Quantum Information

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

We investigate topological order on fractal geometries embedded in n dimensions. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that Z_N topological order cannot survive on any fractal embedded in 2D. For fractal lattice models embedded in 3D or higher spatial dimensions, Z_N topological order survives if the boundaries of the interior holes condense only loop or membrane excitations. Moreover, for a class of models containing only loop or membrane excitations, and are hence self-correcting on an n-dimensional manifold, we prove that topological order survives on a large class of fractal geometries independent of the type of hole boundaries. We further construct fault-tolerant logical gates using their connection to global and higher-form topological symmetries. In particular, we have discovered a logical CCZ gate corresponding to a global symmetry in a class of fractal codes embedded in 3D with Hausdorff dimension asymptotically approaching D_H=2+ϵ for arbitrarily small ϵ, which hence only requires a space-overhead Ω(d^(2+ϵ)) with d being the code distance. This in turn leads to the surprising discovery of certain exotic gapped boundaries that only condense the combination of loop excitations and gapped domain walls. We further obtain logical C^pZ gates with p≤n−1 on fractal codes embedded in nD. In particular, for the logical C^{n−1}Z in the nth level of Clifford hierarchy, we can reduce the space overhead to Ω(d^(n−1+ϵ)). Mathematically, our findings correspond to macroscopic relative systoles in fractals.

Zoom Link: https://pitp.zoom.us/j/96893356441?pwd=cnlxTVIwd0U5TW9uZDMweXRSa3oydz09

Designing encoding and decoding circuits to reliably send messages over many uses of a noisy channel is a central problem in communication theory. When studying the optimal transmission rates achievable with asymptotically vanishing error it is usually assumed that these circuits can be implemented using noise-free gates. While this assumption is satisfied for classical machines in many scenarios, it is not expected to be satisfied in the near term future for quantum machines where decoherence leads to faults in the quantum gates. As a result, fundamental questions regarding the practical relevance of quantum channel coding remain open. By combining techniques from fault-tolerant quantum computation with techniques from quantum communication, we initiate the study of these questions. As our main result, we prove threshold theorems for quantum communication, i.e. we show that coding near the (standard noiseless) classical or quantum capacity is possible when the gate error is below a threshold.

It is known that continuous variable quantum information cannot be protected against naturally occurring noise using Gaussian states and operations only. Noh et al. (PRL 125:080503, 2020) proposed bosonic oscillator-to-oscillator codes relying on non-Gaussian resource states as an alternative, and showed that these encodings can lead to a reduction of the effective error strength at the logical level as measured by the variance of the classical displacement noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N-mode unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states.
Here we ask if - in analogy to qubit error-correcting codes - there are families of oscillator-to-oscillator codes with the following threshold property: They allow to convert physical displacement noise with variance below some threshold value to logical noise with variance upper bounded by any (arbitrary) constant. We find that this is not the case if encoding unitaries involving a constant amount of squeezing and maximum likelihood error decoding are used. We show a general lower bound on the logical error probability which is only a function of the amount of squeezing and independent of the number of modes. As a consequence, any physically implementable family of oscillator-to-oscillator codes combined with maximum likelihood error decoding does not admit a threshold.

We study the problem of learning a Hamiltonian given copies of its Gibbs state at a known inverse temperature. Anshu et al. recently studied the sample complexity (number of copies of the Gibbs state needed) of this problem for geometrically local Hamiltonians. In the high-temperature regime, their algorithm has sample complexity polynomial in the system size, temperature, and accuracy. Their algorithm can also be implemented with polynomial, but suboptimal, time complexity. Here, we study the same question for a more general class of Hamiltonians and present an algorithm that solves this problem with improved sample complexity and time complexity linear in the sample size. Furthermore, we prove a matching lower bound showing that our algorithm's sample complexity is optimal, and hence our time complexity is also optimal. Joint work with Robin Kothari and Ewin Tang.

With the race for quantum computers in full swing, researchers became interested in the question of what happens if we replace a supervised machine learning model with a quantum circuit. While such "supervised quantum models" are sometimes called "quantum neural networks", their mathematical structure reveals that they are in fact kernel methods with kernels that measure the distance between data embedded into quantum states. This talk gives an informal overview of the link, and discusses the far-reaching consequences for quantum machine learning.

In recent years, random quantum circuits have played a central role in the theory of quantum computation. Much of this prominence is due to recent random quantum circuit sampling experiments which have constituted the first claims of "quantum supremacy".   While random quantum circuits enjoy certain advantages that make them ideal for implementation by near-term quantum experiments, it is unclear a priori why they should be difficult to simulate classically. While we know several examples of quantum algorithms which attain exponential speedups over classical computation, they all seem to rely on highly structured circuits (such as quantum Fourier transforms) which are far from typical. Why then should we expect a generic quantum circuit to realize a large computational advantage?

In this talk we will explain the complexity theoretic basis for the classical hardness of random circuit sampling.

This talk will be based on joint work with Adam Bouland, Chinmay Nirkhe, and Umesh Vazirani (https://arxiv.org/abs/1803.04402), as well as Adam Bouland, Yunchao Liu and Zeph Landau (https://arxiv.org/abs/2102.01738).

Zoom Link: https://pitp.zoom.us/j/92408058752?pwd=b1JoeTVISlpjaXk3ZjBoSi9pYjNUZz09

We evaluate the usefulness of holographic stabilizer codes for practical purposes by studying their allowed sets of fault-tolerantly implementable gates. We treat them as subsystem codes and show that the set of transversally implementable logical operations is contained in the Clifford group for sufficiently localized logical subsystems. As well as proving this concretely for several specific codes, we argue that this restriction naturally arises in any stabilizer subsystem code that comes close to capturing certain properties of holography. We extend these results to approximate encodings, locality-preserving gates, certain codes whose logical algebras have non-trivial centers, and discuss cases where restrictions can be made to other levels of the Clifford hierarchy. A few auxiliary results may also be of interest, including a general definition of entanglement wedge map for any subsystem code, and a thorough classification of different correctability properties for regions in a subsystem code.

With gate error rates in multiple technologies now below the threshold required for fault-tolerant quantum computation, the major remaining obstacle to useful quantum computation is scaling, a challenge greatly amplified by the huge overhead imposed by quantum error correction itself. I’ll discuss a new fault-tolerant quantum computing scheme that can nonetheless be assembled from a small number of experimental components, potentially dramatically reducing the engineering challenges associated with building a large-scale fault-tolerant quantum computer. The architecture couples a single controllable qubit to a pair of delay lines which terminate in a detector. Below a threshold value for the error rate associated with the controllable qubit, the logical error rate decays exponentially with the square root of the delay line coherence time. The required gates can be implemented using existing technologies in quantum photonic and phononic systems. With continued incremental improvements in only a few components, we expect these systems to be promising candidates for demonstrating fault-tolerant quantum computation with comparatively modest experimental effort.

Recent work has defined what it means for one quantum system to simulate the full physics of another, and demonstrated that—within a very demanding definition of simulation —there exist families of local Hamiltonians that are universal, in the sense that they can simulate all other quantum Hamiltonians. This rigorous mathematical framework of Hamiltonian simulation not only gave a theoretical foundation for describing analogue Hamiltonian simulation. It also unified many previous Hamiltonian complexity results, and implied new ones. It has even found applications in constructing the first rigorous holographic dualities between local Hamiltonians, providing richer toy models of AdS/CFT duality in quantum gravity.

All previous constructions of universal Hamiltonians have relied heavily on using perturbation gadgets, and constructing complicated ‘chains’ of simulations to prove that simple models are indeed universal. In recent work we developed a new method for proving universality. Unlike perturbation- gadget approaches, this directly leverages the ability to encode computation into the ground states of QMA-hard Hamiltonians. With this technique we are able to derive necessary and sufficient complexity-theoretic conditions characterising universal Hamiltonians. We also use our new simulation method to provide a simple construction of two new universal models. Both of these are translationally invariant systems in 1D, and we show that one of these constructions is efficient in terms of the number of spins in the universal construction (but not in terms of the norm of the simulating Hamiltonian). This is the first translationally invariant universal model which is efficient in terms of system size overhead.

Based on joint work with Stephen Piddock, Johannes Bausch and Toby Cubitt (arXiv:2003.13753, arXiv:2101.12319)

A brief introduction to entanglement of multipartite pure quantum states will be given. As the Bell states are known to be maximally entangled among all two-qubit quantum states, a natural question arises: What is the most entangled state for the quantum system consisting of N sub-systems with d levels each? The answer depends on the entanglement measure selected, but already for four-qubit system, there is no state which displays maximal entanglement with respect to all three possible splittings of the systems into two pairs of qubits.

To construct strongly entangled multipartite quantum states one can use various mathematical techniques involving combinatorial designs, topological methods related to knot theory or the Majorana (stellar) representation of permutation symmetric quantum states.
 

We present two classical algorithms for the simulation of universal quantum circuits on n qubits constructed from c instances of Clifford gates and t arbitrary-angle Z-rotation gates such as T gates. Our algorithms complement each other by performing best in different parameter regimes. The Estimate algorithm produces an additive precision estimate of the Born rule probability of a chosen measurement outcome with the only source of run-time inefficiency being a linear dependence on the stabilizer extent (which scales like ≈1.17^t for T gates). Our algorithm is state-of-the-art for this task:

as an example, in approximately 25 hours (on a standard desktop computer), we estimated the Born rule probability to within an additive error of 0.03, for a 50 qubit, 60 non-Clifford gate quantum circuit with more than 2000 Clifford gates. The Compute algorithm calculates the probability of a chosen measurement outcome to machine precision with run-time O(2^t−r(t−r)t) where r is an efficiently computable, circuit-specific quantity. With high probability, r is very close to min{t,n−w} for random circuits with many Clifford gates, where w is the number of measured qubits. Compute can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+T circuits with n=55, w=5, c=105 and t=80 T-gates, and then compute the Born rule probability with a run-time consistently less than 104 seconds using a single core of a standard desktop computer. We provide a C+Python implementation of our algorithms.