Did these results run on a quantum computer?

No. Every result comes from simulation on a classical computer: an ideal emulator, gate-level circuits with mid-circuit measurement and reset, and noisy density-matrix runs. The noise study shows what hardware would need: for half of the nine-cycle runs to be error-free, even with the cheapest circuit, two-qubit gate errors near 0.02%. The slides put the best current gates at about 0.1%, although a few prototypes have reported lower.

Does the rodeo algorithm beat a classical computer here?

Not at this size. Exact diagonalisation solves the five-site ring in milliseconds, and Lanczos methods reach systems several times larger for the lowest levels, although interior energy levels are much harder for them. Both exact ways of compiling the quantum circuit also need the full unitary matrix calculated on a classical computer first.

What drives the cost?

The controlled time evolution, almost entirely. Per cycle it took 1,004 two-qubit gates with exact synthesis, 1,743 with second-order Trotterisation and 370 with a fitted fixed-depth circuit. The fixed-depth circuit is cheapest at five spins, but both exact methods grow about as fast as 4 to the power n, so Trotterisation is the cheapest from about seven spins.

Rodeo Algorithm Simulation Results: Cost and Limits (PDF)

What the slides cover

Our research note on the rodeo algorithm explained the method and estimated its cost on paper. These slides report what happened when we built it. The test problem is the same: a five-site Heisenberg ring in a magnetic field (J = 1, h = 3), searched between energies of −13 and −9. Exact diagonalisation puts three levels there: −11.236 and −10.472, each a doublet, and −10.000, a singlet.

Every result is from simulation on a classical computer. No quantum hardware was used.

The headline result

Run as gate-level circuits with six qubits, nine mid-circuit measurements and 33,000 shots, the scan found all three levels. The results agree with the ideal emulator (χ²/dof = 1.19).

Exact level	Estimate	Measured overlap (exact)
−11.236	−11.226 ± 0.007	0.136 (0.133)
−10.472	−10.464 ± 0.010	0.112 (0.116)
−10.000	−9.992 ± 0.005	0.329 (0.333)

What the slides show

How one cycle works. The controlled time evolution writes each level's energy into the phase of one extra qubit, a phase gate subtracts the energy being scanned, and keeping only the runs where that qubit reads 1 lets the matching energy through. Random times stop false peaks, and resetting the qubit mid-circuit means one extra qubit serves any number of cycles.
Why it converges. Each cycle keeps a wrong level with probability ½ on average and ¼ in a typical run, so wrong levels fade like 2^−N to 4^−N over N cycles. The measured peak widths matched the predicted resolution to within 3% from six cycles on.
The starting state decides what you can see. Each filter cycle built with the exact evolution commutes with the Hamiltonian, the magnetisation and the ring's translations, so the filter can only re-weight what the starting state already holds. The five-site version of the Néel state used in the original paper has no weight in this window. An equal superposition of all basis states has 3.1% and misses both doublets. A three-term state prepared with one two-qubit gate puts 58% in the window, across all three levels.
Precision and clean states. Zooming in, with the evolution time ten times longer at each stage, the uncertainty fell from 0.007 to 0.0006 to 0.00005: the uncertainty falls as one over the evolution time, the Heisenberg limit. After nine cycles the prepared states reached a fidelity above 0.97. The price is that the share of runs kept falls to the state's overlap with the target.
Ghost peaks from first-order Trotter. Our note predicted that first-order Trotterisation would shift the peaks. Instead it split each doublet into two ghost peaks, because the leading error term cancels on single levels but separates states of opposite momentum. Second order fixes it: the error falls with the square of the step size (fitted slope 2.00) and is only 0.027 at a step of 0.1. Fixing the number of steps instead of the step size smears the −10.47 level away.
An exact fixed-depth circuit. A fixed-depth circuit fitted to the evolution becomes exact to machine precision once it has more angles than the dimension of the system's dynamical Lie algebra, which is 40. Seven layers (35 angles) fail, eight (40) work sometimes and nine (45) are exact at every time tested, for 370 two-qubit gates per cycle.
Noise and the classical baseline. A per-cycle error model matched noisy density-matrix runs to within 0.015, and it sets the bar for hardware: two-qubit gate errors near 0.02%. The slides put the best current gates at about 0.1%, although a few prototypes have reported lower. A classical computer still wins at this size.
A repeatable demonstration. A fresh draw of random times (six cycles, 300 shots per energy) runs in under three seconds and recovers the same three levels.

What it costs

Almost all of the cost sits in the controlled time evolution. The slides compare three ways to build it, with Trotterisation at first and second order, each run on the same 40 sets of nine random times. The table counts two-qubit gates per cycle and per nine-cycle run. A hand-derived controlled exchange gate needs 8 of them, against 11 from a standard library and the 12 our note assumed.

Method	Per cycle	Per run	Accuracy
Exact synthesis	1,004	9,036	Exact
First-order Trotter, step 0.05	2,552	22,972	Ghost peaks (0.27)
Second-order Trotter, step 0.1	1,743	15,683	Bias of 0.027
Fixed depth, nine layers	370	3,330	Exact (10⁻¹³)

At five spins that ranking is a small-size effect. Exact synthesis grows about as fast as 4ⁿ with the number of spins n (measured at 52 to 4,140 gates per cycle for three to six spins), and the fixed-depth circuit needs as many angles as the Lie algebra has dimensions, which grows about as fast. Trotterisation grows far more slowly and is the cheapest from about seven spins. Both exact methods also need the full 2ⁿ × 2ⁿ unitary matrix calculated on a classical computer first.

Key points

In simulation the method works as described: exponential filtering with one extra qubit, energy uncertainties down to about 0.00005 and prepared states above 0.97 fidelity.
The controlled evolution is the whole bill, so compiling it well matters more than anything else in the circuit.
It needs three things: a starting state with real overlap, patience for discarded runs (4.7% of runs were kept) and two-qubit gate errors near 0.02%.
Next steps: controlled reversal gates on rings with an odd number of sites, exact local compilation stitched together with Trotter steps, and a variational method (VQE) to raise the starting overlap.

Who it is for

Researchers and technical leaders deciding which quantum algorithms are worth testing on current hardware, and what a first experiment would need in gates, error rates and classical computing.

How we can help

We run feasibility studies and prototypes in quantum computing, from algorithm selection and circuit cost estimates to simulation, and we design custom algorithms for hard optimisation and modelling problems. See our research work, or read Quantum computing for business for where the technology stands today.

Rodeo algorithm simulation results