What the note covers
The rodeo algorithm, introduced by Choi, Lee, Bonitati, Qian and Watkins in 2021, asks a quantum computer one question: does this state have a component at energy E? One extra qubit controls the system's evolution for a random time, and each cycle in which that qubit returns the right answer suppresses every energy away from E. After enough cycles, what survives is the state you were looking for.
The note explains the method, checks it on a small problem with an exact answer, and estimates what it would cost to run.
- How one cycle works. Two Hadamard gates, a controlled time evolution, a phase gate and a measurement act together as a cosine filter centred on the chosen energy.
- Why the times are random. Equal times would let false energies through. Random times remove them, and the chance of an off-target energy passing falls by about a factor of four per cycle.
- Two uses. Scanning the energy to read off the spectrum, and fixing the energy to prepare one state.
- A worked example. A five-site spin model (the Heisenberg model in a magnetic field) searched between energies of −13 and −9. Exact diagonalisation gives the reference levels, and a simulated scan with 3, 6 and 9 cycles shows the peaks settling on them.
- The cost. Three ways to build the controlled evolution, from an exact circuit of about 1,020 two-qubit gates per cycle to shallower approximations that trade some accuracy for depth.
- Strengths and limits. Where it beats phase estimation, variational methods and adiabatic preparation, and where it does not: the starting state's overlap with the target, discarded runs, degenerate levels and approximation bias.
Key points
- The starting state is the key design choice: the algorithm can only sharpen what that state already contains.
- The controlled evolution, not the filter itself, dominates the gate count, so that is where engineering effort pays off.
- It fits near-term hardware: one extra qubit with mid-circuit measurement, no quantum Fourier transform and no classical optimiser.
Who it is for
Researchers and technical leaders evaluating which quantum algorithms are worth testing on current hardware, particularly for chemistry, materials and physics problems that need energy levels beyond the lowest one.
How we can help
We run feasibility studies and prototypes in quantum computing, from algorithm selection and circuit cost estimates to simulation. See our research work, or read Quantum computing for business for where the technology stands today.