Tag - Quantum computing

For any local Hamiltonian H, I construct a local CPT map and stopping condition which converges to the ground state subspace of H. Like any ground state preparation algorithm, this algorithm necessarily has exponential run-time in general (otherwise BQP=QMA), even for gapped, frustration-free Hamiltonians (otherwise BQP is in NP). However, this dissipative quantum eigensolver has a number of interesting characteristics, which give advantages over previous ground state preparation algorithms.

• &bullet  The entire algorithm consists simply of iterating the same set of local measurements repeatedly.
• &bullet  The expected overlap with the ground state subspace increases monotonically with the length of time this process is allowed to run.
• &bullet  It converges to the ground state subspace unconditionally, without any assumptions on or prior information about the Hamiltonian.
• &bullet  The algorithm does not require any variational optimisation over parameters.
• &bullet  It is often able to find the ground state in low circuit depth in practice.
• &bullet  It has a simple implementation on certain types of quantum hardware, in particular photonic quantum computers.
• &bullet  The process is immune to errors in the initial state.
• &bullet  It is inherently error- and noise-resilient, i.e. to errors during execution of the algorithm and also to faulty implementation of the algorithm itself, without incurring any computational overhead: the overlap of the output with the ground state subspace degrades smoothly with the error rate, independent of the algorithm's run-time.

I give rigorous proofs of the above claims, and benchmark the algorithm on some concrete examples numerically

Measurement incompatibility is an important resource in quantum infomation processing tasks such as e.g. quantum key distribution, Bell inequality violation and steering. While resource theories for quantum states have already been widely studied, much less is known about resource quantification for quantum measurements, in particular for sets of quantum measurements. We introduce distance-based quantifiers for this context. These allow to establish a hierarchy between different measurement resources, and to derive certain polygamy inequalities for subsets of multiple measurements.

The capacities of noisy quantum channels capture the ultimate rates of information transmission across quantum communication lines, and the quantum capacity plays a key role in determining the overhead of fault-tolerant quantum computation platforms. In the case of bosonic systems, central to many applications, no closed formulas for these capacities were known for bosonic dephasing channels, a key class of non-Gaussian channels modelling, e.g., noise affecting superconducting circuits or fiber-optic communication channels. Here we provide the first exact calculation of the quantum, private, two-way assisted quantum, and secret-key agreement capacities of all bosonic dephasing channels. We prove that that they are equal to the relative entropy of the distribution underlying the channel to the uniform distribution. Our result solves a problem that has been open for over a decade, having been posed originally by Jiang and Chen.

I will give a gentle introduction to quantum entanglement from the point of view of quantum information, focusing on the fundamental mathematical structures and open problems, both in finite and infinite dimension.

The observation that the entropy of entanglement for groundstates of quantum mechanical systems often scales like the area of the region separating the two rather than the volume. These weak correlations allow a host of quantum techniques such as matrix product states and DMRG to provide accurate estimates. However, existing area law results fail to rigorously hold for systems that have unbounded operators such as those that naturally appear in lattice gauge theories. Here we rectify this problem by proving an entanglement area law for a class of 1D quantum systems involving infinite-dimensional local Hilbert spaces. This class of quantum systems include bosonic models such as the Hubbard-Holstein model, and both U(1) and SU(2) lattice gauge theories in one spatial dimension. Our proof relies on new results concerning the robustness of the ground state and spectral gap to the truncation of Hilbert space, applied within the approximate ground state projector (AGSP) framework from previous work. In establishing this area law, we develop a system-size independent bound on the expectation value of local observables for Hamiltonians without translation symmetry, which may be of separate interest. Our result provides theoretical justification for using tensor network methods to study the ground state properties of quantum systems with infinite local degrees of freedom.