We investigate the most general mechanisms that lead to perfect synchronization of the quantum states of all subsystems of an open quantum system starting from an arbitrary initial state. We provide a necessary and sufficient condition for such synchronization, prove tight lower bounds on the dimension of the environment's Hilbert space in two main classes of quantum-state synchronizers, and give an analytical solution for their construction. The functioning of the found quantum-state synchronizer of two qubits is demonstrated experimentally on an IBM quantum computer and we show that the remaining asynchronicity is a sensitive measure of the quantum computer's imperfection.
We study mechanisms that allow one to synchronize the quantum phase of two qubits relative to a fixed basis. Starting from one qubit in a fixed reference state and the other in an unknown state, we find that, contrary to the impossibility of perfect quantum cloning, the quantum phase can be synchronized perfectly through a joined unitary operation. When both qubits are initially in a pure unknown state, perfect quantum-phase synchronization through unitary operations becomes impossible. In this situation we determine the maximum average quantum-phase synchronization fidelity and the distribution of relative phases and fidelities, and we identify optimal quantum circuits that achieve this maximum fidelity. A subset of these optimal quantum circuits enable perfect quantum-phase synchronization for a class of unknown initial states restricted to the equatorial plane of the Bloch sphere.
The computational power of
QAC, which are constant-depth quantum circuit families consisting of arbitrary single-qubit unitaries and n-qubit generalized Toffoli gates, has gained tremendous focus recently. In this work, we initiate the study of the computational complexity of geometrically local
QAC circuits, where the qubits are arranged in a high-dimensional grid, and all the generalized Toffoli gates act on contiguous qubits along one direction. We prove that
2D-QAC circuits are universal: any
QAC circuit can be exactly simulated by a two-dimensional geometrically local
QAC circuit. Therefore, we focus on the computational power of
2D-QAC circuits with grid width
w. We prove that
PARITY computation has a tight width--depth tradeoff of
w=nΘ(1/d) in the constant-depth regime if the circuit allows arbitrary ancilla. Moreover, for polynomial-size circuits, a width lower bound of
nω(1/d) for
PARITY would imply
PARITY∈/QAC0. We also establish Fourier tail and sensitivity bounds, together with tight width lower bounds and tight depth lower bounds in
1D-QAC circuits for
MAJORITY and
MOD computation.
Our main tool is \emph{local measurement variance}, the maximum variance of normalized sums of observables on separated regions. We use a hybrid argument to show that each layer of a
2D-QAC0 circuit increases this variance by at most a factor depending only on its width, yielding a variance bound for every
2D-QAC0 state. The high local measurement variance of cat and Dicke states then gives the width lower bounds.
Natural and engineered classical systems are replete with examples of synchronization, understood as the adjustment of rhythms of physical systems. Such synchronization is at the heart of the stability of several classical technologies, such as mechanical bridges and electrical networks. Given the advent of quantum simulation and computation technologies, it is natural to study a quantum analogue of synchronization and explore novel applications. This review surveys synchronization in few and many-body quantum systems, measures that quantify them, and their applications to quantum technologies.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.