[Paper Review] An Information-Theoretic Proof of the Constructive Commutative Quantum Lovász Local Lemma

Martin Schwarz, Toby S. Cubitt+1 more|arXiv (Cornell University)|Nov 25, 2013

Quantum Computing Algorithms and Architecture20 references7 citations

TL;DR

This paper presents a quantum algorithm that efficiently prepares a zero-energy ground state for commuting quantum Hamiltonians with local constraints, using an information-theoretic approach inspired by Moser's classical algorithm. It establishes a constructive, efficient quantum algorithm for the symmetric Quantum Lovász Local Lemma in the commuting case, matching the non-constructive bounds of the original lemma with tight runtime and error guarantees.

ABSTRACT

The Quantum Lovász Local Lemma (QLLL) [AKS12] establishes non-constructively that any quantum system constrained by a local Hamiltonian has a zero-energy ground state, if the local Hamiltonian terms overlap only in a certain restricted way. In this paper, we present an efficient quantum algorithm to prepare this ground state for the special case of commuting projector terms. The related classical problem has been open for more than 34 years. Our algorithm follows the breakthrough ideas of Moser's [Moser09] classical algorithm and lifts his information theoretic argument to the quantum setting. A similar result has been independently published by Arad and Sattath [AS13] recently.

Motivation & Objective

• To provide a constructive, efficient quantum algorithm for preparing zero-energy ground states in quantum systems with commuting local Hamiltonian terms.
• To generalize Moser’s classical information-theoretic argument to the quantum setting, specifically for commuting projectors.
• To achieve tight runtime and error bounds matching the non-constructive symmetric Quantum Lovász Local Lemma (QLLL) parameters.
• To resolve a long-standing open problem in quantum Hamiltonian complexity: efficient preparation of highly entangled ground states under local constraints.
• To demonstrate that the classical constructive LLL can be recovered as a special case of the quantum algorithm, validating its generality.

Proposed method

• Lifts Moser’s classical information-theoretic argument to the quantum domain using coherent measurements and quantum compression of bit sequences.
• Employs a recursive quantum algorithm that uses a stack register to track measurement failures and guide state updates.
• Uses a work register $W$ initialized to a maximally mixed state and a randomness register $R$ to generate coherent random bits for measurements.
• Applies the strong converse of the typical subspace theorem to bound error probabilities and ensure convergence.
• Implements a compression subroutine on the recursion log register $L$ to reduce the number of required operations, leveraging entropy bounds.
• Uses a termination register $term$ and flag registers $F$ and $L$ to manage recursion depth and state transitions.

Experimental results

Research questions

• RQ1 Can Moser’s classical constructive LLL algorithm be generalized to the quantum setting for commuting projectors?
• RQ2 What is the minimal runtime and error probability required to prepare a zero-energy ground state in the commuting QLLL setting?
• RQ3 Can information-theoretic compression techniques be used to tighten the runtime bounds in quantum algorithms for local Hamiltonians?
• RQ4 Does the quantum algorithm achieve the same threshold parameters as the non-constructive QLLL, specifically $d \leq 2^k / (re)$?
• RQ5 Can the classical constructive LLL be recovered as a special case of this quantum algorithm?

Key findings

• The algorithm achieves a runtime of $O(m + \log(1/\varepsilon))$ for any $\varepsilon > 0$, matching the optimal classical bound.
• The success probability is $1 - \varepsilon$, with error bounded using the strong converse of the typical subspace theorem.
• The algorithm prepares a quantum state $\sigma$ such that $\operatorname{Tr}(\Pi_i \sigma) = 0$ for all $i$, confirming a zero-energy ground state.
• The method improves upon Moser’s classical argument by making it tight under the same assumptions as the non-constructive symmetric QLLL.
• The runtime bound is derived via a tight analysis of the recursion depth $N$, bounded by $b + 3a(\log(b) + 1)$ with $a = (k - \log(de))^{-1}$ and $b = (m + \log(1/\varepsilon)) / (k - \log(de))$.
• The algorithm is valid for commuting $k$-local projectors of rank at most $r$, with the same threshold $d \leq 2^k / (re)$ as in the original QLLL.