Quantum Information

In this talk, I’ll discuss the deep connection between emergent k-form symmetries and transversal logical gates in quantum low-density parity-check (LDPC) codes. I’ll then present a parallel fault-tolerant quantum computing scheme for families of homological quantum LDPC codes defined on 3-manifolds with constant or almost-constant encoding rate using the underlying higher symmetries in our recent work. We derive a generic formula for a transversal T gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical CCZ gates on any triplet of logical qubits with their logical-X membranes having a Z2 triple intersection at a single point. The triple intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the (Z2) 3 gauge theory. Moreover, the transversal S gate of the color code corresponds to a higher-form symmetry supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical CZ gates. Both symmetries are related to gauged SPT defects in the (Z2) 3 gauge theory. We have then developed a generic formalism to compute the triple intersection invariants for general 3- manifolds. We further develop three types of LDPC codes supporting such logical gates with constant or almost-constant encoding rate and logarithmic distance. Finally, I’ll point out a connection between the gauged SPT defects in the 6D color code and a recently discovered non-Abelian self-correcting quantum memory in 5D. Reference: arXiv:2310.16982, arXiv:2208.07367, arXiv:2405.11719.

In this talk, I will explain how to implement fault-tolerant non-Clifford gates in copies of toric code in two dimensions achieved by transiently switching to a non-Abelian topologically ordered phase by expanding earlier results by Bombin [arXiv.1810.09571] and Brown [SciAdv.aay4929]. This addresses the challenge of performing universal fault-tolerant quantum computation in purely two spatial dimensions and shows a new approach to quantum computation using non-Abelian phases. This talk is based on upcoming work in collaboration with A. Bauer, B.Brown, J. Magdalena de la Fuente, M. Webster and D. Williamson.

Quantum error correction methods for qubit technologies such as ions, photons, or superconducting qubits can appear very different at first glance. Moreover, as more detailed error models are accounted for, the relationship to the abstract models of fault tolerance can appear to become more distant. In this talk I will discuss two unifying frameworks which connect hardware specific models more closely to the underlying code structures, which can help enable QEC development. First I will introduce a unifying framework for fault tolerance based on the ZX calculus (arXiv:2303.08829) and show how it allows us to view circuit-based, measurement-based, fusion-based quantum computation, and Floquet codes as different flavors of the same underlying stabilizer fault-tolerance structure. Secondly I will introduce fusion complexes (arXiv:2308.07844) which allows a topological interpretation of fault tolerance even under circuit level error models. Both of these frameworks are tools that can aid in the design of quantum error correction methods under hardware-focussed models, and I will give some examples of this applied to the design of photonic architectures.

What is the structure of many-body quantum phases and transitions in the presence of non-unitary elements, such as decoherence or measurements? In this talk we explore two new directions. First, recent works have shown that even if one starts with an ideal preparation of topological order such as the toric code, decoherence can lead to interesting mixed states with subtle phase transitions [e.g., Fan et al, arXiv:2301.05689]. Motivated by a recent experimental realization of non-Abelian topological order [Iqbal et al, Nature 626 (2024)], we generalize this to decohered non-Abelian states, based on work with Pablo Sala and Jason Alicea [to appear]. Second, we study whether and how one can prepare pure states which are already detuned from ideal fixed-point cases---with tunable correlation lengths. This turns out to be possible for large classes of tensor network states which can be deterministically prepared using finite-depth measurement protocols. This is based on two recent works with Rahul Sahay [arXiv:2404.17087; arXiv:2404.16753].

Quantum many-body scars (QMBS) are atypical eigenstates of chaotic systems that are characterized by sub-volume or area law entanglement as opposed to the volume law present in the bulk of the eigenstates. The term, QMBS, was coined using heuristic correlations with quantum scars - eigenstates with high probability density around unstable classical periodic orbits in quantum systems with a semiclassical description. Through the study of entanglement in a multi-qubit system with a semiclassical description, quantum kicked top (QKT), we show that the properties of QMBS states strongly correlate with the eigenstates corresponding to the very few stable periodic orbits in a chaotic system as opposed to quantum scars in such systems. Specifically, we find that eigenstates associated with stable periodic orbits of small periodicity in chaotic regime exhibit markedly different entanglement scaling compared to chaotic quantum states, while quantum scar eigenstates demonstrate entanglement scaling resembling that of chaotic quantum states. Our findings reveal that quantum many-body scars and quantum scars are distinct. This work is in collaboration with Cheng-Ju Lin and Amirreza Negari.

Quantum-chaotic systems are known to exhibit eigenstate thermalization and to generically thermalize under unitary dynamics. In contrast, quantum-integrable systems exhibit a generalized form of eigenstate thermalization and need to be described using generalized Gibbs ensembles after equilibration. I will discuss evidence that the entanglement properties of highly excited eigenstates of quantum-chaotic and quantum-integrable systems are fundamentally different. They both exhibit a typical bipartite entanglement entropy whose leading term scales with the volume of the subsystem. However, while the coefficient is constant and maximal in quantum- chaotic models, in integrable models it depends on the fraction of the system that is traced out. The latter is typical in random Gaussian pure states. I will also discuss the nature of the subleading corrections that emerge as a consequence of the presence of abelian and nonabelian symmetries in such models.

We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the Sachdev-Ye-Kitaev (SYK) model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, N, of fermions goes to infinity. For SYK, the distance scales as N^1/2, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. N^.99, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property and show that these models do have trivial low-energy states in the sense of adiabatic continuity. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity

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.

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).

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.

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.

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.