Symmetry enforced entanglement in maximally mixed states
Subhayan Sahu Perimeter Institute for Theoretical Physics
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Unraveling quantum many-body scars: Insights from collective spin models
Meenu Kumari National Research Council Canada (NRC)
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Typical eigenstate entanglement entropy as a diagnostic of quantum chaos and integrability
Marcos Rigol Pennsylvania State University
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Approximate Quantum Codes From Long Wormholes
Brian Swingle Brandeis University
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Defining stable steady-state phases of open systems
Sarang Gopalakrishnan Princeton University
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Certifying almost all quantum states with few single-qubit measurements
Hsin-Yuan Huang California Institute of Technology (Caltech)
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Symmetry enforced entanglement in maximally mixed states
Subhayan Sahu Perimeter Institute for Theoretical Physics
Entanglement in quantum many-body systems is typically fragile to interactions with the environment. Generic unital quantum channels, for example, have the maximally mixed state with no entanglement as their unique steady state. However, we find that for a unital quantum channel that is `strongly symmetric', i.e. it preserves a global on-site symmetry, the maximally mixed steady state in certain symmetry sectors can be highly entangled. For a given symmetry, we analyze the entanglement and correlations of the maximally mixed state in the invariant sector (MMIS), and show that the entanglement of formation and distillation are exactly computable and equal for any bipartition. For all Abelian symmetries, the MMIS is separable, and for all non-Abelian symmetries, the MMIS is entangled. Remarkably, for non-Abelian continuous symmetries described by compact semisimple Lie groups (e.g. SU(2)), the bipartite entanglement of formation for the MMIS scales logarithmically ∼logN with the number of qudits N.
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Quantum spatiotemporal correlations and Bayes' rule
While quantum correlations between two spacelike-separated systems are fully encoded by the bipartite density operator associated with the joint system, what operator encodes quantum correlations across space and time? I will describe the general theory of such "quantum states over time" as well as a canonical example that encodes the expectation values of certain observables measured sequentially in time. The latter extends the theory of pseudo-density matrices to arbitrary dimensions, not necessarily restricted to multi-qubit systems. In addition, quantum states over time admit a natural proposal for a general-purpose quantum Bayes' rule. Our results specialize to many well-studied examples, such as the state-update rule, the two-state vector formalism and weak values, and the Petz recovery map. This talk is based on joint work with James Fullwood and the two papers: arXiv: 2212.08088 [quant-ph] and 2405.17555 [quant-ph].
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Emergent symmetries and their application to logical gates in quantum LDPC codes
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. -
Universal quantum computation in two dimensions by converting between the toric code and a non-abelian quantum double
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. -
Fault tolerance with the ZX-calculus and fusion complexes
Naomi Nickerson PSI Quantum
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. -
Landscape of Measurement-Prepared Tensor Networks and Decohered Non-Abelian Topological Order
Ruben Verresen University of Chicago
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]. -
Unraveling quantum many-body scars: Insights from collective spin models
Meenu Kumari National Research Council Canada (NRC)
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. -
Typical eigenstate entanglement entropy as a diagnostic of quantum chaos and integrability
Marcos Rigol Pennsylvania State University
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. -
Approximate Quantum Codes From Long Wormholes
Brian Swingle Brandeis University
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 -
Defining stable steady-state phases of open systems
Sarang Gopalakrishnan Princeton University
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 almost all quantum states with few single-qubit measurements
Hsin-Yuan Huang California Institute of Technology (Caltech)
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).