Quantum Subspace Correction for Constraints (2310.20191v2)

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.

• The paper introduces quantum subspace correction (QSC) to stabilize constraint-satisfying subspaces in quantum Ising models.
• It details an algorithm that achieves uniform state distribution and reduces circuit depth for efficient adiabatic quantum calculations.
• The method shows potential for advancing fault-tolerant quantum computing, particularly in systems like neutral atom setups with mid-circuit measurements.

Quantum Subspace Correction for Constraints

In the paper entitled "Quantum Subspace Correction for Constraints," the authors present a methodology to stabilize the subspaces of computational problems embedded in their Ising representations. The central technique, denoted as "quantum subspace correction" (QSC), involves the strategic construction of operators that maintain the integrity of these subspaces by detecting violations of imposed constraints and executing recovery operations to reinstate the subspace integrity. This approach is invaluable in managing errors and optimizing computational processes, particularly in the context of constraint satisfaction and fault tolerance within quantum systems.

The paper meticulously details the development of stabilizers for the Independent Set problem, which is pivotal due to its simplicity and ubiquity across various computational tasks. The authors introduce an algorithm that achieves a perfect uniform or weighted distribution over all constraint-satisfying states via quantum analogues of classical sampling methods. This process is notably adaptable to preparing Gibbs distributions on regular graphs, demonstrating sub-linear preparation times below a specific hardness threshold, $\lambda_d^*$.

Significantly, QSC holds potential utility in optimizing quantum algorithms, particularly in fault-tolerant quantum computing (FTQC) environments. By leveraging techniques inspired by quantum error correction, the method aligns intermediate-term algorithm development with long-term FTQC goals. This alignment is particularly valuable for neutral atom quantum computers, which offer high coherence times, native multi-qubit gates, and the ability to perform mid-circuit measurements and rearrangements.

The quantum subspace correction technique extends beyond stabilizing distribution preparations; it also aids in depth-reduction strategies for adiabatic algorithms. These applications underscore the versatility of QSC in maintaining specified constrained subspaces even amidst computational errors, such as those introduced by trotterization in quantum state preparation processes. Moreover, by maintaining fidelity to desired quantum states, this technique opens pathways for enhanced optimization and computational efficiencies.

The implications of the paper are twofold:

1. Practical Implications: Quantum subspace correction aids in the profound reduction of circuit depth necessary for the execution of adiabatic algorithms, thus optimizing quantum computation resources and enhancing the execution speed of quantum algorithms within practical constraints.
2. Theoretical Implications: The successful application of QSC in manipulating subspaces holds the promise of substantial theoretical advancements in quantum algorithm design, set to impact various domains reliant on quantum computing for complex problem-solving.

Looking forward, the paper anticipates further research into refining recovery strategies specific to different problem classes and graph complexities, thereby enhancing the breadth of quantum subspace correction capabilities. As quantum technology continues to evolve, QSC is poised to become an essential component of the quantum computing toolkit, especially as challenges related to fault tolerance and large-scale computation come to the fore.