Toward Generative Quantum Utility via Correlation-Complexity Map

TL;DR

A Correlation-Complexity Map is proposed as a practical diagnostic tool for determining when real-world data distributions are structurally aligned with IQP-type quantum generative models, supporting the use of QCLI/CCI as practical indicators for locating IQP-aligned domains and advancing generative quantum utility.

Abstract

We propose a Correlation-Complexity Map as a practical diagnostic tool for determining when real-world data distributions are structurally aligned with IQP-type quantum generative models. Characterized by two complementary indicators: (i) a Quantum Correlation-Likeness Indicator (QCLI), computed from the dataset's correlation-order (Walsh-Hadamard/Fourier) power spectrum aggregated by interaction order and quantified via Jensen-Shannon divergence from an i.i.d. binomial reference; and (ii) a Classical Correlation-Complexity Indicator (CCI), defined as the fraction of total correlation not captured by the optimal Chow-Liu tree approximation, normalized by total correlation. We provide theoretical support by relating QCLI to a support-mismatch mechanism, for fixed-architecture IQP families trained with an MMD objective, higher QCLI implies a smaller irreducible approximation floor. Using the map, we identify the classical turbulence data as both IQP-compatible and classically complex (high QCLI/high CCI). Guided by this placement, we use an invertible float-to-bitstring representation and a latent-parameter adaptation scheme that reuses a compact IQP circuit over a temporal sequence by learning and interpolating a low-dimensional latent trajectory. In comparative evaluations against classical models such as Restricted Boltzmann Machine (RBM) and Deep Convolutional Generative Adversarial Networks (DCGAN), the IQP approach achieves competitive distributional alignment while using substantially fewer training snapshots and a small latent block, supporting the use of QCLI/CCI as practical indicators for locating IQP-aligned domains and advancing generative quantum utility.

Figures (8)

• Figure 1: Overview of the proposed correlation indicators and the resulting correlation--complexity map. (a) Quantum Correlation--Likeness Indicator (QCLI). Given a dataset of $n$-bit samples, we compute its correlation-order spectrum by aggregating Walsh--Hadamard (Fourier--Walsh) power over subset sizes $k=|s|$. QCLI, $I_{\text{QCLI}}$, is defined as the Jensen--Shannon divergence between this empirical order spectrum and the binomial reference spectrum induced by the uniform (classically random) distribution. (b) Classical Correlation--Complexity Indicator (CCI). For the same dataset, we compute the total correlation (multi-information) and the dependence captured by the optimal Chow--Liu tree, obtained by maximizing $\sum_{(i,j)\in T} I(X_i;X_j)$ over spanning trees $T$. CCI, $I_{\text{CCI}}$, is the fraction of total correlation not explained by the optimal tree-structured model. (c) Correlation--Complexity Map. Datasets are shown in the $(I_{\text{QCLI}}, I_{\text{CCI}})$ plane, separating regimes that are locally structured and classically expressible (low $I_{\text{CCI}}$), quantum-like but classically easy, high-order but non-IQP-aligned, and the high-$I_{\text{QCLI}}$/high-$I_{\text{CCI}}$ regime that is most compatible with IQP-type generative inductive biases. The turbulence dataset is highlighted as residing in this IQP-compatible quadrant, motivating the IQP-based generative modeling experiments. (d) Latent adaptation for temporal turbulence synthesis. Starting from a trained IQP generator, we fix a shared core parameter block and assign (adapt) a low-dimensional latent block to represent different time snapshots. Interpolating or extrapolating the learned latent trajectory enables generation of unseen turbulence snapshots while keeping the circuit architecture and core parameters fixed.
• Figure 2: Correlation-order spectra and $I_{\text{QCLI}}$ on D-Wave 100Q datasets scriva2023accelerating. Left: the empirical correlation-order spectrum $m_k$ (spectral mass aggregated over Walsh–Hadamard parities of order $|s|=k$) for three D-Wave 100Q datasets collected at anneal times $\tau\in\{1,10,100\}\,\mu\text{s}$, with the corresponding $I_{\text{QCLI}}$ shown in the titles. Right: the absolute deviation $|m_k-b_k|$ from the i.i.d. ideal binomial baseline spectrum $b_k$. As $\tau$ increases, $I_{\text{QCLI}}$ rises and the spectra become visibly more structured, exhibiting sharper peaks and pronounced cancellations across correlation orders, consistent with stronger constructive/destructive interference patterns.
• Figure 3: Empirical probe of the QCLI–MMD support-mismatch mechanism. Scatter plots show the MMD loss achieved when fitting a fixed-architecture IQP learner to a collection of IQP-generated datasets spanning a wide range of $I_{\text{QCLI}}$ values, under a fixed training budget. Each panel corresponds to a different learner expressivity setting (gate count $G_n\in\{50,100,150\}$, with restricted locality), thereby enforcing a controlled architecture–data mismatch. Dashed curves indicate the estimated upper and lower envelopes of the observed MMD as a function of $I_{\text{QCLI}}$. Details of the envelope estimation are provided in Appendix \ref{['app:envelopes']}.
• Figure 4: $I_{\text{QCLI}}$ versus $I_{\text{CCI}}$ for locality-2 IQP circuits with 8Q, 12Q, and 16Q, each using 150 random gates. Circuit parameters are optimized solely to maximize $I_{\text{QCLI}}$, yet higher $I_{\text{QCLI}}$ consistently coincides with elevated $I_{\text{CCI}}$ across all system sizes, indicating an emergent coupling between IQP representational alignment and global multivariate classical correlations.
• Figure 5: Correlation–Complexity Map. Each dataset is positioned by $(I_{\text{QCLI}}, I_{\text{CCI}})$, measuring (x-axis) parity-structured deviation of the correlation-order spectrum from an i.i.d. binomial baseline (QCLI) and (y-axis) the fraction of total dependence not captured by the optimal Chow--Liu tree ($I_{\text{CCI}}$). The shaded region shows an empirical IQP envelope obtained by optimizing $16$-qubit IQP circuits for high $I_{\text{QCLI}}$ and $I_{\text{CCI}}$. Datasets in the upper-right quadrant exhibit both interference-like spectral structure and beyond-pairwise dependence, forming the regime most structurally compatible with IQP-type generators. Point colors indicate data provenance: orange denotes datasets produced by classical processes (e.g., simulations or classical benchmarks), whereas teal denotes datasets generated by quantum processes or quantum hardware.

Theorems & Definitions (5)

• Definition 2.1: Order-$d$ IQP circuit
• Theorem 2.2: Lower Bounding the MMD Loss by $I_{\text{QCLI}}$
• proof : Proof.
• Theorem H.1: Hardness preservation under efficient invertible maps
• proof