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arXiv:1909.10538 · Physical Review A

Quantum digital cooling

Stefano Polla, Yaroslav Herasymenko, Thomas E. O’Brien

Published Jul 15, 2021• 34 citations
arXivHTMLPDF

Abstract

We introduce a method for digital preparation of ground states of simulated Hamiltonians, inspired by cooling in nature and adapted to leverage the capabilities of digital quantum hardware. The cold bath is simulated by a single ancillary qubit, which is reset periodically and coupled to the system nonperturbatively. Studying this cooling method on a 1-qubit system toy model, we optimize two cooling protocols based on weak-coupling and strong-coupling approaches. Extending the insight from the 1-qubit system model, we develop two scalable protocols for larger systems. The LogSweep protocol extends the weak-coupling approach by sweeping energies to resonantly match any targeted transition. We test LogSweep on the 1D transverse-field Ising model, demonstrating approximate ground-state preparation with an error that can be made polynomially small in the computation time for all three phases of the system. The BangBang protocol extends the strong-coupling approach, and exploits a heuristics for local Hamiltonians to maximize the probability of deexciting system transitions in the shortest possible time. Although this protocol does not promise long-time convergence, it allows for a rapid cooling to an approximation of the ground state, making this protocol appealing for near-term demonstrations.

DOI
10.1103/physreva.104.012414

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We introduce a method for digital preparation of ground states of simulated Hamiltonians, inspired by cooling in nature and adapted to leverage the capabilities of digital quantum hardware. The cold bath is simulated by a single ancillary qubit, which is reset periodically and coupled to the system nonperturbatively. Studying this cooling method on a 1-qubit system toy model, we optimize two cooling protocols based on weak-coupling and strong-coupling approaches. Extending the insight from the 1-qubit system model, we develop two scalable protocols for larger systems. The LogSweep protocol extends the weak-coupling approach by sweeping energies to resonantly match any targeted transition. We test LogSweep on the 1D transverse-field Ising model, demonstrating approximate ground-state preparation with an error that can be made polynomially small in the computation time for all three phases of the system. The BangBang protocol extends the strong-coupling approach, and exploits a heuristics for local Hamiltonians to maximize the probability of deexciting system transitions in the shortest possible time. Although this protocol does not promise long-time convergence, it allows for a rapid cooling to an approximation of the ground state, making this protocol appealing for near-term demonstrations.
Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes O~(d 3/2/ε)\tilde{\mathcal{O}}(d^{\,3/2}/ε)O~(d3/2/ε) queries to the time-evolution operator of the system and O~(d 3/2)\tilde{\mathcal{O}}(d^{\,3/2})O~(d3/2) queries to a block-encoding of the perturbation, for ddd cooling steps and an εεε-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.
We present a cooling algorithm for ground state preparation of fermionic Hamiltonians. Our algorithm makes use of the Hamiltonian simulation of the considered system coupled to an ancillary fridge, which is regularly reset to its known ground state. We derive suitable interaction Hamiltonians that originate from ladder operators of the free theory and initiate resonant gaps between system and fridge. We further propose a spectroscopic scan to find the relevant eigenenergies of the system using energy measurements on the fridge. With these insights, we design a ground state cooling algorithm for fermionic systems that is efficient, i.e. its runtime is polynomial in the system size, as long as the initial state is prepared in a low-energy sector of polynomial size. We achieve the latter via a pseudo-adiabatic sweep from a parameter regime whose ground state can be easily prepared. We estimate that our algorithm has a polynomial runtime for systems where the spectral gap decreases at most polynomially in system size, and is faster than the adiabatic algorithm for a large range of settings. We generalize the algorithm to prepare thermal states and demonstrate our findings on the Fermi-Hubbard model.
Simulating many-body systems is one of the most promising applications of near-term quantum computers. An important open question is how to efficiently prepare the ground states of arbitrary fermionic Hamiltonians, especially those with nontrivial topology. Here, we propose an efficient protocol for preparing low-energy states of fermionic Hamiltonians on a noisy bosonic quantum simulator by adiabatic cooling using a simulated bath. We arrange the couplings such that the simulated system and bath together obtain a local fermionic description in which fermionic excitations can be extracted individually, via coherent hopping to the bath, rather than in pairs as would otherwise be required by fermion parity conservation. This approach thus achieves a linear (rather than quadratic) scaling of the cooling rate vs. excitation density at low densities. We show that certain topological phases such as the chiral (non-Abelian) phase of the Kitaev honeycomb model can be prepared efficiently using our protocol. Our protocol performs favorably in the presence of noise, making it suitable for execution on near-term quantum devices. Simulating many-body systems is a promising application of near-term quantum computers; however, preparing ground states of fermions on a quantum computer is challenging, especially if the ground state is topologically non-trivial. The authors propose a protocol to prepare the ground state of arbitrary fermionic Hamiltonians, which is particularly suitable for topological chiral states.
Designing cooling protocols is believed to require knowledge of the system spectrum. In contrast, cooling in nature occurs whenever the system is coupled to a cold bath. How does nature know how to cool? A natural cold bath can be mimicked with a reservoir of "meter" qubits that are initialized in their ground state. We show that a quantum system can be cooled without knowledge of system details when system-meter interactions and meter splittings are chosen randomly. For sufficiently small interaction strengths and long interaction times, the protocol ensures that resonant energy-exchange processes, leading to cooling, dominate over heating. Effectively, the dynamics is then captured by the rotating-wave approximation, which we identify as the basic mechanism for robust and scalable cooling of complex quantum systems through generic, structure-independent protocols. This offers a versatile universal framework for controlling quantum matter far from equilibrium, in particular, for quantum computing and simulation.
Probing correlated states of many-body systems is one of the central tasks for quantum simulators and processors. A promising approach to state preparation is to realize desired correlated states as steady states of engineered dissipative evolution. A recent experiment with a Google superconducting quantum processor [X. Mi , Science 383, 1332 (2024)] demonstrated a cooling algorithm utilizing auxiliary degrees of freedom that are periodically reset to remove quasiparticles from the system, thereby driving it towards its ground state. In this work, we develop a kinetic-theory framework to describe quasiparticle cooling dynamics, and employ it to compare the efficiency of different cooling algorithms. In particular, we introduce a protocol where coupling to auxiliaries is modulated in time to minimize heating processes, and demonstrate that it allows a high-fidelity preparation of ground states in different quantum phases. We verify the validity of the kinetic-theory description by an extensive comparison with numerical simulations for the examples of a one-dimensional transverse-field Ising model, the transverse-field Ising model with an additional integrability-breaking field, and a nonintegrable antiferromagnetic Heisenberg spin ladder. In all cases, we are able to efficiently cool into the many-body ground state. The effects of noise, which limits efficiency of variational quantum algorithms in near-term quantum processors, are investigated through the lens of the kinetic theory: we show how the steady-state quasiparticle populations depend on the noise rate, and we establish maximum noise values for achieving high-fidelity ground states. This work establishes quasiparticle cooling algorithms as a practical, robust method for many-body state preparation on near-term quantum processors.
Simulated cooling is a robust method for preparing low-energy states of many-body Hamiltonians on near-term quantum simulators. In such schemes, a subset of the simulator's spins (or qubits) are treated as a "bath" that extracts energy and entropy from the system of interest. However, such protocols are inefficient when applied to systems whose excitations are highly nonlocal in terms of the microscopic degrees of freedom, such as topological phases of matter; such excitations are difficult to extract by a local coupling to a bath. We explore a route to overcome this obstacle by encoding the microscopic degrees of freedom into those of the quantum simulator in a nonlocal manner. To illustrate the approach, we show how to efficiently cool the ferromagnetic phase of the quantum Ising model, whose excitations are domain walls, via a "gauged cooling" protocol in which the Ising spins are coupled to a Z_{2} gauge field that simultaneously acts as a reservoir for removing excitations. We show that our protocol can prepare the ground states of the ferromagnetic and paramagnetic phases equally efficiently. The gauged cooling protocol naturally extends to (interacting) fermionic systems, where it is equivalent to cooling by coupling to a fermionic bath via single-fermion hopping.