Empirical Quantum Advantage in Constrained Optimization from Encoded Unitary Designs

TL;DR

This work advances constrained quantum optimization by embedding the problem into a block-wise one-hot space and using a depth-efficient, ancilla-free encoder together with a constant-gap XY mixer to form Constraint-Enhanced QAOA (CE-QAOA). By establishing an exact unitary-1-design via block permutation twirls and an ε-approximate unitary-2-design through per-block XY layers plus cross-block diagonal entanglers, the authors derive moment-based guarantees and anticoncentration results that support a polynomial-shot, instance-robust optimization framework. The Polynomial-time Hybrid Quantum–Classical (PHQC) solver combines fixed-parameter quantum sampling with a deterministic classical checker, yielding a guaranteed optimum detection with a shot complexity that scales polylogarithmically with confidence. Empirical results on TSP-like problems from QOPTLib show global optima recovery at depth $p=1$ with relatively small shot budgets, and the analysis reveals how permutation twirls and heavy outputs can yield significant practical speedups over classical baselines. Overall, the paper provides a pathway to unconditional quantum advantage for constrained combinatorial problems via encoded unitary designs and problem-algorithm co-design, with concrete guidance for hardware-aware implementations.

Abstract

We introduce the Constraint-Enhanced Quantum Approximate Optimization Algorithm (CE-QAOA), a shallow, constraint-aware ansatz that operates inside the one-hot product space of size [n]^m, where m is the number of blocks and each block is initialized with an n-qubit W_n state. We give an ancilla-free, depth-optimal encoder that prepares a W_n state using n-1 two-qubit rotations per block, and a two-local XY mixer restricted to the same block of n qubits with a constant spectral gap. Algorithmically, we wrap constant-depth sampling with a deterministic classical checker to obtain a polynomial-time hybrid quantum-classical solver (PHQC) that returns the best observed feasible solution in O(S n^2) time, where S is the number of shots. We obtain two advantages. First, when CE-QAOA fixes r >= 1 locations different from the start city, we achieve a Theta(n^r) reduction in shot complexity even against a classical sampler that draws uniformly from the feasible set. Second, against a classical baseline restricted to raw bitstring sampling, we show an exp(Theta(n^2)) separation in the minimax sense. In noiseless circuit simulations of TSP instances ranging from 4 to 10 locations from the QOPTLib benchmark library, we recover the global optimum at depth p = 1 using polynomial shot budgets and coarse parameter grids defined by the problem sizes.

Figures (5)

• Figure 1: Histogram produced in QiskitQiskit2023 for a TSP problem on 3 locations. Each of the $3^{3}=27$ block–one-hot bit-strings carries probability $1/27$ verifying that the state–preparation algorithm populates all basis states with equal amplitude while leaving probability mass strictly inside the one-hot subspace. The six strings whose block indices form a permutation are coloured green.
• Figure 2: Depth-$p=3$ CE-QAOA for $m=3$ blocks of $n=4$ qubits. Each layer applies a global cost $U_C(\gamma_\ell)$ over all $mn$ wires, followed by parallel block-local XY mixers $U_M^{(j)}(\beta_\ell)$.
• Figure 3: Heavy outputs vs. baselines. The red dotted line is the twirling baseline of $\mathbb{E}\,|\langle x|U|\phi\rangle|^2=1/D$. The light yellow region denotes $\mathbb{E}\,|\langle x|U|\phi\rangle|^2>c/D$ existence results from Thm. \ref{['thm:exist-params']} and Cor. \ref{['cor:feasible-optimum']}. The green region indicate $\mathbb{E}\,|\langle x|U|\phi\rangle|^2 \approx n^{-k}$ for a fixed constant $k$. Here we write the constant numerator in Thm. \ref{['thm:exist-params']}$c$ as $c(n)$ to emphasize that it is instance dependent. It becomes large when constructive interference arises in absence of the permutation twirling. These higher values overlap with $n^{-k}$ region shaded in green. If the probability of the optimum approaches $n^{-k}$, then to achieve success probability $\ge 1-\delta$ it suffices that $S\ \ge\ \frac{\ln(1/\delta)}{p_\star}$ as outline in Cor. \ref{['cor:shots']}. Our simulation results with $\ln(1/\delta)=10$ is reported in Fig. \ref{['fig:heavy-outputs']} where the instance dependent heavy outputs lead to overlap probabilities in the shaded green region.
• Figure 4: Single Layer CE-QAOA circuits on noiseless Aer simulator Qiskit2023 on two TSP instances ( wi5 (25 qubits) and wi6 (36 qubits)) taken from QOptLib benchmark set Osaba2024Qoptlib. Histograms show the counts of feasible bit-strings (permutation indices) out of $50\times n^{4}$ shots for a problem of size $n$ locations. The green bars mark two degenerate best tours. Several grid points already identify the optimum with nontrivial probability well above the lower bound. The dashed line marks the $1/D$ baseline with $D=n^n$. The purple bars show when the best bitsring is also the most frequent.
• Figure 5: Heavy outputs vs. finite-shot thresholds. Shaded green shows the finite-shot region $p \ge \tfrac{1}{10n^3}$. Theorem-level guarantees (yellow boundary in schematic figures) place us above this curve, while instance-dependent heavy outputs (blue points/curve) push into the finite-shot regime Cf. \ref{['fig:heavy']}.

Theorems & Definitions (41)

• Theorem 1: Gate–count optimality on a line
• proof
• Remark 2: Hardware-friendly alternatives
• Proposition 3: ${W_n}$ is non-Clifford bravyi_gosset_2016
• Proposition 4: Spectral gap of one-block XY mixer
• proof
• Proposition 5: Ergodicity of the angle-averaged XY mixer on $\mathcal{H}_1$
• proof
• Definition 6: Encoding isometry (qubit $\leftrightarrow$ qudit)
• Proposition 7: Invariance and quditization of the block–XY mixer