Quantum Information

Minimizing the energy of a many-body system is a fundamental problem in many fields. Although we hope a quantum computer can help us solve this problem better than classical computers, we have a very limited understanding of where a quantum advantage may be found. In this talk, I will present some recent theoretical advances that shed light on quantum advantages in this domain. First, I describe rigorous analyses of the Quantum Approximate Optimization Algorithm applied to minimizing energies of classical spin glasses. For certain families of spin glasses, we find the QAOA has a quantum advantage over the best known classical algorithms. Second, we study the problem of finding a local minimum of the energy of quantum systems. While local minima are much easier to find than ground states, we show that finding a local minimum under thermal perturbations is computationally hard for classical computers, but easy for quantum computers.

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Zoom link TBA

As quantum technologies are expected to provide us with unprecedented benefits, identifying what quantum-mechanical properties are useful is a pivotal question. Quantum resource theories provide a unified framework to analyze such quantum properties, which has been successful in the understanding of fundamental properties such as entanglement and coherence. While these are examples of convex resources, for which quantum advantages can always be identified, many physical resources are described by a non-convex set of free states and their interpretation has so far remained elusive. In this work, we address the fundamental question of the usefulness of quantum resources without convexity assumption, by providing two operational interpretations of the generalized robustness resource measure in general resource theories. On the one hand, we characterize the generalized robustness in terms of a non-linear resource witness and reveal that any state is more advantageous than a free one in some multi-copy channel discrimination task. On the other hand, we consider a scenario where a theory is characterized by multiple constraints and show that the generalized robustness coincides with the worst-case advantage in a single-copy channel discrimination setting. We further extend these results to the weight resource measure and QRTs for quantum channels and quantum instruments. Based on these characterizations, we conclude that every quantum resource state shows a qualitative and quantitative advantage in discrimination problems in a general resource theory even without any assumption on the structure of the free sets. This talk is based on arXiv:2310.09154 and arXiv:2310.09321.

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Zoom link https://pitp.zoom.us/j/97730859535?pwd=VExLK0hNN2FHNVFWUW12RUM3d05UUT09

The last few years have seen rapid progress in the development of quantum low-density parity-check (LDPC) codes. LDPC codes, defined by their constant weight check operators, can have much better parameters than their topological counterparts like the surface code. In particular, a series of pivotal works culminated in the discovery of asymptotically good LDPC codes--those with essentially optimal rate and distance scalings. These codes allow for the possibility fault-tolerant quantum computation with very low overhead. However, for a code to be used in practice, it is necessary to efficiently identify errors from measurement outcomes to get back into the codespace. In this talk, I will present a linear-time decoder for a family of asymptotically good codes called quantum Tanner codes. Furthermore, I will show that quantum Tanner codes support single-shot decoding, which means that one measurement round suffices to perform reliable quantum error correction, even in the presence of measurement errors. These results can be seen as a step toward making quantum LDPC codes more practical.

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Zoom link https://pitp.zoom.us/j/94286584094?pwd=Q21IekhHZXI4Qlk4Y1B3MnNobmR6UT09

Uncertainty relations are a familiar part of any introductory quantum mechanics course. In this talk, I will summarize how uncertainty relations have been re-interpreted and re-expressed in the language of information theory, leading to connections with the geometry of quantum state space and the limits of computational and information processing efficiency. As two particular examples, I will discuss how uncertainty relations allow one to design information-theoretically optimal measurement protocols for function estimation in networks of quantum sensors and how they enable one to bound the speed at which analog quantum computers can possibly perform optimization tasks. Based primarily on arXiv:2110.07613 and arXiv:2210.15687. 

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Zoom link https://pitp.zoom.us/j/98258695315?pwd=Q2pEcmg5MGhLWmFlR1FPako0NVFlQT09

Entanglement renormalization circuits are quantum circuits that can be used to prepare large-scale entangled states. For years, it has remained a mystery whether there exist scale-invariant entanglement renormalization circuits for chiral topological order. In this paper, we solve this problem by demonstrating entanglement renormalization circuits for a wide class of chiral topologically ordered states, including a state sharing the same topological properties as Laughlin's bosonic fractional quantum Hall state at filling fraction 1/4 and eight states with Ising-like non-Abelian fusion rules. The key idea is to build entanglement renormalization circuits by interleaving the conventional multi-scale entanglement renormalization ansatz (MERA) circuit (made of spatially local gates) with quasi-local evolution. Given the miraculous power of this circuit to prepare a wide range of chiral topologically ordered states, we refer to these circuits as MERA with quasi-local evolution (MERAQLE).\

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Zoom link https://pitp.zoom.us/j/98523529456?pwd=dUlNbUQyemZGNFlCUGFJNStPU0xxdz09

Quantum error-correcting codes are a key pillar of quantum computing. They allow for the recovery of quantum information in the presence of noise. There are three main ways to depict a quantum code: by the list of codewords, by an encoding circuit, or by a stabilizer tableau. Although the latter is used most often, all three of these have pros and cons. In this talk, I will showcase ZX calculus, a graphical language that can be used to talk about quantum states, circuits, measurements, and importantly, codes. I will demonstrate its power and versatility and showcase a canonical form for states, circuits, and codes. The power of this theorem allows us to compile these quantum objects to a unique, elegant, and simple form which serves as a much better depiction than any of the three methods above.

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Zoom link https://pitp.zoom.us/j/91431057660?pwd=cW9xKzNXejdIekp3amFacFZ5OHN5dz09

We develop the theory of quantum scars for quantum fields. By generalizing the formalisms of Heller and Bogomolny from few-body quantum mechanics to quantum fields, we find that unstable periodic classical solutions of the field equations imprint themselves in a precise manner on bands of energy eigenfunctions. This indicates a breakdown of thermalization at certain energy scales, in a manner that can be characterized via semiclassics. As an explicit example, we consider time-periodic non-topological solitons in complex scalar field theories. We find that an unstable variant of Q-balls, called Q-clouds, induce quantum scars. Some technical contributions of our work include methods for characterizing moduli spaces of periodic orbits in field theories, which are essential for formulating our quantum scar formula. We further discuss potential connections with quantum many-body scars in Rydberg atom arrays. Based on work in arXiv:2212.01637 with Jordan Cotler.

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Zoom link https://pitp.zoom.us/j/91572728134?pwd=Q0Jzb0lwQW5VU0ptRnRWL2tOTTdLdz09

Measuring the temperature of your coffee should not change the amount of coffee in your cup. This holds because the operators representing the coffee’s energy and volume commute. The intuitive assumption that conserved quantities, also known as charges, commute, underpins basic physics derivations, like that of the thermal state's form and Onsager coefficients. Yet, operators' failure to commute plays a key role in quantum theory, e.g. underlying uncertainty relations. Lifting this assumption has spawned a growing subfield of quantum many-body physics [1].

How can one argue that charges’ noncommutation caused a result? To isolate the effects of charges’ noncommutation, we created analogous models that differ in whether their charges commute and discovered more entanglement in the noncommuting-charge model [2]. We further introduce noncommuting charges (an SU(2) symmetry) into monitored quantum circuits, circuits with unitary evolutions and mid-circuit projective measurements. Numerically, we find that the SU(2)-symmetric model has a critical phase in place of the area-law phase typically found in these circuits [3]. I will focus on the results from Ref 2 and 3. Time permitting, I'll briefly explain how one can use Lie Algebra theory to build the Hamiltonians necessary for testing the predictions of noncommuting charge physics [4].

[1] Majidy et al. "Noncommuting conserved charges in quantum thermodynamics and beyond." Nat Rev Phys (2023)
[2] Majidy et al. "Non-Abelian symmetry can increase entanglement entropy.” PRB (2023)
[3] Majidy et al. "Critical phase and spin sharpening in SU(2)-symmetric monitored quantum circuits." PRB (2023)
[4] Yunger Halpern and Majidy “How to build Hamiltonians that transport noncommuting charges in quantum thermodynamics” npj QI (2022)

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Zoom link https://pitp.zoom.us/j/97193579200?pwd=MkdmbWo1S2lUcUZtUFpORk5VbnFBdz09