Quantum Relaxation-Based Policies

• Quantum relaxation-based policies are quantum algorithms that use QRACs to encode multiple classical bits into a single qubit, achieving space efficiency with non-diagonal, entangling Hamiltonians.
• They facilitate variational and sampling-based optimization by applying non-diagonal Hamiltonian constructions and effective rounding techniques to solve combinatorial problems.
• These policies balance QRAC compression ratios with approximation quality, delivering promising benchmarks in applications like MaxCut and TSP through enhanced entanglement and reset-induced acceleration.

Quantum relaxation-based policies constitute a class of quantum algorithms and control protocols that leverage quantum relaxation, typically via quantum random access codes (QRACs), to encode multiple classical variables per qubit and form non-diagonal, entangling Hamiltonians. These policies are designed to maximize space efficiency and exploit quantum effects for optimization, control of Markovian open systems, and sampling-based solution of classical combinatorial problems. They play a prominent role in variational and sampling-based algorithms for near-term quantum (NISQ) devices, as well as in quantum-enhanced Branch-and-Bound frameworks and reset-induced dynamical acceleration.

1. Quantum Relaxation: Definitions and Underlying Formalism

Quantum relaxation (QR) refers to a family of encodings and algorithmic relaxations in which $k$ classical binary variables are encoded into a single qubit, most commonly using a $(k,1,p)$-QRAC. In the canonical $(3,1,p)$-QRAC, three bits $x_1,x_2,x_3 \in \{0,1\}$ are mapped to a single-qubit density operator: $$\rho(x_1, x_2, x_3) = \frac{1}{2}\left(I + \frac{1}{\sqrt{3}}\left( (-1)^{x_1}X + (-1)^{x_2}Y + (-1)^{x_3}Z \right) \right),$$ where $X,Y,Z$ are Pauli matrices. Decoding any bit is performed by measurement in the corresponding Pauli basis, with bit recovery probability $$p = \frac{1}{2}(1 + \frac{1}{\sqrt{3}}) \approx 0.79$$ for $(3,1,p)$-QRAC, $p \approx 0.85$ for $(2,1,p)$-QRAC, and $p \approx 0.908$ for the $(3,2,p)$-QRAC that encodes three bits into two qubits (Matsuyama et al., 17 Apr 2025, Teramoto et al., 2023).

The motivation arises from qubit-limited NISQ hardware: QRAC-based QR reduces the number of qubits by a factor up to three, at the cost of introducing off-diagonal Hamiltonian terms, since the decoded observables are no longer diagonal.

2. Hamiltonian Construction and Policy Mechanisms

For unconstrained binary quadratic optimization problems (QUBO), standard mapping yields an Ising Hamiltonian diagonal in the computational basis: $$H_{\text{Ising}} = \sum_{(i,j) \in E} J_{ij}Z_iZ_j + \sum_i h_i Z_i.$$ Quantum relaxation, via QRAC, replaces each bit $s_i$ by a Pauli operator $P_i \in \{X_i, Y_i, Z_i\}$ (for $(3,1)$-QRAC), leading to the quantum-relaxed Hamiltonian: $$H_{\text{QR}} = -\sum_{(i,j) \in E} \frac{1}{2}\left(1 - 3P_iP_j\right),$$ where the factor of 3 normalizes the Bloch vector. The resulting $H_{\text{QR}}$ is non-diagonal and generally entangling. The ground-state energy of $H_{\text{QR}}$ provides a relaxation lower bound on the classical optimum, i.e. $\min_{\rho} \Tr[H_{\text{QR}} \, \rho ] \leq \operatorname{OPT}_{\text{classical}}$ (Matsuyama et al., 17 Apr 2025, Matsuyama et al., 2024).

Encoding with QRACs of varying compression ratios yields a fundamental trade-off:

• $(2,1)$-QRAC: 2× space compression, approximation ratio bound $5/8$;
• $(3,1)$-QRAC: 3× compression, approximation ratio bound $5/9$;
• $(3,2)$-QRAC: 1.5× compression, approximation ratio bound $13/18$ (Teramoto et al., 2023).

3. Variational and Sampling-Based Optimization Algorithms

Two major algorithmic paradigms instantiate quantum relaxation-based policies: a) Variational Quantum Optimization (VQA/QRAO):

Variational algorithms (e.g., Quantum Random Access Optimizer, QRAO) use parameterized quantum circuits with QRAC-based non-diagonal Hamiltonians. Rounding to a classical solution is performed by measuring Pauli expectations (Pauli rounding) or by “magic-state” basis selection, which mimics average depolarizing channels (Teramoto et al., 2023).

b) Sampling-Based Quantum Optimization (SQOA-QR):

SQOA-QR (Sampling-based Quantum Optimization Algorithm with Quantum Relaxation) uses a $p$-layer Quantum Alternating Operator Ansatz (QAOAnsatz) circuit with a phase-separation unitary $U_C(\gamma) = \exp(-i\gamma H_{\text{QR}})$ and a single-Pauli mixer. Parameters are fixed a priori via a linear policy (LINXFER), enabling transfer from small to large problem instances without retraining. After $M$ circuit executions, the $R$ most frequent bitstrings define a subspace, within which $H_{\text{QR}}$ is classically diagonalized (“quantum-classical split”). The final solution is decoded by Pauli rounding (Matsuyama et al., 17 Apr 2025).

The general workflow is illustrated below:

Step	QR-Relaxed VQA (QRAO)	Sampling-Based Optimization (SQOA-QR)
Hamiltonian	Non-diagonal $H_\text{QR}$ via QRAC	Non-diagonal $H_\text{QR}$ via QRAC
Parameters	Trained per instance (VQA)	Fixed (LINXFER: 4 parameters)
Solution recovery	Pauli/magic rounding	Pauli rounding after subspace diag.
Scalability	Space-efficient (3-1 QRAC)	Up to 40 qubits/$\sim$120 bits (3-1 QRAC)

4. Quantum Relaxation in Branch-and-Bound Frameworks

QR-based relaxations can be directly integrated as lower bounds in classical Branch-and-Bound (BnB) frameworks. In the Quantum Relaxation-based Branch-and-Bound (QR-BnB) method, at each node:

• The current subproblem is encoded as a QRAC-relaxed Hamiltonian;
• Ground energy or variational minimum is computed (via exact diagonalization or VQE);
• The solution candidate is reconstructed using Pauli expectations;
• Variable selection for branching exploits expectation values: “least-fractional” variables (closest to classical) yield the fastest convergence;
• For constraint-saturated problems (e.g., TSP with one-hot constraints), “onehot” branching dramatically reduces tree depth (Matsuyama et al., 2024).

QR-BnB experiments on MaxCut and TSP showed that Pauli-expectation-based variable selection cut relaxation calls by 30–50% versus random selection, and onehot constraint-aware branching yielded over threefold speed-up. The quantumness gap, defined as $$(\operatorname{OPT} - \tilde z_{\text{QR}})/\operatorname{OPT}$$, corresponds to the relaxation tightness and governs BnB performance.

5. Theoretical Performance Guarantees and Space–Quality Trade-offs

The maximum expected classical cut value recovered from QRAC-based relaxations admits rigorous bounds. For MaxCut on $G=(V,E)$ with $m$ edges:

• $(3,1)$-QRAC: expected approximation ratio at least $5/9 \approx 0.555$, due to depolarizing “magic rounding” channels;
• $(2,1)$-QRAC: bound is $5/8 = 0.625$;
• $(3,2)$-QRAC: improved to $13/18 \approx 0.722$ (Teramoto et al., 2023).

These bounds hold under the condition that the expected energy under the relaxed state equals or exceeds the combinatorial optimum; otherwise, performance is governed by the gap between quantum and classical maxima.

A conjectured general bound states that for an $(m,n)$-QRAC one can expect $E[\gamma] \approx \frac{1}{2}(1 + (n/m)^2)$. Space efficiency is maximized at the cost of precision: higher compression lowers the expected cut ratio below that of the Goemans-Williamson classical SDP result ($\alpha_{\text{GW}}\approx0.878$), unless additional subspace diagonalization or hybridization is applied (Matsuyama et al., 17 Apr 2025).

6. Entanglement and Quantum Effectiveness

Quantum relaxation-based optimization algorithms differ from diagonal Ising encodings (e.g., QAOA) in requiring and leveraging entanglement for best performance. The relaxed Hamiltonian’s maximally excited eigenstate may be entangled, and the entanglement gap $$\Delta_{\mathrm{ent}} = E_{\mathrm{glob}} - E_{\mathrm{sep}} > 0$$ certifies that product-state ansatzes are strictly suboptimal.

Empirical studies showed that:

• Inclusion of CNOT layers (entangling gates) in the QRAO ansatz increases both the coverage of optimal solutions and the normalized relaxed energy;
• Approximation ratios after Pauli rounding can reach $\gamma \in [0.85,1.0]$ for entangled ansatzes, compared to $[0.7,0.9]$ for unentangled;
• Space savings enable solution of larger problems on NISQ hardware, with $(3,1)$-QRAC encoding fit for 48-bit MaxCut instances on 16 qubits (Teramoto et al., 2023).

7. Reset-Induced Quantum Relaxation Acceleration

Quantum reset protocols provide a policy class for exponential acceleration of relaxation in open quantum systems. The dynamical equation

$$\dot\rho = \mathcal{L}[\rho] + r(\mathrm{Tr}(\rho)\rho_\delta - \rho)$$

where $\mathcal{L}$ is a Lindbladian generator and $\rho_\delta$ is a reset target, yields a rigid spectral shift of nonstationary eigenmodes: $\lambda_k(r) = \lambda_k - r$ for $k\geq 2$.

By choosing appropriate reset rates and durations, one can eliminate the slowest relaxation mode, accelerating convergence by orders of magnitude (limited by the next-slowest mode), and realize quantum analogues of the Mpemba effect (initialization farther from equilibrium may relax faster). Applications include rapid qubit reset, engineering faster state preparation, and shortening thermalization in quantum thermodynamics (Bao et al., 2022).

8. Extensions, Open Questions, and Outlook

Quantum relaxation-based policies extend to arbitrary QUBO and constrained optimization via suitable non-diagonal Hamiltonian design. Outstanding research directions include:

• Hardware-efficient realization of strongly non-diagonal unitaries on NISQ devices;
• Optimal mixer Hamiltonians for faster convergence in QAOAnsatz and related circuits;
• Formal bounds relating subspace diagonalization cost and quantum depth in hybrid algorithms;
• Extensions to alternative encodings (e.g., SIC-POVM-based relaxations) and potential quantum complexity barriers analogous to Unique Games Conjecture in classical complexity (Matsuyama et al., 17 Apr 2025, Teramoto et al., 2023).

Theoretical and empirical analysis converge on the utility of quantum relaxation-based policies for space-efficient, approximation-guaranteed, and entanglement-enabled quantum optimization and control, with a spectrum of algorithmic and hardware-level policy frameworks adaptable to both NISQ and open-system dynamical settings.