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arXiv:1706.01995 · Physical Review A

Symmetry-protected dissipative preparation of matrix product states

Leo Zhou, Soonwon Choi, Mikhail D. Lukin

Published Sep 22, 2021• 26 citations
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Abstract

We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size $n$. Furthermore, we demonstrate that this scaling can be improved to $O({log}^{2}n)$ by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.

DOI
10.1103/physreva.104.032418

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We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size nnn. Furthermore, we demonstrate that this scaling can be improved to O(log2n)O({log}^{2}n)O(log2n) by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.
The controlled preparation of quantum many-body states is a central challenge for quantum technologies. While preparing target states coherently is possible in principle, it is often fragile and experimentally demanding. Dissipative state preparation, where engineered environments drive the system toward desired steady states, provides a complementary and powerful paradigm. This thesis explores dissipative methods for preparing complex many-body states along three distinct directions. First, we investigate the dissipative stabilization of injective matrix product states (MPS). Injective MPS are particularly well suited for dissipative preparation because they arise as ground states of frustration-free parent Hamiltonians, allowing for tailored jump operators to steer the system directly into the target state. We showcase the surprising result that such state preparation works even in the dilute limit where steering only acts on a single link. Moreover, we introduce a formalism for engineering additional coherent interactions, extending the set of accessible states. Second, and complementary to steering, we develop and analyze a universal cooling scheme capable of preparing ground states even in frustrated quantum systems. In contrast to steering methods, universal cooling can overcome the glass floor that forbids steering methods to prepare such frustrated ground states. A key insight here is that the presented cooling scheme is completely system-agnostic making it extremely robust. Finally, we pursue dressing of the system eigenstates by means of either classical drives or coherent interactions as a means to influence the action of dissipation on the system. Focusing first on time-dependent systems, we propose a generalized numerical implementation of non-adiabatic master equations suitable for the use of optimal control tools. Additionally, we investigate how the collective coupling between a spin ensemble and a cavity can steer the system into a spin-squeezed steady state. Together, these three avenues illustrate how tailored dissipative dynamics can extend the frontier of many-body state preparation, from stabilizing injective MPS to cooling complex frustrated systems and generating non-classical steady states in the dressed framework.
The preparation of tensor network states is a fundamental prerequisite for a wide range of quantum simulation tasks. While many unitary protocols for preparing these states have been investigated, dissipative state preparation provides a powerful alternative since it can be robust to noise and initialization errors. In this paper, we construct both continuous-time and discrete-time geometrically local dissipative processes whose unique steady state is a given injective tensor network state. Our method prepares all injective matrix product states on NNN sites to an error ε\varepsilonε in O(log⁡(N/ε))O(\log (N/\varepsilon))O(log(N/ε)) time, yielding an exponential improvement over previously known dissipative preparation schemes. For two and higher-dimensional tensor network states, we prove that when the tensors of the state are \emph{highly injective}, the constructed dissipative processes are rapid-mixing i.e., they prepare a state ε\varepsilonε-close to the NNN-site target state in O(log⁡(N/ε))O( \log (N/\varepsilon))O(log(N/ε)) time. For these states, our approach provides a polynomial speedup over known unitary methods for states defined on lattices and an exponential speedup for states on general bounded-degree graphs. We corroborate our theoretical results with numerical studies that indicate that the dissipative protocol can rapidly prepares states outside the high-injectivity assumption.
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Matrix product states (MPS) serve as a key tool for studying quantum systems from chemistry and condensed-matter physics, making their preparation on quantum computers an important task in interfacing classical and quantum simulation. Many systems of interest have U(1)U(1)U(1)-symmetries induced by particle number and spin projection conservation, allowing to restrict the MPS tensors to be of block-sparse form, a property widely used in the implementation of classical algorithms such as the density matrix renormalization group. We reduce the cost of fault-tolerantly preparing block-sparse MPS within the standard ancilla-assisted linear-depth approach by implementing row and column permutations that transform the block-sparse matrices into block-diagonal form. These block-diagonal unitaries are then implemented via unitary synthesis, with the cost being determined by the size of the largest block. In this context, we modify the unitary synthesis approach of Berry et al. in order to reduce the Toffoli cost for real-valued unitaries by a factor of 2\sqrt{2}2​. In numerical benchmarks, we achieve Toffoli cost improvement factors of 10−3010 - 3010−30 compared to the state-of-the-art for MPS of various molecular systems.
We present and analyze a protocol for driven-dissipatively preparing and stabilizing a manifold of quantum many-body entangled states with symmetry-protected topological order. Specifically, we consider the experimental platform consisting of superconducting transmon circuits and linear microwave resonators. We perform theoretical modeling of this platform via pulse-level simulations based on physical features of real devices. In our protocol, transmon qutrits are mapped onto spin-1 systems. The qutrits' sharing of nearest-neighbor dispersive coupling to a dissipative microwave resonator enables elimination of state population in the Stotal=2{S}^{\mathrm{total}}=2Stotal=2 subspace for each adjacent pair, and thus, the stabilization of the many-body system into the Affleck, Kennedy, Lieb, and Tasaki state up to the edge mode configuration. We also analyze the performance of our protocol as the system size scales up to four qutrits, in terms of its fidelity as well as the stabilization time. Our work shows the capacity of driven-dissipative superconducting cQED systems to host robust and self-corrected quantum many-body states that are topologically nontrivial.
Quantum many-body physics simulations with matrix product states can often be accelerated if the quantum symmetries present in the system are explicitly taken into account. Conventionally, quantum symmetries have to be determined beforehand when constructing the tensors for the matrix product state algorithm. In this work, we present a matrix product state algorithm with an adaptive U(1)U(1)U(1) symmetry. This algorithm can take into account or benefit from U(1)U(1)U(1) or Z2{Z}_{2}Z2​ symmetries when they are present or analyze the nonsymmetric scenario when the symmetries are broken without any external alteration of the code. To give some concrete examples we consider an XYZ model and show the insight that can be gained by (i) searching the ground state and (ii) evolving in time after a symmetry-changing quench. To show the generality of the method, we also consider an interacting bosonic system under the effect of a symmetry-breaking dissipation.