We propose a simple, robust protocol to prepare a low-energy state of an arbitrary Hamiltonian on a quantum computer or programmable quantum simulator. The protocol is inspired by the adiabatic demagnetization technique, used to cool solid-state systems to extremely low temperatures. A fraction of the qubits (or spins) is used to model a spin bath that is coupled to the system. By an adiabatic ramp down of a simulated Zeeman field acting on the bath spins, energy and entropy are extracted from the system. The bath spins are then measured and reset to the polarized state, and the process is repeated until convergence to a low-energy steady state is achieved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol's performance in the presence of noise and show how the information from the measurement of the bath spins can be used to monitor the cooling process. The performance of the algorithm depends on the nature of the excitations of the system; systems with non-local (topological) excitations are more difficult to cool than those with local excitations. We explore the possible mitigation of this problem by trapping topological excitations.
We propose a simple, robust protocol to prepare a low-energy state of an arbitrary Hamiltonian on a quantum computer or programmable quantum simulator. The protocol is inspired by the adiabatic demagnetization technique, used to cool solid-state systems to extremely low temperatures. A fraction of the qubits (or spins) is used to model a spin bath that is coupled to the system. By an adiabatic ramp down of a simulated Zeeman field acting on the bath spins, energy and entropy are extracted from the system. The bath spins are then measured and reset to the polarized state, and the process is repeated until convergence to a low-energy steady state is achieved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol's performance in the presence of noise and show how the information from the measurement of the bath spins can be used to monitor the cooling process. The performance of the algorithm depends on the nature of the excitations of the system; systems with non-local (topological) excitations are more difficult to cool than those with local excitations. We explore the possible mitigation of this problem by trapping topological excitations.
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.
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.
Inspired by simulated annealing algorithm, we propose a quantum cooling protocol which includes an annealing process. This protocol can be universally and efficiently applied to various quantum simulators, driving the system from an arbitrary initial state to the ground state with high fidelity. We have described the cooling process based on perturbation theory, validated the advantages of bath under time-modulated Zeeman field compared to bath under static one, and provided a justification for the necessity of an annealing process when the system to be cooled is unknown. We applied tensor network methods to numerically simulate our cooling protocol, using the transverse field Ising model (TFIM) as an example to verify the effectiveness of the protocol in cooling one-dimensional systems, two-dimensional systems, and systems with quantum noise. We compared the overall performance of cooling protocols with and without the annealing process on a test set generated with random parameters gP. The results indicate that the cooling protocol with annealing process can achieve both accuracy and efficiency. Our results also show that the cooling protocol's resistance to noise depends on the type of quantum noise.
We consider the problem of preparing thermal equilibrium states at finite temperature on quantum computers. Assuming thermalization, we show that states that are locally at thermal equilibrium can be prepared by evolving adiabatically an initial thermal Gibbs state of a simple Hamiltonian with an interpolating time-dependent Hamiltonian, identically to adiabatic ground state preparation. We argue that the entropy density of local density matrices is conserved during the adiabatic evolution in the thermodynamic limit, so that both the entropy and energy of the final state can be computed, and thus the final temperature too. We show that in the presence of hardware noise, the entropy created by the noisy evolution can be precisely benchmarked with mirror circuits. We give numerical evidence that the resulting thermal state preparation protocol is noise-resilient for depolarizing noise, in the sense that the energy-temperature curve measured on a noisy quantum computer is remarkably insensitive to the amplitude of depolarizing noise in the state preparation. We finally propose a protocol to estimate the lack of adiabaticity in a given actual Trotter implementation of the dynamics. We test our protocol on Quantinuum’s H1-1 ion-trap device. We measure that a circuit with 640 two-qubit gates implemented on hardware generates an entropy per site of 0.166 ± 0.0045, giving a benchmark metric for this state preparation. We report the preparation of a thermal state with temperature 2.56 ± 0.26 of the Ising model in size 5 × 4.
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.
We compare the performance of representative cooling, adiabatic, and optimization algorithms for ground-state preparation in the presence of noise. Using an exactly solvable family of quadratic fermionic Hamiltonians subject to depolarizing noise, we derive the scaling of the achievable relative energy as a function of the noise rate and support these results with numerical simulations. The Hamiltonian exhibits two phases, separated by a quantum phase transition. As expected, the performance of the different algorithms depends on the phase: adiabatic evolution is favorable in the trivial phase, while a multi-frequency cooling algorithm, as proposed in [1], becomes competitive or superior in the topological phase, where gap-closing limits adiabatic protocols. We further present numerical results for the quantum approximate optimization algorithm [2], showing that it performs competitively with cooling in the trivial phase but is typically outperformed in the topological regime. Finally, we show that for this model the cooling protocol exhibits enhanced robustness to parameter imperfections, highlighting its potential advantage for realistic implementations of noisy quantum state preparation.
The analytical approach developed here, in conjunction with numerical validation, establishes an extendable approach to benchmarking ground-state preparation algorithms.