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arXiv:2105.13350

The Complexity of Approximating Critical Points of Quantum Phase Transitions

James D. Watson, Johannes Bausch

Published May 27, 2021• 5 citations
arXivHTMLPDF

Abstract

Phase diagrams chart material properties with respect to one or more external or internal parameters such as pressure or magnetisation; as such, they play a fundamental role in many theoretical and applied fields of science. In this work, we prove that provided the phase of the Hamiltonian at a finite size reflects the phase in the thermodynamic limit, approximating the critical boundary in its phase diagram to constant precision is $P^{QMA_{EXP}}$-complete. This holds even for translationally-invariant nearest neighbour couplings, and even if the system's phase diagram is promised to have a single critical boundary delineating two phases. For the simpler case of a single parameter, the same problem remains $QMA_{EXP}$-hard. Our results extend the study of quantum phases to systems with more realistic phase diagrams than previously studied. Furthermore, our findings place complexity-theoretic constraints on the effectiveness of (computational or analytic) methods based on finite size criteria, similar in spirit to the Knabe bound, for the task of extrapolating the properties (e.g. gapped/gapless) of a system from finite-size observations to the thermodynamic limit.

DOI
10.48550/arxiv.2105.13350

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Phase diagrams chart material properties with respect to one or more external or internal parameters such as pressure or magnetisation; as such, they play a fundamental role in many theoretical and applied fields of science. In this work, we prove that provided the phase of the Hamiltonian at a finite size reflects the phase in the thermodynamic limit, approximating the critical boundary in its phase diagram to constant precision is PQMAEXPP^{QMA_{EXP}}PQMAEXP​-complete. This holds even for translationally-invariant nearest neighbour couplings, and even if the system's phase diagram is promised to have a single critical boundary delineating two phases. For the simpler case of a single parameter, the same problem remains QMAEXPQMA_{EXP}QMAEXP​-hard. Our results extend the study of quantum phases to systems with more realistic phase diagrams than previously studied. Furthermore, our findings place complexity-theoretic constraints on the effectiveness of (computational or analytic) methods based on finite size criteria, similar in spirit to the Knabe bound, for the task of extrapolating the properties (e.g. gapped/gapless) of a system from finite-size observations to the thermodynamic limit.
Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXP QMA-EXP and is hard for FEXP NEXP . This means that the problem is doubly exponentially hard in the size of the input. As an ingredient in our construction, we study the problem of computing the ground energy of translationally invariant finite 1D chains. A single Hamiltonian term, which is a fixed parameter of the problem, is applied to every pair of particles in a finite chain. In the finite case, the length of the chain is the sole input to the problem and the task is to compute an approximation of the ground energy. No thresholds are provided as in the standard formulation of the local Hamiltonian problem. We show that this problem is contained in FP QMA-EXP and is hard for FP NEXP . Our techniques employ a circular clock structure in which the ground energy is calibrated by the length of the cycle. This requires more precise expressions for the ground energies of the resulting matrices than were required for previous QMA-completeness constructions and even exact analytical bounds for the infinite case which we derive using techniques from spectral graph theory. To our knowledge, this is the first use of the circuit-to-Hamiltonian construction that shows hardness for a function class.
Despite progress in quantum Hamiltonian complexity, little is known about the computational complexity of quantum physics at the thermodynamic limit. Even defining the problem is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally invariant Hamiltonian in the thermodynamic limit, to within a given precision; the number of bits nnn for the precision is the sole input to the problem. The complexity of this problem captures how difficult it is for the physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in $\mbox{FEXP}^{\mbox{QMA-EXP}}$ and is hard for $\mbox{FEXP}^{\mbox{NEXP}}$. This means that the problem is doubly exponentially hard in the size of the input. As an ingredient in our construction, we study the problem of computing the ground energy of translationally invariant finite 1D chains. A single Hamiltonian term, which is a fixed parameter of the problem, is applied to every pair of particles in a finite chain. The length of the chain is the sole input to the problem and the task is to compute an approximation of the ground energy. No thresholds are provided as in the standard formulation of the local Hamiltonian problem. We show that this problem is contained in $\mbox{FP}^{\mbox{QMA-EXP}}$ and is hard for $\mbox{FP}^{\mbox{NEXP}}$. Our techniques employ a circular clock in which the ground energy is calibrated by the length of the cycle. This requires more precise expressions for the ground states of the resulting matrices than was required for previous QMA-completeness constructions and even exact analytical bounds for the infinite case which we derive using techniques from spectral graph theory. To our knowledge, this is the first use of the circuit-to-Hamiltonian construction which shows hardness for a function class.
The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where [Formula: see text], for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian's parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.
Abstract: The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where φ∈R, for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian’s parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.
Phase transitions happen at critical values of the controlling parameters, such as the critical temperature in classical phase transitions, and system critical parameters in the quantum case. However, true criticality happens only at the thermodynamic limit, when the number of particles goes to infinity with constant density. To perform the calculations for the critical parameters, a finite-size scaling approach was developed to extrapolate information from a finite system to the thermodynamic limit. With the advancement in the experimental and theoretical work in the field of ultra-cold systems, particularly trapping and controlling single atomic and molecular systems, one can ask: do finite systems exhibit quantum phase transition? To address this question, finite-size scaling for finite systems was developed to calculate the quantum critical parameters. The recent observation of a quantum phase transition in a single trapped 171 Yb+ ion indicates the possibility of quantum phase transitions in finite systems. This perspective focuses on examining chemical processes at ultra-cold temperatures, as quantum phase transitions—particularly the formation and dissociation of chemical bonds—are the basic processes for understanding the whole of chemistry.
We propose a method to determine the quantum phase boundaries of one-dimensional systems using sine-square deformation (SSD). Based on the proposition, supported by several exactly solved cases though not proven in full generality, that "if a one-dimensional system is gapless, then the expectation value of any local observable in the ground state of the Hamiltonian with SSD exhibits translational symmetry in the thermodynamic limit," we determine the quantum critical point as the location where a local observable becomes site-independent, identified through finite-size scaling analysis. As case studies, we consider two models: the antiferromagnetic Ising chain in mixed transverse and longitudinal magnetic fields with nearest-neighbor and long-range interactions. We calculate the ground state of these Hamiltonians with SSD using the density-matrix renormalization-group algorithm and evaluate the local transverse magnetization. For the nearest-neighbor model, we show that the quantum critical point can be accurately estimated by our procedure with systems of up to 84 sites, or even smaller, in good agreement with results from the literature. For the long-range model, we find that the phase boundary between the antiferromagnetic and paramagnetic phases is slightly shifted relative to the nearest-neighbor case, leading to a reduced region of antiferromagnetic order. Moreover, we propose an experimental procedure to implement the antiferromagnetic J1J_1J1​-J2J_2J2​ Ising couplings with SSD using Rydberg atom arrays in optical tweezers, which can be achieved within a very good approximation. Because multiple independent scaling conditions naturally emerge, our approach enables precise determination of quantum critical points and possibly even the extraction of additional critical phenomena, such as critical exponents, from relatively small system sizes.