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Quantifying quantum states complexity is a key problem in various subfields of science, from quantum computing to black-hole physics. We prove a prominent conjecture by Brown and Susskind about how random quantum circuits complexity increases. Consider constructing a unitary from Haar-random two-qubit quantum gates. Implementing the unitary exactly requires a circuit of some minimal number of gates - the unitarys exact circuit complexity. We prove that this complexity grows linearly in the number of random gates, with unit probability, until saturating after exponentially many random gates. Our proof is surprisingly short, given the established difficulty of lower-bounding the exact circuit complexity. Our strategy combines differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.
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The multiscale entanglement renormalization ansatz describes quantum many-body states by a hierarchical entanglement structure organized by length scale. Numerically, it has been demonstrated to capture critical lattice models and the data of the corresponding conformal field theories with high accuracy. However, a rigorous understanding of its success and precise relation to the continuum is still lacking. To address this challenge, we provide an explicit construction of entanglement-renormalization quantum circuits that rigorously approximate correlation functions of the massless Dirac conformal field theory. We directly target the continuum theory: discreteness is introduced by our choice of how to probe the system, not by any underlying short-distance lattice regulator. To achieve this, we use multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way. This could be a starting point for constructing quantum circuit approximations for more general conformal field theories.
Notions of circuit complexity and cost play a key role in quantum computing and simulation where they capture the (weighted) minimal number of gates that is required to implement a unitary. Similar notions also become increasingly prominent in high energy physics in the study of holography. While notions of entanglement have in general little implications for the quantum circuit complexity and the cost of a unitary, in this note, we discuss a simple such relationship when both the entanglement of a state and the cost of a unitary take small values, building on ideas on how values of entangling power of quantum gates add up. This bound implies that if entanglement entropies grow linearly in time, so does the cost. The implications are two-fold: It provides insights into complexity growth for short times. In the context of quantum simulation, it allows to compare digital and analog quantum simulators. The main technical contribution is a continuous-variable small incremental entangling bound.
The variational quantum eigensolver (VQE) is one of the most promising algorithms to find eigenvalues and eigenvectors of a given Hamiltonian on noisy intermediate-scale quantum (NISQ) devices. A particular application is to obtain ground or excited states of molecules. The practical realization is currently limited by the complexity of quantum circuits. Here we present a novel approach to reduce quantum circuit complexity in VQE for electronic structure calculations. Our algorithm, called ClusterVQE, splits the initial qubit space into subspaces (qubit clusters) which are further distributed on individual (shallower) quantum circuits. The clusters are obtained based on quantum mutual information reflecting maximal entanglement between qubits, whereas entanglement between different clusters is taken into account via a new dressed Hamiltonian. ClusterVQE therefore allows exact simulation of the problem by using fewer qubits and shallower circuit depth compared to standard VQE at the cost of additional classical resources. In addition, a new gradient measurement method without using an ancillary qubit is also developed in this work. Proof-of-principle demonstrations are presented for several molecular systems based on quantum simulators as well as an IBM quantum device with accompanying error mitigation. The efficiency of the new algorithm is comparable to or even improved over qubit-ADAPT-VQE and iterative Qubit Coupled Cluster (iQCC), state-of-the-art circuit-efficient VQE methods to obtain variational ground state energies of molecules on NISQ hardware. Above all, the new ClusterVQE algorithm simultaneously reduces the number of qubits and circuit depth, making it a potential leader for quantum chemistry simulations on NISQ devices.
In this paper, we show how the restriction of the Quantum Geometric Tensor to manifolds of states that can be generated through local interactions provides a new tool to understand the consequences of locality in physics. After a review of a first result in this context, consisting in a geometric out-of-equilibrium extension of the quantum phase transitions, we argue the opportunity and the usefulness to exploit the Quantum Geometric Tensor to geometrize quantum chaos and complexity.
Recently, it is shown that quantum computers can be used for obtaining certain information about the solution of a linear system Ax=b exponentially faster than what is possible with classical computation. Here we first review some key aspects of the algorithm from the standpoint of finding its efficient quantum circuit implementation using only elementary quantum operations, which is important for determining the potential usefulness of the algorithm in practical settings. Then we present a small-scale quantum circuit that solves a 2x2 linear system. The quantum circuit uses only 4 qubits, implying a tempting possibility for experimental realization. Furthermore, the circuit is numerically simulated and its performance under different circuit parameter settings is demonstrated.
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