Physics of Quantum Information

"Optimal fault tolerant error correction thresholds for CCS codes are traditionally obtained via mappings to classical statistical mechanics models, for example the 2d random bond Ising model for the 1d repetition code subject to bit-flip noise and faulty measurements. Here, we revisit the 1d repetition code, and develop an exact “stabilizer expansion” of the full time evolving density matrix under repeated rounds of (incoherent and coherent) noise and faulty stabilizer measurements. This expansion enables computation of the coherent information, indicating whether encoded information is retained under the noisy dynamics, and generates a dual representation of the (replicated) 2d random bond Ising model. However, in the fully generic case with both coherent noise and weak measurements, the stabilizer expansion breaks down (as does the canonical 2d random bond Ising model mapping). If the measurement results are thrown away all encoded information is lost at long times, but the evolution towards the trivial steady state reveals a signature of a quantum transition between an over and under damped regime. Implications for generic noisy dynamics in other CCS codes will be mentioned, including open issues."

Ground states as well as Gibbs states of many-body quantum Hamiltonians have been studied extensively for some time. In contrast, the landscape of mixed states that do not correspond to a system in thermal equilibrium is relatively less explored. In this talk I will motivate a rather coarse characterization of mixed quantum many-body states using the idea of "separability", i.e., whether a mixed state can be expressed as an ensemble of short-range entangled pure states. I will discuss several examples of decoherence-driven phase transitions from a separability viewpoint, and argue that such a framework also provides a potentially new view on Gibbs states. Based on work with Yu-Hsueh Chen. References: 2309.11879, 2310.07286, 2403.06553.

For quantum phases of Hamiltonian ground states, the energy gap plays a central role in ensuring the stability of the phase as long as the gap remains finite. In this talk we introduce Markov length, the length scale at which the quantum conditional mutual information (CMI) decays exponentially, as an equally essential quantity characterizing mixed-state phases and transitions. For a state evolving under a local Lindbladian, we argue that if its Markov length remains finite along the evolution, then it remains in the same phase, meaning there exists another quasi-local Lindbladian evolution that can reverse the former one. We apply this diagnostic to toric code subject to decoherence and show that the Markov length is finite everywhere except at its decodability transition, at which it diverges. This implies that the mixed state phase transition coincides with the decodability transition and also suggests a quasi-local decoding channel.

Long-range entangled mixed states are exotic many-body systems that exhibit intrinsically quantum phenomena despite extensive classical fluctuations. In the first part of the talk, I will show how they can be efficiently prepared with measurements and unitary feedback conditioned on the measurement outcome. For example, symmetry-protected topological phases can be universally converted into mixed states with long-range entanglement, and certain gapped topological states such as Chern insulators can be converted into mixed states with critical correlations in the bulk. In the second part of the talk, I will discuss how decoherence can drive interesting mixed-state entanglement transitions. By focusing on the toric codes in various space dimensions subject to certain types of decoherence, I will present the exact results of entanglement negativity, from which the universality class of entanglement transitions can be completely characterized.

I will show the existence of a universal upper bound on the energy gap of topological states of matter, such as (integer and fractional) Chern insulators, quantum spin liquids and topological superconductors. This gap bound turns out to be fairly tight for the Chern insulator states that were predicted and observed in twisted bilayer transition metal dichalcogenides. Next, I will show a universal relation between the energy gap and dielectric constant of solids. These results are derived from fundamental principles of physics and therefore apply to all electronic materials. I will end by outlining new research directions involving topology, quantum geometry and energy.

Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth circuit. It is conjectured that in two spatial dimensions, two gapped ground states with gappable boundary are in the same phase if and only if they have the same anyon contents, which are described by a unitary modular tensor category. We prove this conjecture for a class of states that obey a strict form of area law. Our main technical development is to transform these states into string-net wavefunctions using constant-depth circuits.

Entanglement in many-body quantum systems is notoriously hard to characterize due to the exponentially many parameters involved to describe the state. On the other hand, we are usually not interested in all the microscopic details of the entanglement attern but only some of its global features. It turns out, quantum circuits of different levels of complexity provide a useful way to establish a hierarchy among many-body entanglement structures. A circuit of a finite depth generates only short range entanglement which is in the same gapped phase as an unentangled product state. A linear depth circuit on the other hand can lead to chaos beyond thermal equilibrium. In this talk, we discuss how to reach the interesting regime in between that contains nontrivial gapped orders. This is achieved using the Sequential Quantum Circuit — a circuit of linear depth but with each layer acting only on one subregion in the system. We discuss how the Sequential Quantum Circuit can be used to generate nontrivial gapped states with long range correlation or long range entanglement, perform renormalization group transformation in foliated fracton order, and create defect excitations inside the bulk of a higher dimensional topological state.

It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit. Here we show that this belief may or may not be true depending on a particular code. To this end, we characterize a tradeoff between the code distance d quantifying the number of correctable errors, and geometric entanglement of logical states quantifying their maximal overlap with product states or more general ``topologically trivial" states. The maximum overlap is shown to be exponentially small in d for three families of codes: (1) low-density parity check (LDPC) codes with commuting check operators, (2) stabilizer codes, and (3) codes with a constant encoding rate. Equivalently, the geometric entanglement of any logical state of these codes grows at least linearly with d. On the opposite side, we also show that this distance-entanglement tradeoff does not hold in general. For any constant d and k (number of logical qubits), we show there exists a family of codes such that the geometric entanglement of some logical states approaches zero in the limit of large code length.

I will discuss recent progress in understanding entanglement-based probes of 2D topological phases of matter. These probes are supposed to extract universal topological information from a many-body ground state. Specifically, I will discuss (1) the topological entanglement entropy, which is supposed to give information about the number of anyon excitations, and (2) the modular commutator, which is supposed to tell us the chiral central charge.

Certifying that an n-qubit state synthesized in the lab is close to the target state is a fundamental task in quantum information science. However, existing rigorous protocols either require deep quantum circuits or exponentially many single-qubit measurements. In this work, we prove that almost all n-qubit target states, including those with exponential circuit complexity, can be certified from only O(n^2) single-qubit measurements. This result is established by a new technique that relates certification to the mixing time of a random walk. Our protocol has applications for benchmarking quantum systems, for optimizing quantum circuits to generate a desired target state, and for learning and verifying neural networks, tensor networks, and various other representations of quantum states using only single-qubit measurements. We show that such verified representations can be used to efficiently predict highly non-local properties that would otherwise require an exponential number of measurements. We demonstrate these applications in numerical experiments with up to 120 qubits, and observe advantage over existing methods such as cross-entropy benchmarking (XEB).

The steady states of dynamical processes can exhibit stable nontrivial phases, which can also serve as fault-tolerant classical or quantum memories. For Markovian quantum (classical) dynamics, these steady states are extremal eigenvectors of the non-Hermitian operators that generate the dynamics, i.e., quantum channels (Markov chains). However, since these operators are non-Hermitian, their spectra are an unreliable guide to dynamical relaxation timescales or to stability against perturbations. We propose an alternative dynamical criterion for a steady state to be in a stable phase, which we name uniformity: informally, our criterion amounts to requiring that, under sufficiently small local perturbations of the dynamics, the unperturbed and perturbed steady states are related to one another by a finite-time dissipative evolution. We show that this criterion implies many of the properties one would want from any reasonable definition of a phase. We prove that uniformity is satisfied in a canonical classical cellular automaton, and provide numerical evidence that the gap determines the relaxation rate between nearby steady states in the same phase, a situation we conjecture holds generically whenever uniformity is satisfied. We further conjecture some sufficient conditions for a channel to exhibit uniformity and therefore stability.