Dissipative ground state preparation and the Dissipative Quantum Eigensolver

TL;DR

The work introduces the Dissipative Quantum Eigensolver (DQE), a universal ground-state preparation method that works for any local Hamiltonian by iterating local weak measurements and employing stopping rules. By leveraging approximate ground-state projectors (AGSPs) and carefully designed stopping criteria, DQE achieves unconditional convergence with circuit-depth scaling linearly with system size, while the total run-time remains exponential for complexity-theoretic reasons. The framework includes rigorous fault- and error-resilience analyses, diverse resampling strategies, and practical circuit constructions, with extensions to stabilizer codes, probabilistic gates, and VDQE/VQE hybrids. This approach broadens the toolbox for ground-state preparation, offering a robust, hardware-friendly alternative to phase estimation, adiabatic schemes, and purely variational methods, with promising implications for quantum chemistry, materials science, and combinatorial optimization.

Abstract

For any local Hamiltonian H, I construct a local CPT map and stopping condition which converges to the ground state subspace of H. Like any ground state preparation algorithm, this algorithm necessarily has exponential run-time in general (otherwise BQP=QMA), even for gapped, frustration-free Hamiltonians (otherwise BQP is in NP). However, this dissipative quantum eigensolver has a number of interesting characteristics, which give advantages over previous ground state preparation algorithms. - The entire algorithm consists simply of iterating the same set of local measurements repeatedly. - The expected overlap with the ground state subspace increases monotonically with the length of time this process is allowed to run. - It converges to the ground state subspace unconditionally, without any assumptions on or prior information about the Hamiltonian. - The algorithm does not require any variational optimisation over parameters. - It is often able to find the ground state in low circuit depth in practice. - It has a simple implementation on certain types of quantum hardware, in particular photonic quantum computers. - The process is immune to errors in the initial state. - It is inherently error- and noise-resilient, i.e. to errors during execution of the algorithm and also to faulty implementation of the algorithm itself, without incurring any computational overhead: the overlap of the output with the ground state subspace degrades smoothly with the error rate, independent of the algorithm's run-time. I give rigorous proofs of the above claims, and benchmark the algorithm on some concrete examples numerically.

Figures (5)

• Figure 1: The evolution of the energy (upper purple solid line) and ground state overlap (lower green solid line) in (part of) a typical run of the DQE \ref{['DQE']} for the 1D Heisenberg chain of length 10, with $\epsilon=0.2$, stopping on the first run of 4 zeros, and using global resampling.footnotemark Although this run of the algorithm took a total of 1016846 iterations before it stopped, the quantum circuit depth required to achieve this is significantly lower. Because global resampling discards the entire state whenever the measurement outcome 1 is obtained, coherence of the quantum state need only be maintained for a maximum of 4 iterations to attain an energy close to the ground state energy (dashed blue line).
• Figure 2: Energy of the DQE output state as a function of maximum run-time $t$ in \ref{['secretary_CPT_map']} with the standard error model (green circles), for the 1D Heisenberg chain of length 5 using the secretary stopping policy of \ref{['secretary_CPT_map']}.footnotemark In the standard error model, a 1- or 2-qubit depolarising channel is applied after each 1-qubit or 2-qubit gate, respectively, here with depolarising parameter $10^{-4}$. For comparison, the energy of the state that would be obtained by starting from the ground state and applying the same $10^{-4}$ 1-qubit depolarising channel across all qubits $t$ times is shown (blue curve). The true ground state energy is also indicated (purple horizontal line). The energy of the DQE output state is clearly seen to be independent of the run-time and remains close in energy to the true ground state (green), illustrating the noise- and fault-resilience discussed in \ref{['sec:fault-resilience']}. Whereas without the DQE process, the state decays over the same run-time to the maximally mixed state, which is far from the ground state.
• Figure 3: Schematic illustration of the DQE \ref{['DQE']} when using the AGSP from \ref{['AGSP_product']}. The qubits are passed through the circuit of \ref{['measurement_circuit']}, repeated for each local term in the Hamiltonian. The top-most qubit is the ancilla qubit that gets measured in \ref{['measurement_circuit']} to produce the classical measurement outcomes, indicated by the double-wires. This ancilla is then reset to $\mathinner{\lvert0\rangle}$ by the controlled-$X$ gate and reused in the subsequent measurement. (Alternatively, a fresh $\mathinner{\lvert0\rangle}$ ancilla could also be used for each measurement.) The qubits are repeatedly cycled through this circuit until the measurement outcomes satisfy the stopping condition of \ref{['DQE']}.
• Figure 4: Schematic illustration of the DQE \ref{['DQE']} when using the AGSP from \ref{['AGSP_mixture']}. The qubits are passed through the circuit of \ref{['measurement_circuit']}, selected at random from each local term in the Hamiltonian. The top-most qubit in any block is the ancilla qubit that gets measured in circuit \ref{['measurement_circuit']}, producing the classical measurement outcomes indicated by the double-wires. This ancilla is then reset to $\mathinner{\lvert0\rangle}$ by the controlled-$X$ gate and reused in the subsequent iteration. (Alternatively, a fresh $\mathinner{\lvert0\rangle}$ ancilla could also be used for each measurement.) The qubits are repeatedly cycled through this circuit until the measurement outcomes satisfy the stopping condition of \ref{['DQE']}.
• Figure :

Theorems & Definitions (93)

• Theorem 2: informal, see \ref{['decaying_CPT_map']} for precise version
• Corollary 3: See \ref{['stopping_time', 'stopped_circuit_depth']} for precise statements.
• Theorem 4: informal; see \ref{['stopped_fault-resilience']} for precise version
• Theorem 5: informal, see \ref{['Chebyshev_fixed-point']} for precise version
• Definition 6: AGSP
• Remark 7
• Lemma 8: AGSP
• Lemma 9
• Proof 1
• Theorem 10