Quantum Information

Quantum position verification (QPV) was first introduced, under the name quantum tagging, in a patent filed in 2004. It was first discussed in the academic literature in 2009-10. The schemes proposed to that point were shown to be breakable by teleportation attacks in 2010, and a general no-go theorem showing all schemes in this class are breakable was subsequently proved. However, in an alternative standard cryptographic security scenario, in which the tag is assumed to be able to keep classical data secret, unconditionally secure schemes were presented in 2010. The various terminologies and security scenarios highlight important points whose theoretical and practical implications still remain underexplored. In practice, one normally wants to verify the location of a person or valuable object, not of an easily replaceable tagging device, for at least two reasons: (i) the device itself is not so valuable, (ii) adversaries can easily construct a replacement device and thereby potentially spoof the scheme. This requires physical assumptions about the integrity of the tag and its attachment, and bounds on the speed with which the tag may be displaced or destroyed and replaced. QPV schemes that do not rely on such assumptions are breakable without teleportation or non-local computation attacks. In the other direction, when such significant physical assumptions are necessary, it may generally be reasonable to include tag security among them. In this overview I review the early history of QPV and describe various security scenarios and their potential applications. I give versions of the secure 2010 scheme designed for efficient practical implementation and discuss the frequency, accuracy and security of position verification attainable for these schemes with present technology. I also discuss the implied constraints on what may be attainable for QPV schemes involving real-time quantum measurement and/or quantum information processing.

Finding ground states of quantum many-body systems is one of the most important---and one of the most notoriously difficult---problems in physics, both in scientific research and in practical applications.  Indeed, we know from a complexity-theoretic perspective that all methods  (quantum or classical) must necessarily fail to find the ground state efficiently in general. The ground state energy problem is already NP-hard even for classical, frustration-free, local Hamiltonians with constant spectral gap. For general quantum Hamiltonians, the problem becomes QMA-hard.

Nonetheless, as ground state problems are of such importance, and classical algorithms are often successful despite the theoretical exponential worst-case complexity, a number of quantum algorithms for the ground state problem have been proposed and studied. From quantum phase estimation-based methods, to adiabatic state preparation, to dissipative state engineering, to the variation quantum eigensolver (VQE), to quantum/probabilistic imaginary-time evolution (QITE/PITE).

Dissipative state engineering was first introduced in 2009 by Verstraete, Cirac and Wolf and by Kraus et al. However, it only works for the restricted class of frustration-free Hamiltonians.

In this talk, I will show how to construct a dissipative state preparation dynamics that provably produces the correct ground state for arbitrary Hamiltonians, including frustrated ones. This leads to a new quantum algorithm for preparing ground states: the Dissipative Quantum Eigensolver (DQE). DQE has a number of interesting advantages over previous ground state preparation algorithms:

• The entire algorithm consists simply of iterating the same set of   simple local measurements repeatedly.
• The expected overlap with the ground state increases monotonically with the length of time this process is allowed to run.
• It converges to the ground state subspace unconditionally, without any assumptions on or prior information about the Hamiltonian (such as spectral gap or ground state energy bound).
• The algorithm does not require any variational optimisation over parameters.
• It is often able to find the ground state in low circuit depth in practice.
• It has a simple implementation on certain types of quantum hardware, in particular photonic quantum computers.
• It is immune to errors in the initial state.
• It is inherently fault-resilient, without incurring any fault-tolerance overhead. I.e.\ not only is it resilient to errors on the quantum state, but also to faulty implementations of the algorithm itself; the overlap of the output with the ground state subspace degrades smoothly with the error rate, independent of the total run-time.

I give a mathematically rigorous analysis of the DQE algorithm and proofs of all the above properties, using non-commutative generalisations of methods from classical probability theory.

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Zoom link https://pitp.zoom.us/j/96022753460?pwd=SWlUVkVta1RyY3dsWUJWckRqOHdNdz09

Recently, Akers et al. proposed a non-isometric holographic map from the interior of a black hole to its exterior. Within this model, we study properties of the black hole S-matrix, which are in principle accessible to observers who stay outside the black hole. Specifically, we investigate a scenario in which an infalling agent interacts with radiation both outside and inside the black hole. Because the holographic map involves postselection, the unitarity of the S-matrix is not guaranteed in this scenario, but we find that unitarity is satisfied to very high precision if suitable conditions are met. If the internal black hole dynamics is described by a pseudorandom unitary transformation, and if the operations performed by the infaller have computational complexity scaling polynomially with the black hole entropy, then the S-matrix is unitary up to corrections that are superpolynomially small in the black hole entropy. Furthermore, while in principle quantum computation assisted by postselection can be very powerful, we find under similar assumptions that the S-matrix of an evaporating black hole has polynomial computational complexity.

The JLMS formula is a cornerstone in our understanding of bulk reconstruction in holographic theories of quantum gravity, best interpreted as a quantum error-correcting code. Moreover, recent work has highlighted the importance of understanding holography as an approximate and perhaps non-isometric code. In this work, we construct an enlarged code subspace for the bulk theory that contains multiple non-perturbatively different background geometries. In such a large holographic code, we carefully derive an approximate version of the JLMS formula from an approximate FLM formula for a class of nice states. We do not assume that the code is isometric, but interestingly find that approximate FLM forces the code to be approximately isometric. Furthermore, we show that the bulk modular Hamiltonian of the entanglement wedge makes important contributions to the JLMS formula and cannot in general be neglected even when the bulk state is semiclassical. Nevertheless, when acting on states with the same background geometry, we find that the modular flow is well approximated by the area flow which takes the geometric form of a boundary-condition-preserving kink transform. We also generalize the results to higher derivative gravity, where area is replaced by the geometric entropy. We conjecture that a Lorentzian definition of the geometric entropy is equivalent to its original, Euclidean definition, and we verify this conjecture in a dilaton theory with higher derivative couplings. Thus we find that the flow generated by the geometric entropy takes the universal form of a boundary-condition-preserving kink transform.

In this paper, we explore the possibility of building a quantum memory that is robust to thermal noise using large N matrix quantum mechanics models. First, we investigate the gauged SU(N) matrix harmonic oscillator and different ways to encode quantum information in it. By calculating the mutual information between the system and a reference which purifies the encoded information, we identify a transition temperature, Tc, below which the encoded quantum information is protected from thermal noise for a memory time scaling as N^2. Conversely, for temperatures higher than T_c, the information is quickly destroyed by thermal noise. Second, we relax the requirement of gauge invariance and study a matrix harmonic oscillator model with only global symmetry. Finally, we further relax even the symmetry requirement and propose a model that consists of a large number N^2 of qubits, with interactions derived from an approximate SU(N) symmetry. In both ungauged models, we find that the effects of gauging can be mimicked using an energy penalty to give a similar result for the memory time. The final qubit model also has the potential to be realized in the laboratory.