Part 4 of the series From quarks to quantum advantage
After the weak-coupling approximation, each lattice site is described by two binary numbers: an incoming string end \(n_i(r)\) and an outgoing string end \(n_o(r)\). This means that the number of qubits is known, but not yet the circuit. A Hamiltonian is a generator of continuous time evolution; a quantum processor outputs a finite series of elementary gates.
The conversion requires four choices: a qubit order, a fermionic mapping, a Trotter decomposition and a gate construction for each Hamiltonia term. Each choice either introduces additional gates or an approach. The final QASM is therefore not a neutral copy of the theory, but a concrete digital implementation with a verifiable origin.
Two qubits per lattice site
The logical mapping is direct:
\[|q_{2r}\rangle=|n_i(r)\rangle,
\qquad
|q_{2r+1}\rangle=|n_o(r)\rangle.
\]
In the measurement order, the incoming and outgoing variables from the same site are therefore next to each other. This zigzag mapping is also the convention of our local observable analyzer. From the measured bits \(n_i(r)\), \(n_o(r)\), the parity-dependent \(n_f(r)\) and the global charges can be built back.
During the circuit, the physical location of the logical variables changes through SWAP gates. The paper therefore chooses an order for the initialization that already absorbs the first necessary SWAP layer. After each complete Trotter step, the mapping returns to the declared zigzag order. That detail is essential: a correct measurement result with an incorrect logical back-mapping gives an incorrect spatial density.
Binary numbers are not yet fermions
A qubit ladder operator can change a zero to a one, but operators on different qubits commute. Fermionic creation and annihilation operators must anticommute. The Jordan-Wigner transform fixes that behavior by placing a string of Pauli (Z) operators in front of the local ladder operator.
The paper uses schematically
\[\chi_i^-(r)=\sigma^+(r)\prod_{r'<r}[-\sigma^z(r’)],
\]
with a corresponding formula for the (o) variables. The precise indexing depends on the chosen fermionic order. For products coupling neighboring fermion modes, many parts of the Jordan-Wigner string cancel each other. After the weak-coupling approximation, this results in a locally executable two-qubit interaction block.
Jordan-Wigner is not noise reduction and not an approximation of physics. It is an exact algebraic mapping for the chosen finite ordering. The extra Pauli strings and the resulting routing do determine the circuit costs.
Continuous evolution in small steps
The desired evolution over a short scaled time \(\delta\tau\) is \(e^{-iW\delta\tau}\). The three parts of \(W\) do not commute, so
\[e^{-i(H_E+\mu H_M+xH_I)\delta\tau}
\ne
e^{-iH_E\delta\tau}e^{-i\mu H_M\delta\tau}e^{-ixH_I\delta\tau}
\]
for a finite step. The paper uses a first-order Trotter approximation:
\[U(\delta\tau)
\approx
e^{-i\widetilde m H_M}
e^{-i\delta\tau H_E}
e^{-icH_I},
\qquad
c=x\delta\tau,
\quad
\widetilde m=\mu\delta\tau.
\]
Because operators from the right act on a state, within a step first the interaction, then the electrical term and finally the mass term are applied. The local error per first-order step starts at order \(\delta\tau^2\); it can accumulate over many steps. The chosen value 0.0015 is small to limit that error, but does not make it exactly zero.
Three errors that are often called Trotter error
It helps to tell three digital abnormalities apart. The real Trotter error arises because exponentials of non-commutating Hamiltonians are multiplied in a chosen order. It already exists in a perfect ideal simulator and can be investigated by reducing \(\delta\tau\) or using a higher order formula.
Next comes gate synthesis. The analytical two-qubit unitaries are decomposed into an available logical gate set. When the angles of rotation are exactly available, that decomposition can be algebraically exact; on hardware, angles and pulses are realized with finite calibration accuracy.
Finally, there is transpiler and hardware failure. A backend can introduce additional routing, different native decomposition, decoherence, and stochastic gate errors. A smaller Trotter step also requires more steps for the same physical time and can therefore accumulate more hardware error. The optimal digital experiment thus balances model, Trotter, and hardware error; just a smaller delta tau is not automatically better on a noisy processor.
flowchart LR
A["initial state"] --> I["interaction: hopping layers"]
I --> S["SWAP to onsite pairs"]
S --> E["electric phase"]
E --> M["staggered Rz mass phase"]
M --> Z["zigzag mapping after one Trotter step"]
Z --> IThe interaction block
The approximate interaction term moves a string end between neighboring modes. On the computational basis \(|00\rangle,|01\rangle,|10\rangle,|11\rangle\), the elementary block has the form
\[U_I(c)=
\begin{pmatrix}
1&0&0&0\\
0&\cos c&i\sin c&0\\
0&i\sin c&\cos c&0\\
0&0&0&1
\end{pmatrix}.
\]
The states \(|01\rangle\) and \(|10\rangle\) are coherently mixed; the empty and double occupied states remain unchanged in this block. It resembles a partial iSWAP or XY rotation and saves the total occupancy of the pair.
In the paper, this block is decomposed into Hadamards, CNOTs and opposite \(R_z(c)\) rotations. The public QASM then contains the concrete basic gates of that dissolution. The exact hardware pulses are only created when the circuit is compiled to the native gates of the chosen IBM processor.
Each non-edge mode must associate with a left and right neighbor. Those interactions cannot all be performed on the same qubit at the same time. The circuit therefore splits them into two layers, similar to an even-odd decomposition. Between the layers, SWAP gates rearrange the relevant logical variables.
The electric phase
The large background flux largely produces a global phase in the approach used. A global phase does not change the measurement probability and can be omitted from the circuit. A configuration-dependent correction remains for a specific onsite pattern of \(n_i,n_o\).
The electric two-qubit block is diagonal. A relative phase \(e^{i\theta}\) is placed on one selected basis state, while the other components remain the same except for a common phase. Two CNOTs and three \(R_z\) rotations realize this controlled phase without an additional flux register.
Here again there is a model approach. The average background flux is not exactly the same in every basis component, but is replaced by a fixed edge flux. The phase block preserves some of the electric-field feedback that would otherwise be completely lost.
The staggered mass phase
The mass term is diagonal and counts the local string ends with an alternating sign:
\[H_M=\sum_r(-1)^r[n_i(r)+n_o(r)].
\]
Therefore, only single-qubit \(R_z\) rotations are needed. At even sites, both site qubits are assigned one direction of rotation; on odd sites the opposite. These gates are applied at the end of each Trotter step.
The mass term is computationally cheap. The depth is mainly determined by the two-qubit interaction, the electric phase and the routing between the required pairs.
Why the depth barely depends on N
The interactions are local and can be performed in parallel over non-overlapping pairs. The SWAPs can also be organized in two parallel layers. A longer grid contains more gates per layer, but not automatically more consecutive layers. As a result, the depth per Trotter step in the ideal logic scheme is independent of the grid length.
For t Trotter steps, the paper reports a two-qubit depth
\[d_2(t)=13t-1.
\]
The subtraction of one is because the first SWAP layer is included in the initialization. At 20 steps is \(d_2=259\); at 25 steps \(d_2=324\). The 25-step circuit contains 17,660 two-qubit gates and a total of 90,955 gates.
That scale is at once impressive and misleading when mentioned without context. A large gate count does not prove that the circuit is classically inaccessible or physically correct. The gate family, desired observable, entanglement and approximations determine the real difficulty.
From logic circuit to IBM Boston
The logical interaction graph is a line. IBM Boston contains a suitable connected chain within its 156 physical qubits. According to the paper, the transpiler is used to select a relatively low-noise circuit and decompose the circuit to the native gates. Because the logical routing already matches the topology, the initial placement does not require additional routing SWAPs on top of the SWAPs of the algorithm itself.
The downloaded QASM files contain the unitary circuits without classical registers and measurements. Our Fire Opal runner keeps those source files unchanged and creates measured copies with a full Z base measurement register. This makes it visible which part comes from upstream and which part has been added locally for implementation.
Small verification before 120 qubits
The authors compare an ideal circuit simulation with exact matrix evolution of the full LSH Hamiltonian for six sites, i.e. twelve qubits. The initial state is the SCV and the total particle density is monitored over time. Agreement at \(x=100\) supports the chosen parameters and circuit construction on a small scale.
That test is necessary, but not sufficient. A small exact match does not guarantee that Trotter, cutoff, and weak link errors remain equally small across 60 sites and over time. Therefore, the large calculation additionally uses TN and Pauli propagation.
What has been demonstrated here?
The approximate LSH Hamiltonian can be converted into a local, parallelizable Trotter circuit with two qubits per site. The origin of each gate block is traceable to the interaction, electrical or mass term, and the small circuit is checked against exact evolution.
What has not yet been proven?
A correct QASM file does not prove that the weak-coupling approximation follows the complete theory at all times. The ideal depth also says nothing about the eventual hardware failure. Those two questions require the classic model baselines and the actual QPU measurement, respectively.
Part 5 therefore follows the circuit through the standalone Fire Opal route to ibm_fez, reconstructs the observables from counts, and shows why readout correction for a 512-shot smoke is not an improvement by itself.
Sources
- Fran Ilčić et al., paper v3, Methods, equations 29-44 and Figure 6.
- Public QASM Circuits.
- Tracker QASM for the step-20 instance.
- Z. Davoudi, A. F. Shaw, and J. R. Stryker, algorithms for non-Abelian Hamiltonian simulation.
- Hadron repository and reproducibility notes.