Nonstabilizerness, also known as magic, quantifies the number of non-Clifford operations needed to prepare a quantum state. As typical measures either involve minimization procedures or a computational cost exponential in the number of qubits $N$ , it is notoriously hard to characterize for many-body states. In this paper, we show that nonstabilizerness, as quantified by the recently introduced stabilizer R\'enyi entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension $\ensuremath{\chi}$ and integer R\'enyi index $n>1$ , we show that the SRE can be expressed in terms of the norm of an MPS with bond dimension ${\ensuremath{\chi}}^{2n}$. For translation-invariant states, this allows us to extract it from a single tensor, the transfer matrix, while for generic MPSs this construction yields a computational cost linear in $N$ and polynomial in $\ensuremath{\chi}$. We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes. We analyze the SRE near criticality and investigate its dependence on the local computational basis, showing that it is, in general, not maximal at the critical point.Nonstabilizerness, also known as magic, quantifies the number of non-Clifford operations needed to prepare a quantum state. As typical measures either involve minimization procedures or a computational cost exponential in the number of qubits $N$, it is notoriously hard to characterize for many-body states. In this paper, we show that nonstabilizerness, as quantified by the recently introduced stabilizer R\'enyi entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension $\ensuremath{\chi}$ and integer R\'enyi index $n>1$, we show that the SRE can be expressed in terms of the norm of an MPS with bond dimension ${\ensuremath{\chi}}^{2n}$. For translation-invariant states, this allows us to extract it from a single tensor, the transfer matrix, while for generic MPSs this construction yields a computational cost linear in $N$ and polynomial in $\ensuremath{\chi}$. We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes. We analyze the SRE near criticality and investigate its dependence on the local computational basis, showing that it is, in general, not maximal at the critical point.

arXiv:2207.13076Physical review. B./Physical review. BJan 27, 2023118 citations

Matrix Product States

Timo Felser, Simone Montangero

Graduate texts in physicsJan 1, 20260 citations

Algebraic Geometry of Matrix Product States

Andrew Critch

We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states.For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions.Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS.We make four conjectures on the identifiability of MPS parameters.We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states.For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions.Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS.We make four conjectures on the identifiability of MPS parameters.

arXiv:1210.2812Symmetry Integrability and Geometry Methods and ApplicationsSep 9, 201422 citations

Matrix Product States: Foundations

Michael L. Wall

Springer thesesJan 1, 20150 citations

Simulations with matrix product states

Ulrich Schollwöck

AIP conference proceedingsJan 1, 20120 citations

Matrix product states approaches to operator spreading in ergodic quantum systems

Kévin Hémery, Frank Pollmann, David J. Luitz

We review different matrix-product-state (MPS) approaches to study the spreading of operators in generic nonintegrable quantum systems. As a common ground to all methods, we quantify this spreading by means of the Frobenius norm of the commutator of a spreading operator with a local operator, which is usually referred to as the out-of-time-order correlation (OTOC) function. We compare two approaches based on matrix-product states in the Schr\"odinger picture: the time-dependent block decimation (TEBD) and the time-dependent variational principle (TDVP), as well as TEBD based on matrix-product operators directly in the Heisenberg picture. The results of all methods are compared to numerically exact results using Krylov space exact time evolution. We find that for the Schr\"odinger picture, the TDVP algorithm performs better than the TEBD algorithm.Moreover, the tails of the OTOC are accurately obtained both by TDVP MPS and TEBD MPO. They are in very good agreement with exact results at short times, and appear to be converged in bond dimensions even at longer times. However, the growth and saturation regimes are not well captured by either of the methods.We review different matrix-product-state (MPS) approaches to study the spreading of operators in generic nonintegrable quantum systems. As a common ground to all methods, we quantify this spreading by means of the Frobenius norm of the commutator of a spreading operator with a local operator, which is usually referred to as the out-of-time-order correlation (OTOC) function. We compare two approaches based on matrix-product states in the Schr\"odinger picture: the time-dependent block decimation (TEBD) and the time-dependent variational principle (TDVP), as well as TEBD based on matrix-product operators directly in the Heisenberg picture. The results of all methods are compared to numerically exact results using Krylov space exact time evolution. We find that for the Schr\"odinger picture, the TDVP algorithm performs better than the TEBD algorithm.Moreover, the tails of the OTOC are accurately obtained both by TDVP MPS and TEBD MPO. They are in very good agreement with exact results at short times, and appear to be converged in bond dimensions even at longer times. However, the growth and saturation regimes are not well captured by either of the methods.

arXiv:1901.05793Physical review. B./Physical review. BSep 11, 201927 citations

N

, it is notoriously hard to characterize for many-body states. In this paper, we show that nonstabilizerness, as quantified by the recently introduced stabilizer R\'enyi entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension $\ensuremath{\chi}$ and integer R\'enyi index

n>1

, we show that the SRE can be expressed in terms of the norm of an MPS with bond dimension ${\ensuremath{\chi}}^{2n}$. For translation-invariant states, this allows us to extract it from a single tensor, the transfer matrix, while for generic MPSs this construction yields a computational cost linear in

N

and polynomial in $\ensuremath{\chi}$. We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes. We analyze the SRE near criticality and investigate its dependence on the local computational basis, showing that it is, in general, not maximal at the critical point.