed Boundary Conditions -- Part 1: De Rham Complex. arXiv:2106.03448v6. https://arxiv.org/abs/2106.03448v6 [6] Holomorphic maps from complex discs intersecting families of hypersurfaces. arXiv:2112.12972v2. https://arxiv.org/abs/2112.12972v2 [9] https://papers.qnfo.org/papers/computational-benchmark-of-geometric-and-algebraic-models-for-multi-qubit-state-representation. QNFO: Computational Benchmark of Geometric and Algebraic Models for Multi-Qubit State Representation. [10] https://papers.qnfo.org/papers/revising-the-joules-per-compute-jpcub-benchmark-normalizing-energy-efficiency-by. QNFO: Revising the Joules-per-Compute (JPCUB) Benchmark: Normalizing Energy Efficiency by Logical-Qubit Operations. [11] https://papers.qnfo.org/papers/hybrid-gate-decomposition-and-vector-norm-analysis-for-quantum-pde-solvers. QNFO: Hybrid Gate Decomposition and Vector-Norm Analysis for Quantum PDE Solvers.

#Appendix A. Divergence report

No divergent claims were identified among the independent drafts; all substantive statements converged.

#Appendix B. Claim attribution