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Quantum computation shows promise for addressing numerous classically intractable problems, such as optimization tasks. Many optimization problems are NP-hard, meaning that they scale exponentially with problem size and thus cannot be addressed at scale by traditional computing paradigms. The recently proposed quantum algorithm arXiv:2206.14999 addresses this challenge for some NP-hard problems, and is based on classical semidefinite programming (SDP). In this manuscript, we generalize the SDP-inspired quantum algorithm to sum-of-squares programming, which targets a broader problem set. Our proposed algorithm addresses degree- polynomial optimization problems with variables (which are representative of many NP-hard problems) using qubits, quantum measurements, and classical calculations. We apply the proposed algorithm to the prototypical Max-SAT problem and compare its performance against classical sum-of-squares, state-of-the-art heuristic solvers, and random guessing. Simulations show that the performance of our algorithm surpasses that of classical sum-of-squares after rounding. Our results further demonstrate that our algorithm is suitable for large problems and approximates the best known classical heuristics, while also providing a more generalizable approach compared to problem-specific heuristics. 

We introduce a quantum information theory-inspired method to improve the characterization of many-body Hamiltonians on near-term quantum devices. We design a new class of similarity transformations that, when applied as a preprocessing step, can substantially simplify a Hamiltonian for subsequent analysis on quantum hardware. By design, these transformations can be identified and applied efficiently using purely classical resources. In practice, these transformations allow us to shorten requisite physical circuit-depths, overcoming constraints imposed by imperfect near-term hardware. Importantly, the quality of our transformations is $tunable$: we define a 'ladder' of transformations that yields increasingly simple Hamiltonians at the cost of more classical computation. Using quantum chemistry as a benchmark application, we demonstrate that our protocol leads to significant performance improvements for zero and finite temperature free energy calculations on both digital and analog quantum hardware. Specifically, our energy estimates not only outperform traditional Hartree-Fock solutions, but this performance gap also consistently widens as we tune up the quality of our transformations. In short, our quantum information-based approach opens promising new pathways to realizing useful and feasible quantum chemistry algorithms on near-term hardware. 

Abstract Distributed quantum computation is often proposed to increase the scalability of quantum hardware, as it reduces cooperative noise and requisite connectivity by sharing quantum information between distant quantum devices. However, such exchange of quantum information itself poses unique engineering challenges, requiring high gate fidelity and costly non-local operations. To mitigate this, we propose near-term distributed quantum computing, focusing on approximate approaches that involve limited information transfer and conservative entanglement production. We first devise an approximate distributed computing scheme for the time evolution of quantum systems split across any combination of classical and quantum devices. Our procedure harnesses mean-field corrections and auxiliary qubits to link two or more devices classically, optimally encoding the auxiliary qubits to both minimize short-time evolution error and extend the approximate scheme’s performance to longer evolution times. We then expand the scheme to include limited quantum information transfer through selective qubit shuffling or teleportation, broadening our method’s applicability and boosting its performance. Finally, we build upon these concepts to produce an approximate circuit-cutting technique for the fragmented pre-training of variational quantum algorithms. To characterize our technique, we introduce a non-linear perturbation theory that discerns the critical role of our mean-field corrections in optimization and may be suitable for analyzing other non-linear quantum techniques. This fragmented pre-training is remarkably successful, reducing algorithmic error by orders of magnitude while requiring fewer iterations. 

Semidefinite programs are optimization methods with a wide array of applications, such as approximating difficult combinatorial problems. One such semidefinite program is the Goemans-Williamson algorithm, a popular integer relaxation technique. We introduce a variational quantum algorithm for the Goemans-Williamson algorithm that uses only $n+1$qubits, a constant number of circuit preparations, and $\text{poly}\left(n\right)$expectation values in order to approximately solve semidefinite programs with up to $N=2^n$variables and $M\sim O\left(N\right)$constraints. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit, a technique known as the Hadamard Test. The Hadamard Test enables us to optimize the objective function by estimating only a single expectation value of the ancilla qubit, rather than separately estimating exponentially many expectation values. Similarly, we illustrate that the semidefinite programming constraints can be effectively enforced by implementing a second Hadamard Test, as well as imposing a polynomial number of Pauli string amplitude constraints. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems, including MaxCut. Our method exceeds the performance of analogous classical methods on a diverse subset of well-studied MaxCut problems from the GSet library. 

Abstract Variational quantum algorithms have the potential for significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, leading the algorithms to converge to local, rather than global, minima and the production of suboptimal solutions. In this work, we introduce a variational quantum algorithm that couples classical Markov chain Monte Carlo techniques with variational quantum algorithms, allowing the former to provably converge to global minima and thus assure solution quality. Due to the generality of our approach, it is suitable for a myriad of quantum minimization problems, including optimization and quantum state preparation. Specifically, we devise a Metropolis–Hastings method that is suitable for variational quantum devices and use it, in conjunction with quantum optimization, to construct quantum ensembles that converge to Gibbs states. These performance guarantees are derived from the ergodicity of our algorithm’s state space and enable us to place analytic bounds on its time-complexity. We demonstrate both the effectiveness of our technique and the validity of our analysis through quantum circuit simulations for MaxCut instances, solving these problems deterministically and with perfect accuracy, as well as large-scale quantum Ising and transverse field spin models of up to 50 qubits. Our technique stands to broadly enrich the field of variational quantum algorithms, improving and guaranteeing the performance of these promising, yet often heuristic, methods.