The Signal
56-Qubit H2-2 Test Reveals Bosonic Encoding Trade-Offs
A study posted October 21 compares three qubit encodings for bosonic simulation and tests unary and Dicke circuits on Quantinuum’s 56-qubit H2-2. Classical simulations favor binary quadrature for its space efficiency, but the team did not r
Quantum.world Team · Written with AI · fact-checked ·
A compact encoding looked best in the researchers’ classical cost comparison, but it was the one they did not test on the quantum device. In a paper posted October 21, Maarten Stroeks, Jelena Mackeprang, Yaroslav Herasymenko and Barbara Terhal compare ways to represent a bosonic mode with qubits, then run unary and Dicke encodings on Quantinuum’s 56-qubit H2-2 trapped-ion computer. Their results put a practical limit on the headline comparison: the device experiments probe only two of the three main encodings.
Binary quadrature wins the classical comparison
The study examines unary Fock encoding, a newly proposed Dicke encoding and binary quadrature encoding. The researchers assess a displacement operation and time evolution under the Dicke model, which couples a bosonic mode to atoms represented by qubits. Their classical simulations compare accuracy and resource costs, including qubit count, circuit depth and CNOT count.
Binary quadrature comes out ahead overall in those simulations, according to the authors, because it uses qubits efficiently while keeping circuit costs relatively low with the quantum Fourier transform. The advantage is not universal: its resource demands depend on the target accuracy and simulation, and the paper says the comparison is not comprehensive. The team also leaves binary Fock encoding out of its benchmarks because implementing displacement with it brings circuit size and depth overhead.
“Our novel Dicke encoding leads to very shallow circuits, but pays in terms of using many more qubits.”
Maarten Stroeks, Jelena Mackeprang, Yaroslav Herasymenko and Barbara Terhal, authors of the paper
Shallow circuits come with a qubit cost
The new Dicke scheme maps bosonic Fock states to permutation-symmetric states of multiple qubits. Its displacement circuit uses only single-qubit gates and has depth one, but keeping the encoding error small requires many more qubits than unary Fock encoding. Unary uses one qubit for each included Fock level; Dicke distributes each excitation across a larger register.
Binary quadrature takes a different route, encoding a discretized position-like coordinate in binary. Its momentum operator uses a quantum Fourier transform, which helps make the encoding space-efficient. But on the tested Dicke-model parameters, the paper estimates that binary quadrature would require 50 Trotter layers and 3,700 CNOTs, with a predicted depth of 3,700. Those figures are theoretical estimates, not device results.
H2-2 tests show the noise trade-off
The researchers ran displacement and Dicke-model experiments using unary and Dicke encodings on the H2-2. For displacement, the unary circuit represented the bosonic mode with 13 qubits and used 11 Trotter layers; the Dicke encoding used 45 qubits. The team set a maximum target error of 0.1 for the measured momentum expectation and collected 500 shots per circuit.
The Dicke displacement circuit, built from single-qubit gates, tracked the noise-free prediction closely. Unary showed larger deviations, which the authors connect to its larger circuits and their growth with displacement size and Trotter layers. The hardware comparison therefore illustrates a trade-off, not an across-the-board winner: the more qubit-hungry Dicke encoding can reduce circuit exposure to noise.
The omitted encoding sets the limit
For Dicke-model time evolution, the device runs used three atom qubits, with three bosonic qubits for unary encoding and five for Dicke encoding. At the final time step, compiled circuit depths were 111 for unary and 56 for Dicke, while the respective two-qubit gate counts were 93 and 100. Unary results drifted from the predicted curve at later times; post-selecting runs that left the unary encoded subspace improved its data, but did not remove the late-time deviation.
Binary quadrature was not run on H2-2: the authors say their limited device access prevented those experiments. The paper’s strongest case for it is therefore a result of classical resource modeling, while the hardware data compare unary with Dicke. That distinction matters for anyone reading the 56-qubit test as a direct device ranking of all three encodings.