Quantum Information

We present a quantum algorithm for simulating the classical dynamics of 2^n coupled oscillators (e.g.,  masses coupled by springs). Our approach leverages a mapping between the Schrodinger equation and Newton's equations for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators. When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in n, almost linear in the evolution time, and sublinear in the sparsity. As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time, for a specification of the problem that we prove is BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers. Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with 2^n modes.

Zoom link:  https://pitp.zoom.us/j/91882209363?pwd=UndJRVdaZW04RGtpL0M2SE52RDJwZz09

Entanglement entropies of two-dimensional gapped ground states are expected to satisfy an area law, with a constant correction term known as the topological entanglement entropy (TEE). In many models, the TEE takes a universal value that characterizes the underlying topological phase. However, the TEE is not truly universal: it can differ even for two states related by constant-depth circuits, which are necessarily in the same phase. The difference between the TEE and the value predicted by the anyon theory is often called the spurious topological entanglement entropy. We show this spurious contribution is always nonnegative, thus the value predicted by the anyon theory provides a universal lower bound. This observation also leads to a definition of TEE which is invariant under constant-depth quantum circuits.

Based on a joint work with Daniel Ranard (MIT), Michael Levin (U Chicago), Ting-Chun Lin (UCSD), and Bowen Shi (UCSD)

Zoom link:  https://pitp.zoom.us/j/93809342984?pwd=QXJkK0ZYR3QzdjlUUWlkMEFXMWRnQT09

For n≥3, we demonstrate the existence of quantum codes which are local in dimension n with V qubits, distance V^{(n−1)/n}, and dimension V^{(n−2)/n}, up to a polylog(V) factor. The distance is optimal up to the polylog factor. The dimension is also optimal for this distance up to the polylog factor. The proof combines the existence of asymptotically good quantum codes, a procedure to build a manifold from a code by Freedman-Hastings, and a quantitative embedding theorem by Gromov-Guth.

Zoom link:  https://pitp.zoom.us/j/96227350980?pwd=TEFtamRmTXg3dnBMNEhKUkpHbmVoZz09

Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing using no or few additional quantum resources, in contrast to fault-tolerant schemes that come along with heavy overheads. Error mitigation has been successfully applied to reduce noise in near-term applications. 

In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively `undone' for larger system sizes. We set up a framework that rigorously captures large classes of error mitigation schemes in use today. The core of our argument combines fundamental limits of statistical inference with a construction of families of random circuits that are highly sensitive to noise.

We show that even at poly loglog depth, a super-polynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. They also impact other near-term applications, constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.

Zoom link:  https://pitp.zoom.us/j/95736148335?pwd=akZLaHE5aStNQVZOeVFETlltNzVwdz09

Recent progress in the development of quantum technologies has enabled the direct investigation of dynamics of increasingly complex quantum many-body systems. This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage. Here we present classical algorithms for approximating the dynamics of local observables and nonlocal quantities such as the Loschmidt echo, where the evolution is governed by a local Hamiltonian. For short times, their computational cost scales polynomially with the system size and the inverse of the approximation error. In the case of local observables, the proposed algorithm has a better dependence on the approximation error than algorithms based on the Lieb–Robinson bound. Our results use cluster expansion techniques adapted to the dynamical setting, for which we give a novel proof of their convergence. This has important physical consequences besides our efficient algorithms. In particular, we establish a novel quantum speed limit, a bound on dynamical phase transitions, and a concentration bound for product states evolved for short times.

Joint work with Dominik S. Wild (arXiv:2210.11490)

Zoom link:  https://pitp.zoom.us/j/97166113711?pwd=UEg4NnJlQkkxQXFnN2xXYTBxS001Zz09

We present an efficient machine learning (ML) algorithm for predicting any unknown quantum process over n qubits. For a wide range of distributions D on arbitrary n-qubit states, we show that this ML algorithm can learn to predict any local property of the output from the unknown process, with a small average error over input states drawn from D. The ML algorithm is computationally efficient even when the unknown process is a quantum circuit with exponentially many gates. Our algorithm combines efficient procedures for learning properties of an unknown state and for learning a low-degree approximation to an unknown observable. The analysis hinges on proving new norm inequalities, including a quantum analogue of the classical Bohnenblust-Hille inequality, which we derive by giving an improved algorithm for optimizing local Hamiltonians. Overall, our results highlight the potential for ML models to predict the output of complex quantum dynamics much faster than the time needed to run the process itself.

Zoom link:  https://pitp.zoom.us/j/93857777354?pwd=c044blZuQVhLS200ME4vN25uaGJudz09

Tom Leinster recently introduced a curious notion of entropy modulo p (https://arxiv.org/abs/1903.06961). While entropy has a certain meaning in information theory and physics, mathematically it is simply a function with certain properties. Stating these as axioms, the function is unique. Surprisingly, Leinster shows that a function obeying the same axioms can also be found for "probability distributions" over a finite field, and this function is unique too.

In quantum information, mutually unbiased bases is an important set of measurements and an example of a quantum design. While in odd prime power dimensions their construction is based on a finite field, in dimension 2^n it relies on an unpleasant Galois ring. I will replace this ring by length-2 Witt vectors whose arithmetic involves only finite field operations and Leinster's entropy mod 2. This expresses qubit mutually unbiased bases entirely in terms of a finite field and allows deriving an explicit unitary correspondence between them and the affine plane over this field.

Zoom link:  https://pitp.zoom.us/j/94032116379?pwd=TTI1RnByQnFuVHp1MytFUlJxckM4Zz09

Kronecker coefficients appear in representation of the symmetric group in the decomposition of tensor products of irreducible representations. They are notoriously difficult to compute and it is a long standing problem to find a combinatorial expression for them. 

We study the problem of computing Kronecker coefficients from quantum computational perspective. First, we show that the coefficients can be expressed as a dimension of a subspace given by intersection of two commuting, efficiently implementable projectors and relate their computation to the recently introduced quantum approximate counting class (QAPC). Using similar construction, we show that deciding positivity of Kronecker coefficients is contained in QMA. We give similar results for a related problem of approximating row sums in a character table of the symmetric group and show that its decision variant is in QMA. We then discuss two quantum algorithms - one that samples a distribution over squared characters and another one that approximates normalized Kronecker coefficients to inverse-polynomial additive error. We show that under a conjecture about average-case hardness of computing Kronecker coefficients, the resulting distribution is hard to sample from classically. 

Our work explores new structures for quantum algorithms and improved characterization of the quantum approximate counting.

Joint work with David Gossett, Sergey Bravyi, Anirban Chowdhury and Guanyu Zhu 

Zoom link:  https://pitp.zoom.us/j/95976938016?pwd=eDV3TXZReHo5UHdvZ0hIbkhXOFcxQT09

Classical simulation algorithms provide a rigorous ground for investigating quantum resources responsible for quantum speedup. In my talk, I will consider one such algorithm provided by Lambda polytopes. These polytopes are defined to be the polar dual of the stabilizer polytopes and can be used to provide a hidden variable model for finite-dimensional quantum theory. This hidden variable model can be turned into a  classical algorithm that can simulate any quantum computation. The efficiency of this algorithm depends on the combinatorial structure of the polytope. In general, which subset of the vertices gives rise to efficient simulation is an open problem. I will describe some of the known classes of vertices and available methods for studying this polytope.

Zoom link:  https://pitp.zoom.us/j/95216680309?pwd=aGlIN2NtZVRtczdHcXl5RzgzQTlOdz09

Experimental realizations of long-range entangled states such as quantum spin liquids are challenging due to numerous complications in solid state materials. Digital quantum simulators, on the other hand, have recently emerged as a promising platform to controllably simulate exotic phases. I will talk about a constructive design of long-range entangled states in this setting, and exploit competing measurements as a new source of frustration to generate spin liquid. Specifically, we consider random projective measurements of the anisotropic interactions in the Kitaev honeycomb model. The monitored trajectories can produce analogues of the two phases in the original Kitaev model: (i) a topologically-ordered phase with area-law entanglement and two protected logical qubits, and (ii) a “critical” phase with a logarithmic violation of area-law entanglement and long-range tripartite entanglement. A Majorana parton description permits an analytic understanding of these two phases through a classical loop model. Extensive numerical simulations of the  monitored dynamics confirm our analytic predictions. This talk is based on https://arxiv.org/abs/2207.02877.

Zoom link:  https://pitp.zoom.us/j/99600719755?pwd=a0pOWlliU0swVDdGYnhxaGFGNkJSdz09

A large class of Λ < 0 cosmologies have big-bang / big crunch spacetimes with time-symmetric backgrounds and asymptotically AdS Euclidean continuations suggesting a possible holographic realization. We argue that these models generically have time-dependent scalar fields, and these can lead to realistic cosmologies at the level of the homogeneous background geometry, with an accelerating phase prior to the turnaround and crunch. We first demonstrate via explicit effective field theory examples that models with an asymptotically AdS Euclidean continuation can also exhibit a period of accelerated expansion without fine tuning. We then show that certain significantly more tuned examples can give predictions arbitrarily close to a ΛCDM model. Finally, we demonstrate via an explicit construction that the potentials of interest can arise from a superpotential, thus suggesting that these solutions may be compatible with an underlying supersymmetric theory.

This talk is based on 2212.00050.

Zoom link:  https://pitp.zoom.us/j/93499736007?pwd=Qmw5cmZERUN3UmtwTzdKcEdXejJ5UT09

Typicality, the feature that almost any microstate living in the microcanonical subspace cannot be locally distinguished from a thermal ensemble, lies at the fundamental part of statistical physics. However, one may wonder if there exists a sufficient amount of orthogonal atypical states to account for the whole entropy. 
In this talk, we show that, in some physical systems, there exists a sufficient amount of certain orthogonal atypical states to account for the leading order of the entropy in the following two scenarios, by finding proper upper bounds of the entanglement of formation (EoF) for each case and applying other techniques from quantum information theory. 
In the first scenario, the physical system under consideration is AdS black holes at the semiclassical limit G_N —> 0. In this case, microcanonical subspace is the subspace formed by the black hole microstates, and typical states are usually considered to have a smooth horizon as well as the black hole interior. We consider a class of atypical states called disentangled states which have large entanglement deficits compared to typical states such that they cannot have smooth horizons. In this scenario, we use a geometric quantity called entanglement wedge cross section to give upper bounds to EoF. 
In the second scenario, we consider generic quantum many-body systems with short-ranged interactions at the standard thermodynamic limit V —> ∞. In this case, it is known that typical microstates have volume law entanglement. We consider area-law entangled microstates as atypical states. We use reflected entropy to give upper bounds to EoF. 
We will also discuss the relations of our results with the additivity conjectures and atypical black hole microstate counting. 
This talk is based on 2211.11787. 

Zoom link:  https://pitp.zoom.us/j/94823687961?pwd=V3QrSjUrTklheG1iU0RsckwzQmRTUT09