Quantum Subspace Correction for Constraints

• URL: http://arxiv.org/abs/2310.20191v2
• Date: Wed, 7 Feb 2024 18:36:37 GMT
• Title: Quantum Subspace Correction for Constraints
• Authors: Kelly Ann Pawlak, Jeffrey M. Epstein, Daniel Crow, Srilekha Gandhari, Ming Li, Thomas C. Bohdanowicz, Jonathan King
• Abstract summary: We show that it is possible to construct operators that stabilize the constraint-satisfying subspaces of computational problems in their Ising representations. We also look at a potential use of quantum subspace correction for fault-tolerant depth-reduction.
• Score: 3.977810072874835
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We demonstrate that it is possible to construct operators that stabilize the constraint-satisfying subspaces of computational problems in their Ising representations. We provide an explicit recipe to construct unitaries and associated measurements given a set of constraints. The stabilizer measurements allow the detection of constraint violations, and provide a route to recovery back into the constrained subspace. We call this technique ''quantum subspace correction". As an example, we explicitly investigate the stabilizers using the simplest local constraint subspace: Independent Set. We find an algorithm that is guaranteed to produce a perfect uniform or weighted distribution over all constraint-satisfying states when paired with a stopping condition: a quantum analogue of partial rejection sampling. The stopping condition can be modified for sub-graph approximations. We show that it can prepare exact Gibbs distributions on $d-$regular graphs below a critical hardness $\lambda_d^*$ in sub-linear time. Finally, we look at a potential use of quantum subspace correction for fault-tolerant depth-reduction. In particular we investigate how the technique detects and recovers errors induced by Trotterization in preparing maximum independent set using an adiabatic state preparation algorithm.

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