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An Information-Minimal Geometry for Qubit-Efficient Optimization

Gordon Ma, Dimitris G. Angelakis

TL;DR

This work addresses the width bottleneck in near-term quantum optimization by embedding a quadratic objective in an information-minimal two-body geometry. It builds a log-width quantum circuit that generates pairwise moments, then explicitly projects these moments onto the SA(2) polytope via a $\rho$-damped IPF step and decodes with a maximum-entropy Ising model using Gibbs sampling. The combination—explicit SA(2) anchoring, differentiable feasibility repair, and a principled probabilistic decoder—yields a complete, end-to-end pipeline with $\tilde{O}(\log N)$ qubits that achieves near-optimal ratios on large Max-Cut graphs (GSET) at shallow depth, outperforming SA(2)-based baselines. The work reframes quantum advantage in optimization as a question of information geometry: up to a point, a minimal two-body representation suffices, and quantum structure becomes essential only when the underlying geometry bends beyond SA(2) into spectrahedral, curvature-driven spaces. This provides a clean baseline and a roadmap for extending qubit-efficient methods toward higher-order relaxations and curved geometries where genuine quantum coherence is required.

Abstract

Qubit-efficient optimization seeks to represent an $N$-variable combinatorial problem within a Hilbert space smaller than $2^N$, using only as much quantum structure as the objective itself requires. Quadratic unconstrained binary optimization (QUBO) problems, for example, depend only on pairwise information -- expectations and correlations between binary variables -- yet standard quantum circuits explore exponentially large state spaces. We recast qubit-efficient optimization as a geometry problem: the minimal representation should match the $O(N^2)$ structure of quadratic objectives. The key insight is that the local-consistency problem -- ensuring that pairwise marginals correspond to a realizable global distribution -- coincides exactly with the Sherali-Adams level-2 polytope $\mathrm{SA}(2)$, the tightest convex relaxation expressible at the two-body level. Previous qubit-efficient approaches enforced this consistency only implicitly. Here we make it explicit: (a) anchoring learning to the $\mathrm{SA}(2)$ geometry, (b) projecting via a differentiable iterative-proportional-fitting (IPF) step, and (c) decoding through a maximum-entropy Gibbs sampler. This yields a logarithmic-width pipeline ($2\lceil\log_2 N\rceil + 2$ qubits) that is classically simulable yet achieves strong empirical performance. On Gset Max-Cut instances (N=800--2000), depth-2--3 circuits reach near-optimal ratios ($r^* \approx 0.99$), surpassing direct $\mathrm{SA}(2)$ baselines. The framework resolves the local-consistency gap by giving it a concrete convex geometry and a minimal differentiable projection, establishing a clean polyhedral baseline. Extending beyond $\mathrm{SA}(2)$ naturally leads to spectrahedral geometries, where curvature encodes global coherence and genuine quantum structure becomes necessary.

An Information-Minimal Geometry for Qubit-Efficient Optimization

TL;DR

This work addresses the width bottleneck in near-term quantum optimization by embedding a quadratic objective in an information-minimal two-body geometry. It builds a log-width quantum circuit that generates pairwise moments, then explicitly projects these moments onto the SA(2) polytope via a -damped IPF step and decodes with a maximum-entropy Ising model using Gibbs sampling. The combination—explicit SA(2) anchoring, differentiable feasibility repair, and a principled probabilistic decoder—yields a complete, end-to-end pipeline with qubits that achieves near-optimal ratios on large Max-Cut graphs (GSET) at shallow depth, outperforming SA(2)-based baselines. The work reframes quantum advantage in optimization as a question of information geometry: up to a point, a minimal two-body representation suffices, and quantum structure becomes essential only when the underlying geometry bends beyond SA(2) into spectrahedral, curvature-driven spaces. This provides a clean baseline and a roadmap for extending qubit-efficient methods toward higher-order relaxations and curved geometries where genuine quantum coherence is required.

Abstract

Qubit-efficient optimization seeks to represent an -variable combinatorial problem within a Hilbert space smaller than , using only as much quantum structure as the objective itself requires. Quadratic unconstrained binary optimization (QUBO) problems, for example, depend only on pairwise information -- expectations and correlations between binary variables -- yet standard quantum circuits explore exponentially large state spaces. We recast qubit-efficient optimization as a geometry problem: the minimal representation should match the structure of quadratic objectives. The key insight is that the local-consistency problem -- ensuring that pairwise marginals correspond to a realizable global distribution -- coincides exactly with the Sherali-Adams level-2 polytope , the tightest convex relaxation expressible at the two-body level. Previous qubit-efficient approaches enforced this consistency only implicitly. Here we make it explicit: (a) anchoring learning to the geometry, (b) projecting via a differentiable iterative-proportional-fitting (IPF) step, and (c) decoding through a maximum-entropy Gibbs sampler. This yields a logarithmic-width pipeline ( qubits) that is classically simulable yet achieves strong empirical performance. On Gset Max-Cut instances (N=800--2000), depth-2--3 circuits reach near-optimal ratios (), surpassing direct baselines. The framework resolves the local-consistency gap by giving it a concrete convex geometry and a minimal differentiable projection, establishing a clean polyhedral baseline. Extending beyond naturally leads to spectrahedral geometries, where curvature encodes global coherence and genuine quantum structure becomes necessary.

Paper Structure

This paper contains 94 sections, 2 theorems, 54 equations, 9 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The qubit bottleneck.
  3. From expressivity to efficiency.
  4. Qubit-efficient compression schemes.
  5. This work: geometric representation anchored to pairwise consistency.
  6. The local view.
  7. Operationalizing Sherali--Adams.
  8. Resolving the local-consistency gap.
  9. Our contributions.
  10. Empirical low-depth performance.
  11. Roadmap -- the geometric transport.
  12. Our Approach: the Informational-Minimal Two-Body Framework
  13. From Parameters to Pseudo-moments
  14. From Pseudo-Moments Towards Locally Feasible Moments
  15. A local repair for global feasibility.
  16. ...and 79 more sections

Key Result

Lemma B.1

Lemma (pairwise table reconstruction). Given $(\mu_i,\mu_j,\nu_{ij})$ with $0\le \mu_i,\mu_j\le 1$, define a $2\times 2$ table Then $P_{ij}$ is a valid joint distribution (nonnegative entries summing to $1$) with $\sum_{x_j}P_{ij}(1,x_j)=\mu_i$ and $\sum_{x_i}P_{ij}(x_i,1)=\mu_j$if and only if the Boole–Fréchet bounds Eq. eq:bf-mu-nu hold. Proof. ($\Rightarrow$) Nonnegativity of each entry implie

Figures (9)

  • Figure 1: The information-minimal two-body framework as geometric transport. The framework treats optimization as a sequence of geometric maps, each conserving information while changing representation. (Left) A quadratic objective $E_Q=\mathbb{E}[x^\top Qx]$ depends only on the one- and two-body moments of its variables. A log-width quantum circuit of $n_q = 2\lceil\log_2 N\rceil {+} 2$ qubits produces a Born distribution $p_\theta(z)$ whose expected local statistics $(\hat{\mu},\hat{\nu})$ define the pseudo-moment space. (Center) A $\rho$-damped, KL-regulated iterative-proportional-fitting (IPF) step softly projects these moments toward the Sherali--Adams level-2 polytope SA(2)---the convex body of locally consistent marginals---yielding near-feasible $(\tilde{\mu},\tilde{\nu})$. The information distance $D_{\mathrm{KL}}((\tilde{\mu},\tilde{\nu})\Vert(\hat{\mu},\hat{\nu}))$ enters the loss, guiding training toward natural feasibility. (Right) A maximum-entropy mapping converts the repaired moments into an Ising surrogate with fields $(h,J)$; Gibbs sampling (dotted arrow) realizes discrete bit-strings $x$ from distribution implied by the learned moments. The lower panel summarizes this information–geometric transport from parameters to bit-strings— linking the variational circuit, convex relaxation, and probabilistic decoding within one minimal representation.
  • Figure 2: Pairwise moment geometry. By a "$2{\times}2$ table" we mean the classical Venn-diagram representation of two Bernoulli events. For binary variables $X_i,X_j\!\in\!\{0,1\}$, $\mu_i=\Pr(A)=\mathbb{E}[X_i]$, $\mu_j=\Pr(B)=\mathbb{E}[X_j]$, and $\nu_{ij}=\Pr(A\cap B)=\mathbb{E}[X_iX_j]$ correspond to the regions shown. Nonnegativity of all four areas—$p^{ij}_{10} =(A\setminus B)$, $p^{ij}_{01} =(B\setminus A)$, $p^{ij}_{11} = (A\cap B)$, and its complement, $p^{ij}_{00}$—yields the inclusion–exclusion (Boole–Fréchet) bounds $\max\{0,\mu_i+\mu_j-1\}\le\nu_{ij}\le\min\{\mu_i,\mu_j\}$.
  • Figure 3: Training dynamics and incumbent ratios on Gset.Left: Representative instance G23 ($N{=}2000$). Orange: training objective $E_{\mathrm{QUBO}}(\tilde{\mu},\tilde{\nu})$ (plotted as $-E_{\mathrm{QUBO}}$ so higher values correspond to better cuts). Purple dots: decoded samples at sampling epochs. Purple line: incumbent curve (running best). Gray: BM-MS/BM-SS. Red: Gurobi (10 min). Blue outline: SA(2)$\!\to$ Gibbs. Horizontal dashed lines: relaxation optimum from the $\mathrm{SA}(2)$--LP and the best-known cut obtained by the Breakout Local Search (BLS) heuristic benlic_breakout_2013. Shaded band: $\approx0.99$ region. (Specs: $n_q{=}24$, $D{=}2$, 46 two-qubit gates.) Right: Incumbent approximation ratios $r^\star$ across Gset ($N{=}800$--$2000$) at the median-optimal depth $D^\ast$ ($D^\ast{=}2$; G35: $D^\ast{=}3$). Gray: BM. Red: Gurobi. Blue outline: SA(2)$\!\to$ Gibbs. Purple: our solver, Two-Body (@$D^\ast$). Vertical rules group instances by $N$. Shaded band: constant-depth plateau. All results use the frozen policy ($\rho{=}0.5$, $\lambda_{\mathrm{KL}}\!:\!0.10{\rightarrow}0.30$); only depth varies on Gset. Gibbs decode only; no local search or classical post-processing is applied.
  • Figure 4: Depth–accuracy and projection–strength effects on Erdos Renyi random graphs. (a) Gap $(1{-}r)$ versus depth $D$ for representative $\rho$ values. Projected runs cross the PCP bound ($1/17{\approx}0.059$) at $D{\le}2$; the unprojected control catches up only at large $D$. (b) Gap versus $\rho$, aggregated across depth bands, showing the characteristic U-shape with a broad optimum near 0.5. (c) At $\rho{=}0.5$, decoded (solid) and training (dashed) objectives converge with depth, marking the transition to self-consistent geometry.
  • Figure 5: Depth–projection trade-offs across graph densities ($N{=}128$). Each column corresponds to one density family (tree-like, sparse, denser; $\alpha{=}1,\ 4\!-\!5,\ 8\!-\!12$). Top: gap $1{-}r$ versus circuit depth $D$ for projection strengths $\rho\!\in[0,1]$; bottom: gap versus projection strength for fixed depths. Each curve aggregates five random seeds per configuration. Trends mirror those in Fig. \ref{['fig:flagship_depth_vs_gap']}: under-relaxed IPF ($\rho\!\approx\!0.3\!-\!0.5$) consistently yields the smallest gaps, with a characteristic U-shaped dependence on $\rho$. Higher-degree graphs display a sharper “knee’’ near $\rho{=}0.5$ and slightly noisier convergence, but the same qualitative pattern—diminishing sensitivity at larger depths once the ansatz saturates. These dynamics guided the choice of $\rho{=}0.5$ for dense instances such as GSET.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma B.1: Pairwise table reconstruction
  • Proposition B.1