A two-step route to spin-specific quantum ground states

Finding a system’s lowest-energy state is a central task in quantum simulation. But sometimes the target is the lowest-energy state within a specified spin sector: one with a chosen total spin magnitude and a chosen value of the spin component along the z-axis. This is called a spin-adapted ground state. It can help researchers study particular magnetic states or compare states with different spin quantum numbers.

One way to select a spin sector is to add penalty terms to the system’s Hamiltonian. These terms make states with the wrong spin energetically costly. In nonvariational algorithms—methods that evolve a quantum state rather than repeatedly adjusting circuit parameters to lower a measured energy—the penalty operators must be implemented as part of the quantum computation. The paper says a straightforward implementation of the total-spin penalty can require a number of gates that grows as the fourth power of the number of spins. That can become costly as a simulated system grows.

Separate the spin choices

Takumi Kobori and colleagues propose splitting the task into two stages. First, they prepare a state with the desired total spin magnitude and its largest possible spin-z value. Then they use a rotation that preserves total spin, followed by a measurement-based projection, to select the desired spin-z value.

The separation lets the authors alter the penalty used in the first stage. By combining information about total spin and spin-z in a modified penalty, they eliminate terms that would otherwise make the gate count scale quartically. The resulting penalty construction has quadratic scaling in the number of spins, according to the paper. The second stage uses a circuit to project onto the desired spin-z sector; the authors describe its depth as scaling as the square of the logarithm of system size for the encoding considered. Alternative encodings can change implementation costs.

The method is designed for spin-rotationally symmetric Hamiltonians: systems whose interactions preserve total spin. The authors test it with two nonvariational approaches, adiabatic time evolution and probabilistic imaginary-time evolution. The latter uses measurement and post-selection, so unsuccessful outcomes can require repeating steps.

What the simulations found

The tests were classical statevector simulations, not runs on quantum hardware. For spin-1/2 Heisenberg rings, the simulations used seven qubits; for a manganese-trimer model, they used ten. The team reports that the proposed method prepared states with the intended spin properties in both settings. In the ring-model tests, gate counts for the penalty terms showed the expected change from quartic to roughly quadratic growth in the tested size range. The trimer calculations examined several total-spin sectors and likewise showed successful preparation in the simulated cases.

The reduction is a statement about the cost of the penalty construction, not a guarantee that every full calculation becomes faster by the same amount. Total runtime also depends on factors such as the system’s energy spectrum, the starting state, and the evolution schedule. The authors note that penalty strengths must be chosen carefully: penalties that are too weak can leave unwanted spin components, while very strong penalties can require smaller time steps and reduce efficiency.

The simulations also reveal trade-offs. In some probabilistic imaginary-time tests, the conventional penalty approach converged faster; in others, convergence was comparable. The cumulative probability of success in post-selected evolution can decline as steps accumulate. The paper also reports that preparing higher-excitation spin sectors with adiabatic evolution became more challenging in its tests. These results caution against reading the lower gate-count scaling as proof of better performance in every setting.

What remains uncertain

The evidence supports a narrower conclusion: the proposed construction reduced the number of penalty terms in the authors’ implementations and worked in the tested model simulations. It does not show experimental operation on quantum hardware, nor establish that the method will outperform alternatives across large or realistic systems. The authors’ cost comparisons simplify details of the system Hamiltonian and depend on how its time evolution is implemented. The second-stage projection’s success probability also affects the expected number of repetitions.

This account is based on the arXiv preprint Accelerated spin-adapted ground-state preparation with nonvariational quantum algorithms (arXiv:2506.04663v2). A preprint is a research manuscript shared before, or independently of, formal journal peer review; its claims should be treated as provisional. The reported numerical results are simulations, not hardware validation. Read the preprint for the full methods and qualifications.