Quantum Goemans-Williamson Algorithm With The Hadamard Test And Approximate Amplitude Constraints

Taylor L. Patti^1,2, Jean Kossaifi², Anima Anandkumar^3,2, and Susanne F. Yelin¹

¹Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
²NVIDIA, Santa Clara, California 95051, USA
³Department of Computing + Mathematical Sciences (CMS), California Institute of Technology (Caltech), Pasadena, CA 91125 USA

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Abstract

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 $text{poly}(n)$ expectation values in order to approximately solve semidefinite programs with up to $N=2^n$ variables and $M sim O(N)$ 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.

Featured image: Diagram of HTAAC-QSDP for the Goemans-Williamson algorithm with $n=3$ non-auxiliary qubits. a) A classical problem of $N$ variables (here an $N$-vertex MaxCut problem where $N=8$). The weight matrix $W$ is used to generate the unitary $U_W$. b) The Hadamard Test is used to efficiently evaluate the objective function and population balancing constraints. The $n$-qubit state $|psirangle = U_V|mathbf{0}rangle$ is prepared with a variational quantum circuit $U_V$. The $n+1$th (auxilary) qubit is initialized as $(|0rangle -i |1rangle)/sqrt{2}$. Subsequently, the Hadamard Test is carried out: $U_W$ is implemented as a controlled-unitary conditioned on the auxilary qubit, which is then measured to compute $langle sigma_{n+1} rangle_{W,t} = text{Im}[langle psi|U_W|psi rangle]$ or $langle sigma_{n+1} rangle_{P,t} = text{Im}[langle psi|U_P|psi rangle]$. c) The $M=2^n$ SDP amplitude constraints constraints are approximately enforced with only $m sim text{poly}(n)$ Pauli string constraints. These are computed by collecting $n$-qubit Pauli-$z$ measurements and using marginal statistics to estimate the $m$ expectation values.

Popular summary

Semidefinite programs allow us to approximate a wide array of hard problems, including NP-hard problems. One such semidefinite program is the Goemans-Williamson algorithm, which can solve hard problems, such as MaxCut. We introduce a variational quantum algorithm for the Goemans-Williamson algorithm that uses only $n{+}1$ qubits, a constant number of circuit preparations, and a polynomial number of expectation values in order to approximately solve semidefinite programs with an exponential number of variables and constraints. We encode the problem into a quantum circuit (or unitary) and read it out on a single auxilary qubit, a technique known as the Hadamard Test. Similarly, we illustrate that the problem constraints can be enforced by 1) a second Hadamard Test and 2) a polynomial number of Pauli string 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.

► BibTeX data

@article{Patti2023quantumgoemans, doi = {10.22331/q-2023-07-12-1057}, url = {https://doi.org/10.22331/q-2023-07-12-1057}, title = {Quantum {G}oemans-{W}illiamson {A}lgorithm with the {H}adamard {T}est and {A}pproximate {A}mplitude {C}onstraints}, author = {Patti, Taylor L. and Kossaifi, Jean and Anandkumar, Anima and Yelin, Susanne F.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1057}, month = jul, year = {2023}<br /> }

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Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints

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