Self-correcting quantum memories store logical quantum information for exponential time in thermal equilibrium at low temperatures. By definition, these systems are slow mixing. This raises the question of how the memory state, which we refer to as the Gibbs state within a logical sector, is created in the first place. In this paper, we show that for a broad class of self-correcting quantum memories on lattices with parity check redundancies, a quasi-local quantum Gibbs sampler rapidly converges to the corresponding low-temperature Gibbs state within a logical sector when initialized from a ground state. This illustrates a dynamical view of self-correcting quantum memories, where the "syndrome sector" rapidly converges to thermal equilibrium, while the "logical sector" remains metastable. As a key application, when initialized from a random ground state, this gives a rapid Gibbs state preparation algorithm for the 4D toric code in $\mathrm{polylog}(n)$ depth. The main technical ingredients behind our approach are new, low-temperature decay-of-correlation properties for these metastable states.Self-correcting quantum memories store logical quantum information for exponential time in thermal equilibrium at low temperatures. By definition, these systems are slow mixing. This raises the question of how the memory state, which we refer to as the Gibbs state within a logical sector, is created in the first place. In this paper, we show that for a broad class of self-correcting quantum memories on lattices with parity check redundancies, a quasi-local quantum Gibbs sampler rapidly converges to the corresponding low-temperature Gibbs state within a logical sector when initialized from a ground state. This illustrates a dynamical view of self-correcting quantum memories, where the "syndrome sector" rapidly converges to thermal equilibrium, while the "logical sector" remains metastable. As a key application, when initialized from a random ground state, this gives a rapid Gibbs state preparation algorithm for the 4D toric code in $\mathrm{polylog}(n)$ depth. The main technical ingredients behind our approach are new, low-temperature decay-of-correlation properties for these metastable states.

arXiv:2507.10976Jul 15, 20250 citations

Quantum memory at nonzero temperature in a thermodynamically trivial system

Yifan Hong, Jinkang Guo, Andrew Lucas

Passive error correction protects logical information forever (in the thermodynamic limit) by updating the system based only on local information and few-body interactions. A paradigmatic example is the classical two-dimensional Ising model: a Metropolis-style Gibbs sampler retains the sign of the initial magnetization (a logical bit) for thermodynamically long times in the low-temperature phase. Known models of passive quantum error correction similarly exhibit thermodynamic phase transitions to a low-temperature phase wherein logical qubits are protected by thermally stable topological order. Here, in contrast, we show that certain families of constant-rate classical and quantum low-density parity check codes have no thermodynamic phase transitions at nonzero temperature, but nonetheless exhibit ergodicity-breaking dynamical transitions: below a critical nonzero temperature, the mixing time of local Gibbs sampling diverges in the thermodynamic limit. Slow Gibbs sampling of such codes enables fault-tolerant passive quantum error correction using finite-depth circuits. This strategy is well suited to measurement-free quantum error correction, and may present a desirable experimental alternative to conventional quantum error correction based on syndrome measurements and active feedback. It has been commonly assumed that self-correcting quantum memories are only possible in systems with finite-temperature phase transitions to topological order. Here the authors show a complete breakdown of this expectation in quantum low-density parity-check codes.Passive error correction protects logical information forever (in the thermodynamic limit) by updating the system based only on local information and few-body interactions. A paradigmatic example is the classical two-dimensional Ising model: a Metropolis-style Gibbs sampler retains the sign of the initial magnetization (a logical bit) for thermodynamically long times in the low-temperature phase. Known models of passive quantum error correction similarly exhibit thermodynamic phase transitions to a low-temperature phase wherein logical qubits are protected by thermally stable topological order. Here, in contrast, we show that certain families of constant-rate classical and quantum low-density parity check codes have no thermodynamic phase transitions at nonzero temperature, but nonetheless exhibit ergodicity-breaking dynamical transitions: below a critical nonzero temperature, the mixing time of local Gibbs sampling diverges in the thermodynamic limit. Slow Gibbs sampling of such codes enables fault-tolerant passive quantum error correction using finite-depth circuits. This strategy is well suited to measurement-free quantum error correction, and may present a desirable experimental alternative to conventional quantum error correction based on syndrome measurements and active feedback. It has been commonly assumed that self-correcting quantum memories are only possible in systems with finite-temperature phase transitions to topological order. Here the authors show a complete breakdown of this expectation in quantum low-density parity-check codes.

arXiv:2403.10599Nature CommunicationsJan 2, 202514 citations

Circuit-Based Characterization of Finite-Temperature Quantum Phases and Self-Correcting Quantum Memory

Anonymous, Vedika Khemani, Shengqi Sang

Quantum phases at zero temperature can be characterized as equivalence classes under local unitary transformations: two ground states within a gapped phase can be transformed into each other via a local unitary circuit. We generalize this circuit-based characterization of phases to systems at finite-temperature thermal equilibrium described by Gibbs states. We construct a channel circuit that approximately transforms one Gibbs state into another provided the two are connected by a path in parameter space along which a certain correlation-decay condition holds. For finite-dimensional systems of linear size L and approximation error ϵ , the locality of the circuit is Polylog ( Poly ( L ) / ϵ ) . The correlation-decay condition, which we specify, is expected to be satisfied in the interior of many noncritical thermal phases, including those displaying discrete symmetry breaking and topological order. As an application, we show that any system in the same thermal phase as a zero-temperature topological code coherently preserves quantum information for a macroscopically long time, establishing self-correction as a universal property of thermal phases. As part of the proof, we provide explicit encoding and decoding channel circuits to encode information into, and decode it from, a system in thermal equilibrium.Quantum phases at zero temperature can be characterized as equivalence classes under local unitary transformations: two ground states within a gapped phase can be transformed into each other via a local unitary circuit. We generalize this circuit-based characterization of phases to systems at finite-temperature thermal equilibrium described by Gibbs states. We construct a channel circuit that approximately transforms one Gibbs state into another provided the two are connected by a path in parameter space along which a certain correlation-decay condition holds. For finite-dimensional systems of linear size L and approximation error ϵ , the locality of the circuit is Polylog ( Poly ( L ) / ϵ ) . The correlation-decay condition, which we specify, is expected to be satisfied in the interior of many noncritical thermal phases, including those displaying discrete symmetry breaking and topological order. As an application, we show that any system in the same thermal phase as a zero-temperature topological code coherently preserves quantum information for a macroscopically long time, establishing self-correction as a universal property of thermal phases. As part of the proof, we provide explicit encoding and decoding channel circuits to encode information into, and decode it from, a system in thermal equilibrium.

arXiv:2509.15204PRX QuantumJul 6, 20260 citations

Quantum Self-Correction in the 3D Cubic Code Model

Sergey B. Bravyi, Jeongwan Haah

A big open question in the quantum information theory concerns the feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction, if the memory is in contact with a cold enough thermal bath. Here we report analytic and numerical evidence for self-correcting behavior in the quantum spin lattice model known as the 3D cubic code. We prove that its memory time is at least L(cβ), where L is the lattice size, β is the inverse temperature of the bath, and c>0 is a constant coefficient. However, this bound applies only if the lattice size L does not exceed a critical value which grows exponentially with β. In that sense, the model can be called a partially self-correcting memory. We also report a Monte Carlo simulation indicating that our analytic bounds on the memory time are tight up to constant coefficients. To model the readout step we introduce a new decoding algorithm, which can be implemented efficiently for any topological stabilizer code. A longer version of this work can be found in Bravyi and Haah, arXiv:1112.3252.A big open question in the quantum information theory concerns the feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction, if the memory is in contact with a cold enough thermal bath. Here we report analytic and numerical evidence for self-correcting behavior in the quantum spin lattice model known as the 3D cubic code. We prove that its memory time is at least L(cβ), where L is the lattice size, β is the inverse temperature of the bath, and c>0 is a constant coefficient. However, this bound applies only if the lattice size L does not exceed a critical value which grows exponentially with β. In that sense, the model can be called a partially self-correcting memory. We also report a Monte Carlo simulation indicating that our analytic bounds on the memory time are tight up to constant coefficients. To model the readout step we introduce a new decoding algorithm, which can be implemented efficiently for any topological stabilizer code. A longer version of this work can be found in Bravyi and Haah, arXiv:1112.3252.

arXiv:1112.3252Physical Review LettersNov 12, 2013292 citations

Rapid mixing of quantum spin chains at any finite temperature

Leeseok Kim

We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in $\mathcal{O}\left(\log (n/\varepsilon)\right)$ time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order $1$ . We further provide $\mathcal{O}(\text{polylog}(n/\varepsilon))$ depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in $\mathcal{O}\left(\log (n/\varepsilon)\right)$ time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order $1$. We further provide $\mathcal{O}(\text{polylog}(n/\varepsilon))$ depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.

quant-pharXiv:2610.01190Oct 2, 20260 citations

Entanglement generation and self-correcting quantum memories

Beat Röthlisberger

Building a working quantum computer that is able to perform useful calculations remains a challenge. With this thesis, we are trying to contribute a small piece to this puzzle by addressing three of the many fundamental questions one encounters along the way of reaching that goal. These questions are: (i) What is an easy way to create highly entangled states as a resource for quantum computation? (ii) What can we do to efficiently quantify states of noisy entanglement in systems coupled to the outside world? (iii) How can we protect and store fragile quantum states for arbitrary long times? \n \nThe first two questions are the subject of part one of this thesis, `Entanglement Measures & Highly Entangled States'. We devise a particular proposal for generating entanglement within a solid-state setup, starting first with the tripartite case and continuing with a generalization to four and more qubits. The main idea there is to realize systems with highly entangled ground states in order for entanglement to be created by merely cooling to low enough temperatures. We have addressed the issue of quantifying entanglement in these systems by numerically calculating mixed-state entanglement measures and maximizing the latter as a function of the external magnetic field strength. The research along these lines has led to the development of the numerical library 'libCreme'. \n \nThe second part of the thesis, 'Self-Correcting Quantum Memories', addresses the question how to reliably store quantum states long enough to perform useful calculations. Every computer, be it classical or quantum, needs the information it processes to be protected from corruption caused by faulty gates and perturbations from interactions with its environment. However, quantum states are much more susceptible to these adverse effects than classical states, making the manipulation and storage of quantum information a challenging task. Promising candidates for such 'quantum memories' are systems exhibiting topological order, because they are robust against local perturbations, and information encoded in their ground state can only be manipulated in a non-local fashion. We extend the so-called toric code by repulsive long-range interactions between anyons and show that this makes the code stable against thermal fluctuations. Furthermore, we investigate incoherent effects of quenched disorder in the toric code and similar systems.Building a working quantum computer that is able to perform useful calculations remains a challenge. With this thesis, we are trying to contribute a small piece to this puzzle by addressing three of the many fundamental questions one encounters along the way of reaching that goal. These questions are: (i) What is an easy way to create highly entangled states as a resource for quantum computation? (ii) What can we do to efficiently quantify states of noisy entanglement in systems coupled to the outside world? (iii) How can we protect and store fragile quantum states for arbitrary long times? \n \nThe first two questions are the subject of part one of this thesis, `Entanglement Measures & Highly Entangled States'. We devise a particular proposal for generating entanglement within a solid-state setup, starting first with the tripartite case and continuing with a generalization to four and more qubits. The main idea there is to realize systems with highly entangled ground states in order for entanglement to be created by merely cooling to low enough temperatures. We have addressed the issue of quantifying entanglement in these systems by numerically calculating mixed-state entanglement measures and maximizing the latter as a function of the external magnetic field strength. The research along these lines has led to the development of the numerical library 'libCreme'. \n \nThe second part of the thesis, 'Self-Correcting Quantum Memories', addresses the question how to reliably store quantum states long enough to perform useful calculations. Every computer, be it classical or quantum, needs the information it processes to be protected from corruption caused by faulty gates and perturbations from interactions with its environment. However, quantum states are much more susceptible to these adverse effects than classical states, making the manipulation and storage of quantum information a challenging task. Promising candidates for such 'quantum memories' are systems exhibiting topological order, because they are robust against local perturbations, and information encoded in their ground state can only be manipulated in a non-local fashion. We extend the so-called toric code by repulsive long-range interactions between anyons and show that this makes the code stable against thermal fluctuations. Furthermore, we investigate incoherent effects of quenched disorder in the toric code and similar systems.

Jan 1, 20120 citations

\mathrm{polylog}(n)

depth. The main technical ingredients behind our approach are new, low-temperature decay-of-correlation properties for these metastable states.

We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in

\mathcal{O}\left(\log (n/\varepsilon)\right)

time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order

1

. We further provide

\mathcal{O}(\text{polylog}(n/\varepsilon))

depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.