Quantum Physics

Recent advances in scaling photonics for universal quantum computation spotlight the need for a thorough understanding of practicalities such as distinguishability in multimode quantum interference.  Rather than the usual second quantized approach, we can bring quantum information concepts to bear in first quantization.  Distinguishability can then be modelled as entanglement between photonic degrees of freedom, where loss of interference is caused by decoherence due to correlations with an environment carried by the particles themselves.  This shows that multiparticle, multimode Fock states can have natural Schmidt decompositions, corresponding to what has been called unitary-unitary duality in the representation theory of many-body physics.  
We present results along two lines: 
(i) Generalised Hong-Ou-Mandel interference as unambiguous state discrimination (arXiv:1806.01236, with S. Stanisic); we find optimised interferometers that discriminate between indistinguishable and interesting distinguishable states by projecting onto non-symmetric irreps of the unitary group of interferometers.  We give analytic and numerical results for up to nine photons in nine modes.
(ii) Quantum simulation of noisy boson sampling (arXiv:1803.03657, with A. Moylett); we provide quantum circuits that simulate bosonic sampling with arbitrarily distinguishable particles, based on the quantum Schur-Weyl transform.  This makes clear how particle distinguishabililty leads to decoherence in the standard quantum circuit model.  We show how ideal samples can be postselected, how photon loss can also be modelled, and how standard results in the classical simulation of noisy quantum computations fail to apply to the boson sampling case.

Extended Clifford circuits straddle the boundary between classical and quantum computational power. Whether such circuits are efficiently classically simulable seems to depend delicately on the ingredients of the circuits. While some combinations of ingredients lead to efficiently classically simulable circuits, other combinations, which might just be slightly different, lead to circuits which are likely not. We extend the results of Jozsa and Van den Nest [Quantum Inf. Comput. 14, 633 (2014)] by studying various further extensions of Clifford circuits. First, we consider how the classical simulation complexity changes when we allow for more general measurements. Second, we investigate different notions of what it means to "classically simulate" a quantum circuit. Third, we consider the class of conjugated Clifford circuits, where one conjugates every qubit in a Clifford circuit by the same single-qubit gate. Our results provide more examples where seemingly modest changes to the ingredients of Clifford circuits lead to "large" changes in the classical simulation complexities of the circuits, and also include new examples of extended Clifford circuits that are potential candidates for "quantum supremacy". Based on Quantum Inf. Comput. 17, 0262 (2017) and proceedings of CCC’18 pp.21:1–21:25 (2018) (joint work with Adam Bouland and Joseph F. Fitzsimons).

In quantum theory, the no-information-without-disturbance and no-free-information principles express that those observables that do not disturb the measurement of another observable and those that can be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these principles do not hold in general. In this way, we obtain characterizations of the probabilistic theories where these principles hold and we show that the two principles are not equivalent.  

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.