An official website of the United States government
Abstract The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.
more »
« less
Cutoff phenomenon and entropic uncertainty for random quantum circuits
Abstract How fast a state of a system converges to a stationary state is one of the fundamental questions in science. Some Markov chains and random walks on finite groups are known to exhibit the non-asymptotic convergence to a stationary distribution, called the cutoff phenomenon. Here, we examine how quickly a random quantum circuit could transform a quantum state to a Haar-measure random quantum state. We find that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform (QFT). Thus the entropic uncertainty of random quantum states has balanced Shannon entropies for the computational basis and the QFT basis. By calculating the Shannon entropy for random quantum states and the Wasserstein distances for the eigenvalues of random quantum circuits, we show that the cutoff phenomenon occurs for the random quantum circuit. It is also demonstrated that the Dyson-Brownian motion for the eigenvalues of a random unitary matrix as a continuous random walk exhibits the cutoff phenomenon. The results here imply that random quantum states could be generated with shallow random circuits.
more »
« less
- Award ID(s):
- 1955907
- PAR ID:
- 10452895
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Electronic Structure
- Volume:
- 5
- Issue:
- 3
- ISSN:
- 2516-1075
- Page Range / eLocation ID:
- Article No. 035004
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Adaptive quantum circuits, which combine local unitary gates, midcircuit measurements, and feedforward operations, have recently emerged as a promising avenue for efficient state preparation, particularly on near-term quantum devices limited to shallow-depth circuits. Matrix product states (MPS) comprise a significant class of many-body entangled states, efficiently describing the ground states of one-dimensional gapped local Hamiltonians and finding applications in a number of recent quantum algorithms. Recently, it has been shown that the Affleck-Kennedy-Lieb-Tasaki state—a paradigmatic example of an MPS—can be exactly prepared with an adaptive quantum circuit of constant depth, an impossible feat with local unitary gates alone due to its nonzero correlation length [Smith , PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this approach and demonstrate that a diverse class of MPS can be exactly prepared using constant-depth adaptive quantum circuits, outperforming theoretically optimal preparation with unitary circuits. We show that this class includes short- and long-ranged entangled MPS, symmetry-protected topological (SPT) and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families of states with tunable correlation length. Moreover, we illustrate the utility of our framework for designing constant-depth sampling protocols, such as for random MPS or for generating MPS in a particular SPT phase. We present sufficient conditions for particular MPS to be preparable in constant time, with global on-site symmetry playing a pivotal role. Altogether, this work demonstrates the immense promise of adaptive quantum circuits for efficiently preparing many-body entangled states and provides explicit algorithms that outperform known protocols to prepare an essential class of states.more » « less
-
ABSTRACT Given a reversible Markov chain on states, and another chain obtained by perturbing each row of by at most in total variation, we study the total variation distance between the two stationary distributions, . We show that for chains withcutoff, converges to 0, is asymptotically at most (with a sequence of perturbations for which convergence to this bound occurs), and converges to 1, respectively, if the product of and the mixing time of converges to 0, , and , respectively. This echoes recent results for specific random walks that exhibit cutoff, suggesting that cutoff is the key property underlying such results. Moreover, we show is maximized byrestart perturbations, for which “restarts” at a random state with probability at each step. Finally, we show thatpre‐cutoffis (almost) equivalent to a notion of “sensitivity to restart perturbations,” suggesting that chains with sharper convergence to stationarity are inherently less robust.more » « less
-
Abstract Most existing quantum algorithms are discovered accidentally or adapted from classical algorithms, and there is the need for a systematic theory to understand and design quantum circuits. Here we develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. Compared to the conventional entanglement description of quantum circuits and states, the unitary dependence picture offers more practical information on the measurement and manipulation of qubits, easier generalization to many-qubit systems, and better robustness upon partitioning of the system. The unitary dependence theory can be applied to systematically understand existing quantum circuits and design new quantum algorithms.more » « less
-
We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A(α→) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy, thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ Hamiltonian using rotation and reflection symmetry and H2 Hamiltonian within Sz=0 subspace using S2 symmetry. We have found the variationally trained unitary gives good results with very low depth circuits and can thus be used to prepare symmetry states within near term quantum computers.more » « less
