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arXiv:2002.09592 · Physical review. B./Physical review. B

Adiabatic ground state preparation in an expanding lattice

Christopher T. Olund, Maxwell Block, Snir Gazit, John McGreevy, Norman Y. Yao

Published Apr 30, 2020• 2 citations
arXivHTMLPDF

Abstract

We implement and characterize a numerical algorithm inspired by the $s$ source framework [B. Swingle and J. McGreevy, Phys. Rev. B 93, 045127 (2016)] for building a quantum many-body ground state wave function on a lattice of size $2L$ by applying adiabatic evolution to the corresponding ground state at size $L$, along with $L$ interleaved ancillae. The procedure can in principle be iterated to repeatedly double the size of the system. We implement the algorithm for several one-dimensional (1D) spin model Hamiltonians, and find that the construction works particularly well when the gap is large and, interestingly, at scale-invariant critical points. We explain this feature as a natural consequence of the lattice expansion procedure. This behavior holds for both the integrable transverse-field Ising model and nonintegrable variations. We also develop an analytic perturbative understanding of the errors deep in either phase of the transverse-field Ising model, and suggest how the circuit could be modified to parametrically reduce errors. In addition to sharpening our perspective on entanglement renormalization in 1D, the algorithm could also potentially be used to build states experimentally, enabling the realization of certain long-range correlated states with low-depth quantum circuits.

DOI
10.1103/physrevb.101.155152

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We implement and characterize a numerical algorithm inspired by the sss source framework [B. Swingle and J. McGreevy, Phys. Rev. B 93, 045127 (2016)] for building a quantum many-body ground state wave function on a lattice of size 2L2L2L by applying adiabatic evolution to the corresponding ground state at size LLL, along with LLL interleaved ancillae. The procedure can in principle be iterated to repeatedly double the size of the system. We implement the algorithm for several one-dimensional (1D) spin model Hamiltonians, and find that the construction works particularly well when the gap is large and, interestingly, at scale-invariant critical points. We explain this feature as a natural consequence of the lattice expansion procedure. This behavior holds for both the integrable transverse-field Ising model and nonintegrable variations. We also develop an analytic perturbative understanding of the errors deep in either phase of the transverse-field Ising model, and suggest how the circuit could be modified to parametrically reduce errors. In addition to sharpening our perspective on entanglement renormalization in 1D, the algorithm could also potentially be used to build states experimentally, enabling the realization of certain long-range correlated states with low-depth quantum circuits.
Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians HI{H}_{\mathrm{I}}HI​ and HF{H}_{\mathrm{F}}HF​. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of HF{H}_{\mathrm{F}}HF​ from the ground state of HI{H}_{\mathrm{I}}HI​. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.
We present a quantum algorithm for implementing ${\ensuremath{\phi}}^{4}$ lattice scalar field theory on qubit computers. The field is represented in the discretized field amplitude basis. The number of qubits and elementary gates required by the implementation of the evolution operator is proportional to the lattice size. The algorithm allows efficient ${\ensuremath{\phi}}^{4}$ state preparation for a large range of input parameters in both the normal and broken-symmetry phases. The states are prepared using a combination of variational and adiabatic evolution methods. First, the ground state of a local Hamiltonian, which includes the ${\ensuremath{\phi}}^{4}$ self-interaction, is prepared using short variational circuits. Next, this state is evolved by switching on the coupling between the lattice sites adiabatically. The parameters defining the local Hamiltonian are adjustable and constitute the input of our algorithm. We present a method to optimize these parameters in order to reduce the adiabatic time required for state preparation. For preparing broken-symmetry states, the adiabatic evolution problems caused by crossing the phase transition critical line and by the degeneracy of the broken-symmetry ground state can be addressed using an auxiliary external field which gradually turns off during the adiabatic process. We show that the time dependence of the external field during the adiabatic evolution is important for addressing the broken-symmetry ground state degeneracy. The adiabatic time dependence on the inverse error tolerance can be reduced from quadratic to linear by using a field strength that decreases exponentially in time relative to one that decreases linearly.
Quantum state preparation is crucial for quantum algorithms, and improved state preparation methods can reduce resource requirements by orders of magnitude. In this work, we introduce a state preparation method for lattice Hamiltonians on digital quantum computers inspired by nucleation. In our method, the exact ground state of a small lattice Hamiltonian is prepared in a quantum circuit, then a series of growth stages are performed via Trotterized adiabatic evolution to increase the size of the lattice to the desired model. Using the adiabatic theorem and Trotter error bounds, we provide sufficient conditions on the total evolution time and number of Trotter steps required for nucleation to prepare an initial state with desired fidelity. In addition, we introduce optimized nucleation which considers variational optimization on top of the nucleation structure. For a two-dimensional Ising model, we show that optimized nucleation can outperform a current leading method for state preparation on digital quantum computers.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
A central challenge in the successful implementation of adiabatic quantum algorithms is to maintain the quantum adiabaticity during the entire evolution. However, the energy gap between the ground and the excited states of interacting many-body systems often decreases quickly with the number of qubits, and the quantum adiabaticity can be severely impaired. Despite numerous previous efforts, a practical method to preserve the quantum adiabaticity has yet to be developed. Here we present a universal approach to enhancing the quantum adiabaticity via cavity or circuit QED. By coupling an adiabatic quantum computer to an ancilla cavity, the coupled system can exhibit a bistable regime with bifurcation points, where the time evolution becomes very slow. Utilizing these generic nonlinear features, we show that the energy gap of the adiabatic quantum computer can be positioned between the bifurcation points, which results in strongly-enhanced quantum adiabaticity. We then apply this method to a quantum two-level system, an Exact Cover problem, and a transverse field Ising model. In contrast to previous works, this approach does not require the spectral knowledge of the quantum system or the construction of unphysical interactions and can be applied to a vast variety of adiabatic quantum processes.
A bang-bang (BB) algorithm prepares the ground state of a lattice quantum many-body Hamiltonian H=H1+H2H=H_1+H_2H=H1​+H2​ by evolving an initial product state alternating between H1H_1H1​ and H2H_2H2​. We optimize the algorithm with tensor networks in one and two dimensions. The optimization has two stages. In stage one, a shallow translationally-invariant circuit is optimized in an infinite lattice. In stage two, the infinite-lattice gate sequence is used as a starting point for a finite lattice where it remains optimal in the bulk. The prepared state requires optimization only at its boundary, within a healing length from lattice edges, and the gate sequence needs to be modified only within the causal cone of the boundary. We test the procedure in the 1D and 2D quantum Ising model near its quantum critical point employing, respectively, the matrix product state (MPS) and the pair-entangled projected state (PEPS). At the boundary already the infinite-lattice sequence turns out to provide a more accurate state than in the bulk.