A Non-Variational Quantum Approach to the Job Shop Scheduling Problem

TL;DR

Addresses NP-hard Just-in-Time Job Shop Scheduling Problem (JIT-JSSP) with quantum heuristics suitable for near-term hardware. Introduces Iterative-QAOA, a non-variational, shallow QAOA variant with a fixed-angle schedule and iterative warm-start updates based on Boltzmann-weighted measurement outcomes. Benchmarks on IonQ trapped-ion hardware and tensor-network simulations show Iterative-QAOA converging to optimal or high-quality solutions up to 97 qubits, outperforming LR-QAOA and demonstrating robustness to noise with error mitigation. The results highlight favorable scaling prospects for fault-tolerant quantum computers and outline directions for improved encodings and hybrid strategies to tackle larger industrial instances.

Abstract

Quantum heuristics offer a potential advantage for combinatorial optimization but are constrained by near-term hardware limitations. We introduce Iterative-QAOA, a variant of QAOA designed to mitigate these constraints. The algorithm combines a non-variational, shallow-depth circuit approach using fixed-parameter schedules with an iterative warm-starting process. We benchmark the algorithm on Just-in-Time Job Shop Scheduling Problem (JIT-JSSP) instances on IonQ Forte Generation QPUs, representing some of the largest such problems ever executed on quantum hardware. We compare the performance of the algorithm against both the Variational Quantum Imaginary Time Evolution (VarQITE) algorithm and the non-variational Linear Ramp (LR) QAOA algorithm. We find that Iterative-QAOA robustly converges to find optimal solutions as well as high-quality, near-optimal solutions for all problem instances evaluated. We evaluate the algorithm on larger problem instances up to 97 qubits using tensor network simulations. The scaling behavior of the algorithm indicates potential for solving industrial-scale problems on fault-tolerant quantum computers.

Figures (14)

• Figure 1: Workflow of the Iterative-QAOA algorithm. $(a)$ A non-variational QAOA ansatz is prepared using a fixed number of layers $p$ and a predetermined schedule for the angles $\{\beta_k\}$ and $\{\gamma_k\}$. The circuit applies alternating cost ($U_C(\gamma_k)=\exp[-i\gamma_kH_C]$) and mixer ($U_M(\beta_k)=\exp[-i\beta_kH_M]$) unitaries to an initial state $|\psi_\text{init}^{(j)}\rangle$. $(b)$ This circuit is executed on quantum hardware (QPU) yielding $(c)$ a set of classical bitstrings. $(d)$ These measurement outcomes are used to compute a statistical bias for each qubit, given by the thermal expectation value $\langle \sigma^q_z \rangle_{T}$. $(e)$ The computed bias determines the angles $\{\phi_q\}$ of the next iteration's initial state $|\psi_\text{init}^{(j+1)}\rangle$, which is prepared by a layer of single-qubit rotations $R_y(\phi_q)$.
• Figure 2: Performance of Iterative-QAOA on JSSP instances of two different sizes executed on IonQ Forte Generation QPU. Shown here are the low-energy sectors of the cost probability distributions for the 32- (top row), 33- (middle row), and 36- (bottom row) qubit instances. The columns display the distribution's evolution at the initial (Iter = 0), an intermediate (Iter = 3), and the final (Iter = 9) iteration. Each panel compares the ideal noiseless simulation results with raw data from the QPU executions and the error-mitigated results after applying DNL. The dashed vertical line indicates the known optimal cost. The algorithm parameters used are $\Delta_{\beta/\gamma} = 0.17$ and a quadratic schedule for $\beta_T\in[0.1,1]$.
• Figure 3: Evolution of the full cost distribution for the 33-qubit JSSP instance under Iterative-QAOA on the IonQ Forte Generation QPU. The panels display the distributions at the initial (Iter = 0), an intermediate (Iter = 3), and the final (Iter = 9) iteration. The algorithm parameters used are $\Delta_{\beta/\gamma} = 0.17$ and a quadratic schedule for $\beta_T\in[0.1,1]$.
• Figure 4: Comparison of LR-QAOA and Iterative-QAOA performance on the 24-qubit JIT-JSSP instance. All panels show the low-energy sector of the final cost distribution from ideal noiseless simulations with 4,000 measurement shots. The first three panels (from left to right) display the results for a standard LR-QAOA circuit with increasing depth of $p=4, \,25,$ and $50$ layers, respectively. The results show that even at a significant depth of $p=50$, the probability of sampling the ground state remains low ($<4\%$). The rightmost panel shows the distribution for $p=4$ Iterative-QAOA after 10 iterations, resulting in a ground state population of 97.92%. The algorithm parameters used are $\Delta_{\beta/\gamma} = 0.17$ and a quadratic schedule for $\beta_T\in[0.1,1]$.
• Figure 5: Performance of Iterative-QAOA on 50- and 97-qubit JIT-JSSP instances with an ideal MPS simulator and bond dimension $\chi=256$. Each panel shows the low-energy sectors of the cost probability distributions at the initial (Iter = 0), an intermediate (Iter = 3), and the final (Iter = 9) iteration. The number of layers for the 50-qubit instance was $p=6$ while for the 97-qubit instance we show the two cases of $p=6$ and 7. The algorithm parameters used are $\Delta_{\beta/\gamma} = 0.17$ and a quadratic schedule for $\beta_T\in[0.1,1]$.