Schedule for: 22w5111 - Probability and Quantum Information Science (Online)

Google supremacy experiments have raised new statistical questions about random quantum circuits. After presenting some of these, we delve into questions related to a particular random process, the so-called averaging process, which is studied at least from the 80s. Recently Sourav Chatterjee, Persi Diaconis, Allan Sly and Lingfu Zhang have established a sharp convergence cutoff of \( 1/(2 * \log 2) * n * \log n + O(n * \sqrt(\log n)) \) when we run the process on the complete graph. We prove that the complete graph is an extreme case: we cannot get a quicker convergence on any other graph. We also resolve a conjecture of Sam Spiro: the convergence in the case of the the cycle graph occurs in time \( \Theta(n^3) \). We also make several other observations about the process. Joint with Ramis Movassagh and Guanyang Wang.

Quantifying quantum states' complexity is a key problem in various subfields of science, from quantum computing to black-hole physics. We prove a prominent conjecture by Brown and Susskind about how random quantum circuits' complexity increases. Consider constructing a unitary from Haar-random two-qubit quantum gates. Implementing the unitary exactly requires a circuit of some minimal number of gates—the unitary's exact circuit complexity. We prove that this complexity grows linearly with the number of random gates, with unit probability, until saturating after exponentially many random gates. Our proof is surprisingly short, given the established difficulty of lower-bounding the exact circuit complexity. Our strategy combines differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits. References 1) Haferkamp, Faist, Kothakonda, Eisert, and NY H, accepted by Nat. Phys. (in press) arXiv:2106.05305. 2) NY H, Kothakonda, Haferkamp, Munson, Faist, and Eisert, arXiv:2110.11371 (2021).

For single-particle quantum systems described by Schrödinger operators, Anderson localization is a phenomenon of having dense purely point spectrum with exponentially decaying eigenfunctions. One can relate is with absence of transport and presence of disorder, such as random disorder in the Anderson model. In the one-dimensional case, the tight binding Anderson model demonstrates localization regardless of the strength of the coupling at the disorder term. In 1980s, it was discovered that models with ergodic non-random disorder, such as quasiperiodic operators, demonstrate Anderson localization at large coupling. However, the case of small coupling for analytic quasiperiodic operators is very different and one can show the presence of absolute continuous spectrum and transport. In the present talk, I will focus on a large class of quasiperiodic operators with monotone or locally monotone potentials which, under relatively mild regularity conditions, have much stronger localization properties than those with analytic potentials. In particular, one can obtain localization at small coupling in ID, such as in random models, and uniform localization that does not appear in other models, which allows to argue that these operators demonstrate more "random-like" behavior than analytic quasiperiodic operators. In some regimes, one can prove localization by directly constructing converging perturbation series for eigenvalues and eigenvectors, while in other regimes the methods are very indirect.

The Quantum search algorithm (also known as Grover's algorithm) lays the foundation of many other quantum algorithms. It demonstrates an advantage (for unstructured search) over the classical algorithm. Although it is very simple, its implementation is limited on the noisy intermediate-scale quantum (NISQ) processors. The limitation is due to the circuit depth, rather than the number of qubits. For the first time, we present detailed and completed benchmarks of the five-qubit quantum search algorithm on different quantum processors, including IBMQ, IonQ and Honeywell quantum devices. Besides Grover's algorithm, we also present the implementations of a recently proposed optimized quantum search algorithm [Phys. Rev. A 101, 032346 (2020)]. Noisy simulations are also applied to interpret the results. We report the highest success probability of the five-qubit search algorithm compared to previous works. Our study shows that different quantum processors, with different levels of errors, have different optimal ways to realize the quantum search algorithms. The original Grover's algorithm might not be the best choice. To maximize the power of NISQ computers, designing error-aware algorithms is necessary.

In the design of quantum algorithms, randomness can play an essential role. In this talk, I will describe how quantum random circuits can be used to sample quantum states from a Gibbs distribution. I will describe two novel algorithms in particular. The first is a universal algorithm that can be applied to any local Hamiltonian. The second algorithm is an efficient way to prepare Gibbs states for Hamiltonians exhibiting eigenstate thermalization hypothesis. I will provide an overview of the advantages of these algorithms for noisy quantum hardware and illustrate how optimizing over initially random parameters can mitigate noise. (joint work with Ramis Movassagh)

Panel discussion: Vladimir Korepin, Seth Lloyd, Jarrod McClean. Moderator: Ramis Movassagh

The celebrated Lieb-Robinson bound asserts the existence of a maximal propagation speed for the quantum dynamics of lattice spin systems. Analogous general bounds are not available for most bosonic lattice gases due to their unbounded local interactions. Here we establish a general ballistic upper bound on macroscopic particle transport in the paradigmatic Bose-Hubbard model. The bound is the first to cover a broad class of initial states with positive density including Mott states, which resolves a longstanding problem. It applies to Bose-Hubbard type models on any lattice with not too long-ranged hopping. The proof rests on controlling the time evolution of a new kind of adiabatic spacetime localization observable via iterative differential inequalities. Joint work with J. Faupin and I.M. Sigal.

Matrix Product Operators (MPO) are an ubiquitous mathematical structure in the area of quantum information. Despite their apparent simplicity, they encode very complex problems. In this talk, I will focus on their use to understand exotic (topological) phases of matter.

With the rapid development of quantum technology, one of the leading applications is the simulation of chemistry. Interestingly, even before full scale quantum computers are available, quantum computer science has exhibited a remarkable string of results that directly impact what is possible in chemical simulation with any computer. Some of these results even impact our understanding of chemistry in the real world. In addition, quantum technology has the potential to revolutionize how we acquire and process experimental data to learn about the physical world. We show that an experimental setup that transduces data from a physical system to a stable quantum memory, and processes that data using a quantum computer, could have significant advantages over conventional experiments in which the physical system is measured and the outcomes are processed using a classical computer. We prove that, in various tasks, quantum machines can learn from exponentially fewer experiments than those required in conventional experiments. The exponential advantage holds in predicting properties of physical systems, performing quantum principal component analysis on noisy states, and learning approximate models of physical dynamics. Conducting experiments with up to 40 superconducting qubits and 1300 quantum gates, we demonstrate that a substantial quantum advantage can be realized using today's relatively noisy quantum processors.

Quantum channel discrimination is a fundamental problem in quantum information science. In this study, we consider general quantum channel discrimination problems, and derive the lower bounds of the error probability. Our lower bounds are based on the triangle inequalities of the Bures angle and the trace distance. As a consequence of the lower bound based on the Bures angle, we prove the optimality of Grover's search if the number of marked elements is fixed to some integer k. This result generalizes Zalka's result for k=l. We also present several numerical results in which our lower bounds based on the trace distance outperform recently obtained lower bounds.

We consider two families of quantum spin chains with $O(n)$-invariant nearest-neighbor interactions and discuss the ground state phase diagram of this class of models. Using a graphical representation for the partition function, we provide a proof of a spectral gap above the ground states and spontaneous breaking of the translation symmetry for an open region in the phase diagram, for all sufficiently large values of n. (Joint work with Jakob Bjoernberg, Peter Muehlbacher, and Daniel Ueltschi).

Random quantum circuits (RQCs) are valuable constructions in both quantum information and quantum many-body physics. They are a solvable model of strongly-interacting quantum dynamics, efficient implementations of quantum pseudorandomness, and have been the central focus of recent demonstrations of quantum computational advantage. In this talk I’ll discuss some very useful techniques for studying RQCs involving both a statistical-mechanical mapping to lattice model as well as bounding the spectral gap of a local Hamiltonian. Using these approaches, we’ll compute the depth at which RQCs form approximate unitary designs (probability distributions which 'look like' Haar random unitaries) and study how RQCs scramble local noise.

Open quantum systems are everywhere in real life, whether it is systems exposed to temperature, electric field, or anything that causes nonequilibrium decay and/or driving. We find that the study of open quantum systems has particular promise in the area of physics. We will discuss practical models we have run on quantum hardware, which turns out to be an excellent platform for such simulations, being inherently robust against quantum hardware errors even with deep circuits. We give two examples: 1) we simulate one thousand steps of time evolution for an infinite (U=0) Hubbard model in a driving electric field while cooled by a temperature bath, and calculate the current through the system; and 2) we simulate the thermalization of two interacting electrons in an orbital, a Hubbard atom, in a magnetic field. These problems were solved using circuits containing up to two thousand entangling gates on quantum computers made available by IBM, showing no signs of decreasing fidelity at long times. Our results demonstrate that our algorithms for simulating open quantum systems, analysis and execution of very long runs on current quantum computers and associated error correction, are able to far outperform similarly complex non-dissipative algorithms on noisy hardware. Our two examples are the basic building blocks of many condensed matter physics systems, and we anticipate their demonstrated robustness to hold with increasing complexity of open quantum problems. Quantum computing is an inherently probabilistic process, and I will discuss some applications to quantum information as well.

Error-mitigation techniques can enable access to accurate estimates of physical observables that are otherwise biased by noise in pre-fault-tolerant quantum computers. One particularly general error-mitigation technique is probabilistic error cancellation (PEC), which effectively inverts a well-characterized noise channel to produce noise-free estimates of observables. Experimental realizations of this technique, however, have been impeded by the challenge of learning correlated noise in large quantum circuits. In this work, we present a practical protocol for learning a sparse noise model that scales to large quantum devices and is efficient to learn and invert. These advances enable us to demonstrate PEC on a superconducting quantum processor with crosstalk errors, thereby revealing a path to error-mitigated quantum computation with noise-free observables at larger circuit volumes. (https://arxiv.org/abs/2201.09866)