The Quantum Formalization Drive

Wilson's numerical renormalization group (NRG) computes the low-energy states of quantum impurity models, such as the Kondo and Anderson models, by diagonalizing a semi-infinite chain one site at a time and keeping only the lowest multiplets after each step (Wilson 1975; review: Bulla, Costi, Pruschke 2008). Dynamical quantities (spectral functions, Green's…

A quantum LDPC code is a CSS code whose two check matrices $H_X$ and $H_Z$ have rows and columns of bounded weight, so that every check measures only a few qubits. Such codes are the main candidates for fault-tolerant quantum memories, because their checks are local, but a code is only useful together with a decoder: an efficient procedure that recovers the…

The Quantum Approximate Optimization Algorithm (QAOA) of Farhi, Goldstone and Gutmann (arXiv:1411.4028) prepares a parametrized quantum state whose measurement yields a candidate solution of a combinatorial optimization problem. Its level-one version alternates a single phase operator and a single mixing operator and is simple enough to analyse by hand. In a…

Many recent quantum algorithms for linear algebra and optimization take their input as a block-encoding: a unitary circuit whose top-left block, after projecting a set of ancilla qubits onto $|0\rangle$, approximates the matrix of interest. The quantum semidefinite-programming solver of van Apeldoorn and Gilyén (arXiv:1804.05058) introduces the quantum…

A quantum computer cannot perform gate operations perfectly, so a small constant error per operation is unavoidable. A priori this could mean that a noisy quantum computer can apply only a bounded number of gates before the computation is destroyed by noise, which would rule out long quantum computations in practice. The threshold theorem (or quantum…

Quantum circuits built from Clifford gates alone are classically simulable in polynomial time. Universality is recovered by adding copies of a magic state, and the fastest known classical simulators of such circuits work by writing the magic-state input as a short linear combination of stabilizer states. The length of the shortest such combination -- the sta…

Quantum key distribution (QKD) lets two parties, Alice and Bob, extract a secret key from a noisy quantum channel controlled by an eavesdropper, Eve. In continuous-variable QKD the signals are bosonic modes with Gaussian statistics. Coherent-state protocols use standard optical components and tolerate large losses (Grosshans–Grangier 2002, Grosshans et al.…

Entanglement between two parties is not an all-or-nothing property. For a pure bipartite state, the Schmidt rank counts how many product terms are needed to write the state, and local operations with classical communication cannot increase it. Terhal and Horodecki (arXiv:quant-ph/9911117v4) extend this count to mixed states: the Schmidt number of a density…

In binary quantum hypothesis testing a quantum system is prepared in one of two known states, the null hypothesis $\rho_0$ or the alternative $\rho_1$, and an observer must decide which by a measurement on $M$ copies. In many applications the two errors are not equally costly: a false negative in a medical or security test may be much worse than a false positive. …

In quantum statistical mechanics an equilibrium state at inverse temperature $\beta$ of a system with dynamics $\tau$ is a $(\tau,\beta)$-KMS state $\omega_\beta$. Its correlation functions $t \mapsto \omega_\beta(A\tau_t(B))$ continue analytically into the strip $0 \le \Im t \le \beta$, and the values in imaginary time (the Schwinger functions) are the objects that…

Entanglement is treated as a resource in quantum information theory, and the unit of account is the ebit, one singlet $\Phi=(|01\rangle-|10\rangle)/\sqrt2$ shared by two parties. That accounting is justified only if entanglement in one form can be converted into another without paying in a different currency. For bipartite pure states, Bennett, Bernstein,…

Mixed entangled states of two parties come in two kinds. Free entanglement (FE) can be converted into nearly pure singlets by local quantum operations and classical communication (LQCC); this conversion is distillation. Bound entanglement (BE) is entanglement that cannot be distilled (Horodecki, Horodecki, Horodecki 1998). Bound entangled states were first…

Topological codes are ground spaces of gapped local Hamiltonians, and they protect information because local operators cannot distinguish their code states. A natural question is whether the low-energy spaces of other physical many-body systems have the same property. Gschwendtner, König, Şahinoğlu and Tang (arXiv:1902.02115) study this for error detection,…

Classical shadows (Huang, Kueng and Preskill, arXiv:2002.08953) estimate many expectation values $\operatorname{tr}(O\rho)$ of an unknown $n$-qubit state $\rho$ from few copies. Each copy is rotated by a random unitary and measured in the computational basis, and the outcome is inverted through a linear map, the measurement channel. The number of copies needed…

A quantum error correcting code protects quantum information by spreading it over many physical qubits. The textbook notion is a subspace code: the information lives in a subspace, and recovery must return every corrupted state to that subspace exactly. Knill, Laflamme and Viola, and Kribs, Laflamme and Poulin observed that this asks too much. Information…

For a quantum channel $\Phi$ and an input state $\rho$, the entropy gain $H(\Phi[\rho]) - H(\rho)$ measures how much von Neumann entropy the channel creates. Its infimum over states, the minimal entropy gain $G(\Phi)$, was studied by Alicki for channels on a $d$-dimensional space (arXiv:quant-ph/0402080), who showed $-\log d \le G(\Phi) \le 0$ and additivity under…

A quantum channel models every physical process a quantum system can undergo: noise in a fibre, a gate in a circuit, a measurement, a preparation. Some channels destroy every correlation of the input with the outside world that is not classical. These entanglement-breaking channels are the channels that can be simulated by measuring the input and preparing…

We construct a new secret-key assisted polar coding scheme for private classical communication over a quantum or classical wiretap channel. The security of our scheme rests on an entropic uncertainty relation, in addition to the channel polarization effect. Our scheme achieves the symmetric private information rate by synthesizing "amplitude" and "phase"…

We present a quantum CISC compiler and show how to assemble complex instruction sets in a scalable way. Enlarging the toolbox of universal gates by optimised complex multi-qubit instruction sets thus paves the way to fight decoherence for realistic settings. Compiling a quantum module into the machine code for steering a concrete quantum hardware device…

We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules…

We consider the secret key generation problem when sources are randomly excited by the sender and there is a noiseless public discussion channel. Our setting is thus similar to recent works on channels with action-dependent states where the channel state may be influenced by some of the parties involved. We derive single-letter expressions for the secret…

We propose a quantum algorithm that emulates the action of an unknown unitary transformation on a given input state, using multiple copies of some unknown sample input states of the unitary and their corresponding output states. The algorithm does not assume any prior information about the unitary to be emulated or the sample input states. To emulate the…

We study the general framework of quantum channel simulation, that is, the ability of a quantum channel to simulate another one using different classes of codes. First, we show that the minimum error of simulation and the one-shot quantum simulation cost under no-signalling assisted codes are given by semidefinite programs. Second, we introduce the…

The challenge of equality in the strong subadditivity inequality of entropy is approached via a general additivity of correlation information in terms of nonoverlapping clusters of subsystems in multipartite states (density operators). A family of tripartite states satisfying equality is derived.

We extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation…