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We show that there are two distinct obstacles to an efficient
classical simulation of a general quantum circuit. The first obstacle, which
has been well-studied, is the difficulty of efficiently estimating the
probability associated with a particular measurement outcome. We show that
this alone does not determine whether a quantum circuit can be efficiently
simulated. The second obstacle is that, in general, there can be an
exponential number of `relevant' outcomes that are needed for an accurate
simulation, and so efficient simulation may not be possible even in
situations where the probabilities of individual outcomes can be efficiently
estimated. We show that these two obstacles are distinct, overcoming the
former obstacle being necessary but not sufficient for simulability whilst
overcoming the pair is jointly sufficient for simulability. Specifically, we
prove that a family of quantum circuits is efficiently simulable if it
satisfies two properties: one related to the efficiency of Born rule
probability estimation, and the other related to the sparsity of the outcome
distribution. We then prove a pair of hardness results (using standard
complexity assumptions and a variant of a commonly-used average case
hardness conjecture), where we identify families of quantum circuits that
satisfy one property but not the other, and for which efficient simulation
is not possible. To prove our results, we consider a notion of simulation of
quantum circuits that we call epsilon-simulation. This notion is less
stringent than exact sampling and is now in common use in quantum hardness
proofs.

I will review recent and ongoing developments in path integral optimization and path integral complexity in 2d CFTs. After, I will discuss the connection with geometric approach to circuit complexity and point out several open problems.

A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size, without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional onsite symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D `cluster state' model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally. Joint work with Stephen Bartlett.

Quantum computers have the potential to significantly outperform classical computers at certain tasks. However, many applications of quantum computers require a fault-tolerant quantum computer. Such a device would function correctly even in the presence of noise. State-of-the art quantum computing architecture proposals require at least hundreds of thousands of high quality qubits to achieve fault-tolerance. These requirements are far beyond today's technology. In this talk, I will present results about a novel quantum computing architecture which is based on 3-d surface codes (a family of quantum error-correcting codes). This architecture may have smaller resource requirements than the current leading architectures for certain experimental parameters. I will show that the 3-d surface code has a transversal CCZ gate and I will discuss a cellular automaton which can be used to decode 3-d surface codes. Time permitting, I will also discuss the relationship between 3-d surface codes and 3-d colour codes. 

Recent years have shown that error correction is one of the most fundamental ingredients of various physical phenomena, from topological order to holography. However, only toy models and fixed point ground states could have been studied, even though the error-correcting properties are expected to hold generally in the low energy subspace.

In this talk, we will employ Matrix Product State formalism and see how low energy eigenstates of 1D translationally invariant Hamiltonians can form quantum error-correcting codes. Before diving into the results, we will review the necessary basics of quantum error correction and matrix product states.

Joint work with Martina Gschwendtner, Robert Koenig and Eugene Tang.

In this talk I will discuss generalising the mapping from quantum error correcting codes to stat. mech. models (originally due to Dennis, Kitaev, Landahl and Preskill) to the case of correlated Pauli noise. This mapping connects the error correcting threshold to a phase transition, allowing us to use methods for studying stat. mech. models to estimate code thresholds. Using this, I will present numerical estimates of toric code thresholds under correlated noise. Time permitting, I will also discuss how this mapping can be used to give a tensor network-based algorithm for approximate optimal decoding of 2d topological codes, generalising the decoder of Bravyi, Suchara and Vargo.

What does it mean for quantum state to be genuinely fully multipartite? Some would say, whenever the state cannot be decomposed as a mixture of states each of which has no entanglement across some partition. I'll argue that this partition-centric thinking is ill-suited for the task of assessing the connectivity of the network required to realize the state. I'll introduce a network-centric perspective for classifying multipartite entanglement, and it's natural device-independent counterpart, namely a network-centric perspective for classifying multipartite nonclassicality of correlations. Time permitting, we can then explore semidefinite programming (SDP) algorithms for convex optimization over k-partite-entangled states and k-partite-nonlocal correlations relative to the network-centric classification. Joint work with Denis Rosset and others. We will compare the new quantum-inflation techniques to the classical inflation of arXiv:1609.00672. I'll share a few results made possible by these SDPs, while being openly critical about some disappointing apparent limitations.

The spectral gap problem consist in deciding, given a local interaction, whether the corresponding translationally invariant Hamiltonian on a lattice has a spectral gap independent of the system size or not. In the simplest case of nearest-neighbour frustration-free qubit interactions, there is a complete classification. On the other extreme, for two (or higher) dimensional models with nearest-neighbour interactions this problem can be reduced to the Halting Problem, and it is therefore undecidable.

There are a lot of indications that one dimensional spin chain are relatively simpler than their counterparts in higher dimensions. Nonetheless, I will present a construction of a family of nearest-neighbour, translationally invariant Hamiltonians on a spin chain, for which the spectral gap problem is undecidable.

Ubiquitous in the behavior of physical systems is the competition between an energy term E and an entropy term S of their free energy F = E - beta S. These concepts are also relevant for error correction, where the `energy` E is the number of qubits afflicted by an error, the `entropy' S(E) is the logarithm of the number of energy-E failing errors, and beta relates to the probability of each qubit's error. Error-correction schemes with larger minimum free energy have better performance. Often distance d (which correct all errors with energy less than d/2) is used as a proxy for a code's performance since it increases the minimal E and therefore tends to increases the minimal F. However, a sufficiently large entropy S can counteract a large d and reduce the free energy (negatively impacting a code's performance). A great example of these principles is the surface code, which is at present the leading architecture for fault tolerant quantum computing. Rotating a square lattice geometry over the surface of the torus can increase the distance of the code by a factor of root two, but at the cost of increased entropy. We obtain exact expressions for this entropic effect in the low error regime, and introduce an analytical model that qualitatively describes the behavior for error rates all the way up to threshold. Our predictions are corroborated by numerical estimates of the low error failure rate, using the splitting method algorithm introduced by Bravyi et al. We find that although the rotated lattice outperforms the non-rotated lattice with the same number of qubits for low error rates, the two codes have very similar performance for error rates which are an appreciable fraction of the threshold error rate. Surprisingly, we also find some system sizes and error rates for which the non-rotated lattice has marginally better performance. 

Violations of Bell inequalities have traditionally been used to refute a local-realistic description of the world. Not surprisingly, under the assumption that the world is quantum, they can be used to certify quantum devices. What is surprising is that in some cases this characterisation turns out to be (almost) complete, i.e.~we can determine (almost) everything about the devices and this phenomenon is known as self-testing of quantum systems. Although the first self- testing results can be traced back to the works of Tsirelson published in the 80's, the topic has remained largely unknown until the seminal work of Mayers and Yao in 1998. It has received further exposure with the advent of device-independent quantum cryptography to which it is closely connected. In this talk I will give a brief introduction to the topic of self-testing and discuss some recent developments, e.g.~robust self-testing, weak self-testing, self-testing of entangled measurements, self-testing of high-dimensional systems or self-testing in prepare-and-measure scenarios.

Despite considerable effort, magic state distillation remains one of the leading candidates to achieve universal fault-tolerant quantum computation. However, when analyzing magic state distillation schemes, it is often assumed that gates belonging to the Clifford group can be implemented perfectly. In many current quantum technologies, two-qubit Cliffords gates are amongst the noisiest components of quantum computers. In this talk I will present a new scheme for preparing magic states with very low overhead that uses flag qubits. I will then compare our scheme to leading magic state distillation methods and show that the overhead can be reduced by orders of magnitude.

I will present a method for the implementation of a universal set of fault-tolerant logical gates using homological product codes. In particular, I will show how one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.