Warm-Starting Pauli Correlation Encoding

• The paper introduces Warm-PCE, which integrates a GW-derived bias into the variational loss to achieve polynomial qubit compression and delay barren plateau onset.
• Warm-PCE leverages Pauli Correlation Encoding to efficiently capture multi-qubit correlations, enabling robust application to problems like MaxCut and TSP.
• Empirical benchmarks on 5-city TSP instances show that Warm-PCE significantly improves success rates and approximation ratios compared to standard PCE across varying circuit depths.

Warm-Starting Pauli Correlation Encoding (Warm-PCE) is an advanced variational quantum algorithmic technique designed to enhance the efficiency and performance of quantum combinatorial optimization, particularly on near-term hardware. Warm-PCE extends Pauli Correlation Encoding (PCE) by incorporating a classical bias derived from the Goemans–Williamson (GW) randomized rounding algorithm directly into the variational loss function. This method achieves polynomial reductions in qubit requirements, mitigates barren plateau issues for deep quantum circuits, and demonstrates strong empirical gains on problems such as MaxCut and the Traveling Salesman Problem (TSP) (Carmo et al., 17 Sep 2025).

1. Principles of Pauli Correlation Encoding

PCE encodes classical binary variables $x_i\in\{-1,1\}$ via the expectation values (correlators) of multi-qubit Pauli strings. Given an $n$-qubit parameterized quantum state $\lvert\Psi(\theta)\rangle$ and a set of Pauli strings $$\Pi_i=P_{i,1}\otimes P_{i,2}\otimes\cdots\otimes P_{i,n}$$ ($P_{i,k}\in\{I,X,Y,Z\}$, $i=1,\dots,m$), the encoding is

$s_i(\theta) = \langle\Psi(\theta)|\Pi_i|\Psi(\theta)\rangle,\quad x_i = \sgn(s_i(\theta)).$

Pairwise correlations—such as those needed in MaxCut—are captured by Pauli strings like $\Pi_{(i,j)}=Z_i\otimes Z_j\otimes I^{\otimes(n-2)}$.

Relative to standard one-hot quantum encodings, which require one qubit per binary variable, PCE achieves polynomial qubit compression. For $n$ qubits, the number of representable independent correlations is $m\le 4^n-1$. For two-body correlators ($k=2$), this yields a quadratic bit-to-qubit compression. This feature also delays the onset of barren plateaus: for $k=2$ PCE the critical circuit depth at which gradients vanish exponentially shifts to $\mathcal O(n\log n)$ rather than $\mathcal O(n)$, as reported in Sciorilli et al., 2025.

2. Warm-Start Extension: Goemans–Williamson Bias

Warm-PCE augments PCE by embedding information from a classical GW SDP solution into the variational loss. For each variable, the GW randomized rounding yields a bit $c^\star_i\in\{0,1\}$, which is regularized to $\widehat c_i^\star$ within $[\varepsilon,1-\varepsilon]$ to prevent saturation ($\varepsilon\in(0,0.5)$).

The Warm-PCE loss function modifies the classical objective by up-weighting edges in the problem graph that the GW solution places across the cut: $$\mathcal L_{\rm Warm\text{-}PCE}(\theta) = \sum_{(i,j)\in E} W_{ij}\,[1+|\widehat c_i^\star - \widehat c_j^\star|]\,s_i(\theta)\,s_j(\theta) + \mathcal L^{(\rm reg)}$$ with $s_i(\theta)=\tanh(\alpha\,\langle\Pi_i\rangle)$. The regularization term $\mathcal L^{(\rm reg)}$ penalizes excessively large correlator values. Alternatively, the loss is expressible as a standard PCE Hamiltonian plus a GW-informed linear soft-bias term: $$\mathcal L_{\rm warm} = \langle H_{\rm PCE}\rangle + \lambda\sum_i b_i\,\langle Z_i\rangle,\quad b_i = \widehat c_i^\star,\,\lambda\simeq \alpha\langle W\rangle.$$ The parameter $\varepsilon$ tunes the bias strength: $\varepsilon\to 0$ heavily biases toward exact GW recovery, while $\varepsilon =0.5$ reverts to standard unbiased PCE.

3. Problem Mapping: QUBO-to-MaxCut Transformation for TSP

Warm-PCE is made applicable to the TSP by mapping the standard QUBO formulation of TSP to an instance of weighted MaxCut. For $N$ cities, binary variables $x_{i,t}$ encode city-slot assignments. The QUBO Hamiltonian is

$$\begin{aligned} H_{\rm cost}&=\sum_{i,j=1}^{N-1}\sum_{t=1}^{N-2} W_{ij}\,x_{i,t}\,x_{j,t+1} +\sum_{i=1}^{N-1}W_{0i}(x_{i,1}+x_{i,N-1}),\ H_{\rm row}&=\sum_{i=1}^{N-1}\left(\sum_{t}x_{i,t}-1\right)^2,\ H_{\rm col}&=\sum_{t=1}^{N-1}\left(\sum_{i}x_{i,t}-1\right)^2,\ H_{\rm QUBO}&=A\,H_{\rm row}+B\,H_{\rm col}+H_{\rm cost}, \end{aligned}$$

with $A, B \gg \max W_{ij}$ enforcing one-hot constraints.

Each $x_i$ is mapped to $x_i = (1-z_i)/2$, $z_i\in\{-1,1\}$, and the quadratic and linear QUBO terms are rewritten as MaxCut-like edge and root penalties: $$W_{ij}^{(\rm MC)} = \frac12\,Q_{ij},\quad Q_{ij}=\text{off-diagonal of }H_{\rm QUBO}$$ producing a MaxCut Hamiltonian suitable for application of PCE (and thus Warm-PCE).

4. Quantum Circuit Architecture and Initialization

Experiments utilize a “TwoLocal” quantum ansatz of depth $p=5$ on $n=4$ qubits, enabling encoding of $17$ binary variables via $k=2$ correlators. Each layer $\ell=1,\dots,5$ comprises:

• Single-qubit rotations $$\bigotimes_{i=1}^n R_X(\theta_{\ell,i})\,R_Z(\phi_{\ell,i})$$;
• A controlled-$Z$ entangling block between all qubit pairs or in a ring configuration.

Each layer introduces $2n=8$ parameters (for $40$ total). All parameters are randomly initialized in $[0,2\pi)$ for both PCE and Warm-PCE. While GW-based initialization of $R_Y$ angles is possible, empirical results reported for Warm-PCE rely solely on loss bias without such preprocessing (Carmo et al., 17 Sep 2025).

5. Empirical Performance on 5-City TSP Instances

Extensive benchmarking was performed on $50$ random 5-city TSP instances, using $10$ independent parameter initialization seeds per instance, circuit depths $p=1$–$5$, and the COBYLA optimizer (maximum $1000$ steps). GW bias used the best-of-100 roundings, with a default $\varepsilon=0.2$.

Empirical metrics:

Depth $p$	Warm-PCE Success Rate	PCE Success Rate
1	28%	4%
2	46%	12%
3	54%	16%
4	60%	4%
5	64%	26%

The mean approximation ratio $r=E/E_{\mathrm{opt}}$ for Warm-PCE increases monotonically from $r\approx 0.83$ at $p=1$ to $r\approx 0.88$ at $p=5$, while standard PCE remains nearly flat at $r\approx 0.82$ across all $p$. Additionally, error bars (IQR) for Warm-PCE decrease as $p$ increases.

Comparative alignment between Warm-PCE and PCE in best-of-10 optimizations per instance shows Warm-PCE wins in 32-41 instances (across $p$), ties in 5-13, and loses in 4-10, depending on the circuit depth (Carmo et al., 17 Sep 2025).

6. Interpretation, Significance, and Future Directions

Warm-PCE outperforms standard PCE both in the probability of obtaining the true optimal solution (up to a 15-fold gain) and in average approximation ratio, particularly for $p\geq 3$ layers. The GW-derived bias enables more effective exploitation of deeper variational circuits, guiding the optimizer toward promising subspaces identified by leading classical approximation algorithms.

The combined polynomial qubit compression of PCE and the performance enhancement of the warm-start can potentially enable direct quantum variational solution approaches for moderate-sized TSP and QUBO problems on sub-20-qubit devices. It is notable that warm-starting reduces the circuit depth required for high-quality solutions.

Directions for future research include expanding to larger TSP instances, investigating higher-order correlators ($k>2$), deeper circuits, noise resilience, and systematic comparisons on large-scale MaxCut benchmarks. Warm-PCE’s properties suggest broad applicability for quantum combinatorial optimization on near-term quantum hardware (Carmo et al., 17 Sep 2025).