Resource-efficient variational quantum solver for the travelling salesman problem and its silicon photonics implementation

TL;DR

This work introduces a resource-efficient variational quantum algorithm for the traveling salesman problem (TSP) that uses two maximally entangled registers, encoding routes in a correlation matrix $X$ and requiring only $2\,\lceil\log_2 N\rceil$ qubits. The algorithm forms a quantum route adjacency matrix via $X_{ij}(\boldsymbol{\alpha})=2^n\mathrm{Tr}[\rho(\boldsymbol{\alpha})\hat{P}_{ij}]$, which is doubly-stochastic and combined with subtour-elimination yields a convex-cost function $C(\boldsymbol{\alpha})=\sum_{ij} D_{ij} X_{ij}(\boldsymbol{\alpha}) - A_{\rm sub}\sum_{S} \sum_{i\in S} \sum_{j\notin S}X_{ij}(\boldsymbol{\alpha})$. This formulation leverages the Birkhoff–von Neumann decomposition to interpret $X$ as a convex combination of permutation matrices, enabling direct optimization over feasible routes. The authors validate the approach experimentally by solving four-city TSPs on a room-temperature silicon photonic integrated circuit, mapping departure/arrival indices to path-encoded qudits and reconstructing $X$ from coincidence measurements (with observed overlaps to optimal routes around 90–95%). The work demonstrates a promising, qubit-efficient path for near-term quantum devices to tackle NP-hard combinatorial problems while highlighting current limitations in optimization convergence and subtour-term scaling, and it discusses prospects for scalability with modular photonic architectures.

Abstract

The travelling salesman problem is a well-known example of computationally-hard combinatorial problem for classical machines. Here, we propose a novel variational quantum algorithm to solve it. The method is based on the preparation of two maximally entangled quantum registers whose correlations are assigned to different paths between pairs of cities. For $N$ cities, this encoding requires $2 \lceil\log_2 N\rceil$ qubits and the solution to the problem is directly found in the correlation matrix of the two registers composing the overall trial state. As a proof-of-concept experiment, we implement this algorithm for generic problems with four cities on a reconfigurable room-temperature silicon photonic circuit with integrated photon-pair sources, used to initialize maximally entangled path-encoded single-photon states.

Figures (6)

• Figure 1: Schematic workflow of the new VQA for TSPs. Like any VQA, the algorithm involves a reconfigurable quantum Hardware, on the left, and a PC, on the right. The first is able to prepare quantum trial states, and in our case it is based on a silicon photonic integrated circuit Baldazzi2025. The generic trial state is a bipartite maximally entangled system and its measurement produces the correlation matrix $\emph{X}$. In our photonic implementation, the system is composed of a pair of correlated photons generated on-chip. The PC evaluates the cost function given the correlation matrix $\emph{X}$ and the distance matrix $\emph{D}$ of the specific TSP, and then it executes an optimization routine which updates the setting used to prepare the trial state. The variational algorithm ends when the cost function converges to its extremal value and the Hardware is trained to prepare the associated quantum state.
• Figure 1: Four-dimensional unitary transformation and its equivalent set of projective measurements.(a) Decomposition of a unitary transformation of dimension four clements_optimal_2016 in terms of SU$(2)$ matrices given in Eq. \ref{['eq:su2_matrix']}. On the left of the scheme we can find the spatial modes labels, while on the right the symbol for an MZI, whose action is described by an SU$(2)$ matrix. (b) On the left, utilized decomposition of a unitary transformation $U_d$ satisfying the condition in Eq. \ref{['eq:cond_UU_app']}(left). On the right, the triangular decomposition whose action is equivalent through multiple projective measurements, i.e. $\left\{U_i^{(j)}\right\}_{j\in (1\ldots 4)}$. (c) On the left, utilized decomposition of a unitary transformation $U_a$ satisfying the condition in Eq. \ref{['eq:cond_UU_app']}(right). On the right, the triangular decomposition whose action is equivalent through multiple projective measurements, i.e. $\left\{U_s^{(j)}\right\}_{j\in (1\ldots 4)}$.
• Figure 2: Travelling salesman problem with four cities.(a) Fully connected directed weighted graph associated with the travelling salesman problem with four cities. The parameters $D_{ij}$ quantify the distance of the path departing from city $i$ and arriving at city $j$. Paths with the same departing and arriving city, associated with diagonal terms of the distance matrix $\emph{D}$, have not been represented: these terms can be chosen large enough in such a way to avoid solutions not satisfying the constraint $\textbf{(0)}$ in Eq. \ref{['eq:class_constraints']}. (b) Route adjacency matrices $\emph{x}$, defined in Eq. \ref{['eq:class_bin_var']}, representing all the routes among four cities satisfying all the constraints in Eq. \ref{['eq:class_constraints']}. The associated route is reported above each matrix. Starting and ending with city 1 is just a choice: any cyclic permutation of the route does not alter the associated $\emph{x}$ and the route length, because of the cyclicity of the problem.
• Figure 3: Quantum circuit to prepare the trial states for our VQA for TSPs. Gate representation of the quantum circuit able to prepare the generic trial state for the presented VQA associated with TSPs for $N$ cities. First of all, $2n = 2 \lceil\log_2 N\rceil$ qubits are collected in two registers labelled with $d$ for "departures" and $a$ for "arrivals", and they are initialized to the zero state. Then, Hadamard gates (H) are applied to each qubit in the arrival register, and controlled-NOT gates are applied pairwise to couple of qubits, one in the departure and one in the arrival register, taking the second as the control. At this point, the two registers form a maximally entangled state, Eq. \ref{['eq:max_ent']}. Finally, two independent unitary transformations $U_{d/a}$ are applied to the departure and arrival registers. The final state, reported in Eq. \ref{['eq:final_prep_state']}, can be used as a trial state for the presented VQA associated with the cost function in Eq. \ref{['eq:cost_funct']}.
• Figure 4: Block scheme of the photonic circuit able to implement the VQA for four-city TSPs.(a) The Si-PIC is composed of four stages: (I) pump splitting, (II) sources of photon pair, (III) separation and routing, and (IV) independent linear manipulation $U_{i/s}$ of the generated photon pair Baldazzi2025. By equally pumping all four sources, after the stage (III), the state of the two photons can encode two maximally entangled ququarts. In particular, the state can be mapped to the maximally entangled state in Eq. \ref{['eq:max_ent']} with $n=2$. Finally, after stage (IV), the state is precisely the one shown in Eq. \ref{['eq:final_prep_state']}. Since we utilized just two outputs and two detectors, one for each component of the bipartite system, the transformation $U_i\otimes U_s$ performs multiple projective measurements equivalent to the action of the transformation $U_d\otimes U_a$ in Figure \ref{['fig:quantum_circuit']}. (b) Graphical representation of the triangular network of MZIs, contained in the stage (I) for pump splitting (left) and in the stage (IV) for the linear manipulations $U_{i/s}$ (right). (c) From left to right, symbols for MZI, spontaneous-four-wave-mixing-based photon pair source of stage (II) (pump laser in green, signal and idler photons as red and blue dots), and asymmetric MZI (AMZI) of stage (III).