Quantum Information

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

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

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

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

Recent advances in mechanical sensing technologies have led to the suggestion that heavy dark matter candidates around the Planck mass range could be detected through their gravitational interaction alone. The Windchime collaboration is developing the necessary techniques, systems, and experimental apparatus using arrays of optomechanical sensors that operate in the regime of high-bandwidth force detection, i.e., impulse metrology.  Today's sensors can be limited by the added noise due to the act of measurement itself. Techniques to go beyond this limit include squeezing of the light used for measurement and backaction evading measurement by estimating quantum non-demolition operators — typically the momentum of a mechanical resonator well above its resonance frequency. In this talk, we will discuss the theoretical limits to noise reduction using such quantum enhanced readout techniques for these optomechanical sensors.

 In holography, spacetime is emergent and its properties depend on the entanglement structure of the dual theory. An interesting question is how changes in the entanglement structure affect the bulk dual description. In this talk, I will describe how local projective measurements performed on a subregion  of the boundary theory modify the bulk dual spacetime. The post-measurement bulk is cut off by end-of-the-world branes and is dual to the complementary unmeasured region . Using a bulk calculation in —which involves a phase transition triggered by the measurement—and tensor network models of holography, I will show that the portion of bulk preserved after the measurement depends on the size of  and the state we project on. Interestingly, the post-measurement bulk includes regions that were part of the entanglement wedge of  before the measurement. Our results indicate that the effect of a measurement performed on a subregion  of the boundary is to teleport part of the bulk information contained in  into the complementary region . Finally, I will comment on applications to the eternal black hole in JT gravity (dual to the SYK thermofield double state) and the relationship between measurements and traversable wormholes.

I will present a general bulk reconstruction technique in AdS/CFT suitable for addressing a facet of the black hole information problem: How to unambiguously predict the results of measurements performed by an infalling observer in the black hole interior.
 
I will explicitly apply the method in the AdS_2/SYK correspondence. My proposal provides an internal notion of time for quantum gravitational systems that may be useful for cosmology.

Quantum theory has a temporal composition, which is expressed under many different operational frameworks. Here, points in time are imbued with a Hilbert space structure, and quantum states are passed between times through a series of experimental interventions. A multi-time quantum process, therefore, carries the same complex properties as a many-body quantum state. This invites the question: to what extent can temporal correlations be as interesting as spatial ones, and how can we access them? One particular avenue through which this structure manifests is in open quantum systems. System-environment dynamics can precipitate non-Markovian processes by which correlations persist between different times. Recently, the advent of high-fidelity quantum devices has made it possible to probe coherent quantum systems. In this talk, I will discuss my recent work in which we show how this serves as a novel test bed to capture many-time physics. We build frameworks to extract generic spatiotemporal properties of quantum stochastic processes, show how process complexity may be manipulated, and elevate user-control into the theory to make it self-consistent. Remarkably, many of these complex features are already present in naturally occurring noise, and hence the results have direct application to the development of fault-tolerant quantum devices. I will also briefly discuss some of my future research goals: the existence of exotic temporal phenomena and how emergent spatiotemporal features can be captured through renormalisation group approaches; the learnability of spacetime quantum correlations and avenues here to quantum advantage; and the taming of correlated noise in quantum devices through bespoke error suppression and error correction. 

Quantum neural networks have been widely studied in recent years, given their potential practical utility and recent results regarding their ability to efficiently express certain classical data. However, analytic results to date rely on assumptions and arguments from complexity theory. Due to this, there is little intuition as to the source of the expressive power of quantum neural networks or for which classes of classical data any advantage can be reasonably expected to hold. In this talk, I will discuss my recent results (arXiv:2209.14353) studying the relative expressive power between a broad class of neural network sequence models and a class of recurrent models based on Gaussian operations with non-Gaussian measurements. We explicitly show that quantum contextuality is the source of an unconditional memory separation in the expressivity of the two model classes. Additionally, we use this intuition to study the relative performance of our introduced model on a standard translation data set exhibiting linguistic contextuality and show that the quantum model outperforms state-of-the-art classical models even in practice. I will also briefly discuss connections to my previous work studying the trainability of variational quantum algorithms (arXiv:2109.06957, arXiv:2205.05786).

Constraint satisfaction problems are known to always be "easy" or "hard", in the sense of being either solvable in P or being NP-complete, with no intermediate difficulty levels. The quantum analog of constraint problems, frustration-free Hamiltonians, are known to exhibit at least two more levels of complexity: QMA (for arbitrary local Hamiltonians) and MA (for stoquastic Hamiltonians). Wondering if other complexity classes can occur, we answer in the affirmative: there are interactions which can be freely arranged on qubits in any arrangement, such that the resulting frustration problem is BQP-complete, and captures exactly the difficulty of quantum computation. Simple modifications of this construction show that quantum constraint problems can be complete for QCMA and BPP as well. Based on https://arxiv.org/abs/2101.08381