Classical simulation methods

Part 6 of the series From quarks to quantum advantage

What is the classical competitor to a 120-qubit quantum circuit? The obvious answer, a classical simulation of 120 qubits, is too coarse. There are at least three different objects we can simulate: the full LSH Hamiltonian, the approximate ideal circuit, and just the measured local observables. Each choice leads to a different method, error source and calculation time.

The paper therefore does not use any classical truth. Exact diagonalization controls small systems. Tensor networks approximate the full physical Hamiltonian. Pauli propagation approximates the noise-free digital circuit. Our project adds Aer MPS, ITensor QASM-MPS, a small LSH-MPO validation and a Majorana feasibility test.

The validation card

Method	Primary object	Strongest contribution	Main border
Exact diagonalization	complete small Hamiltonian	reference without Trotter error	exponential Hilbert space
TN/TDVP	full LSH Hamiltonian	physical model reference	flux cutoff and bond dimension
Pauli propagation	ideal circuit and local observable	circuit reference	growth of Pauli terms
Aero MPS	compiled QASM	local circuit smoke	MPS convergence and deployment
ITensor QASM-MPS	compiled QASM	second circuit engine	bond dimension and runtime
Local LSH-MPO	full small LSH construction	Hamiltonian validation	no paper-scale TDVP yet
Majorana/Gaussian	quadratic fermion circuits	fast exact route if applicable	circuit is not Gaussian

flowchart TD
    H["full LSH Hamiltonian"] --> ED["exact diagonalization, small N"]
    H --> TN["MPS/MPO plus TDVP"]
    H --> A["weak-coupling approximation"]
    A --> Q["ideal QASM circuit"]
    Q --> PP["Pauli propagation"]
    Q --> MPS["Aer and ITensor QASM-MPS"]
    Q --> HW["QPU hardware"]

The arrows are more important than the names. TN and PP are not two implementations of exactly the same calculation. Their mutual agreement actually tests the step from complete theory to approximate circuit.

Exact diagonalization: small but decisive

With exact diagonalization, the Hamiltonian is constructed as a complete matrix in a finite basis. The time evolution then follows from matrix functions or from the complete eigendecomposition. Within the chosen cutoff there is no Trotter error, no MPS truncation and no Pauli term cutoff.

The price is exponential growth. Even after unphysical sectors are removed, the number of allowed LSH configurations grows rapidly with the number of sites and the flux cutoff. The paper therefore uses six sites and twelve qubits as proof of concept. The exact full LSH evolution is compared there with an ideal Qiskit simulation of the Trotter circuit.

Our local dense route does additional structure checks for \(N=2,3,4\) with \(n_{l,\max}=2\). It controls Hermiticity, the Abelian Gauss Law, global sectors and links from the SCV. For \(N=4\), the chosen \(Q=4,q=0\) sector has 88 states. That's great for unit tests, but not a performance baseline for \(N=60\).

Precise small systems therefore have a special function: they can unmask an incorrect sign, link constraint or incorrect gate block at an early stage. They cannot demonstrate that a classical method remains fast at 60 sites.

Tensor networks: the full LSH physics

A matrix product state writes the many-body state as a chain of local tensors. The internal indices have bond dimension \(D\). If the entanglement remains limited, a relatively small \(D\) can describe the state much more compactly than the full Hilbert space.

For the paper, the local physical index contains the full LSH state \(|n_l,n_i,n_o\rangle\). The Hamiltonian is constructed as a matrix product operator. Global symmetries \(B\) and \(q\) are incorporated into block-sparse tensors. The time evolution uses 2-site TDVP with ITensors.jl.

The most important settings are:

• full LSH Hamiltonian, not the simplified qubit model;
• bosonic cutoff \(j_{\max}=5/2\);
• maximum bond dimension \(D_{\max}=200\);
• separate MPSs for SCV and central meson;
• observables are subtracted from each other after evolution.

TDVP does not perform a first-order Trotter decomposition of the paper QPU here. The TN route therefore avoids exactly that digital error and serves as a physical reference. But full Hamiltonian does not mean exact result: the flux cutoff and finite bond dimension remain approximations.

In real-time evolution, entanglement grows. Once the required bond dimension exceeds 200, the MPS must discard information. The paper reports that the TN results at \(x=100\) are no longer sufficiently controlled for reporting beyond about 20 steps. With \(x=200\) the practical deterioration occurs earlier.

Our local LSH-MPO route

The project includes a separate ITensor construction of the full small LSH Hamiltonian. The link operators correspond to the dense reference on the AGL sector up to approximately \(10^{-15}\). For \(N=4\), \(n_{l,\max}=2\) and time 0.001 the dense-versus-MPO smoke gives:

• energy difference \(1.11\times10^{-11}\);
• norm difference \(2.50\times10^{-13}\);
• RMSE of the local \(Q\) contribution \(1.20\times10^{-8}\).

That's good physical implementation validation. It's not a useful timing claim. The generic OpSum -> MPO build plus TDVP took about 171 seconds for this small case, while dense evolution took 0.019 seconds. This ratio mainly says that the current generic local implementation is not optimized.

A slow prototype should never be used as an artificially weak classical competitor for quantum advantage. The paper TN figures remain the relevant published baseline for this.

Pauli propagation: simulate the observable

Pauli propagation works in the Heisenberg view. Instead of evolving the entire state forward, the method starts at the operator to be measured, for example \(Z_q\), and propagates it backward through the circuit:

\[
\langle Z_q(t)\rangle
=\langle\psi_0|U^\dagger Z_q U|\psi_0\rangle.
\]

The operator is written as a sum of Pauli strings. Clifford gates send a Pauli string to one other Pauli string and therefore do not grow the number of terms. Non-Clifford rotations can branch a term. In the worst case, the number of terms required grows exponentially with circuit depth.

The CPU and GPU routes in the paper control that growth differently. Budgets, caps, or coefficient thresholds are used to limit insignificant terms. The estimated final numbers for \(x=50,100,200\) are approximately 9,400, 12,500, and 66,000 Pauli terms, respectively. The two implementations therefore provide mutual control over the truncation strategy.

PP simulates the ideal digital circuit including the chosen weak-coupling approximation and Trotterization. Agreement with TN supports the physical validity of that circuit. Deviation between PP and QPU mainly indicates hardware and measurement errors.

Circuit MPS with Aer and ITensor

A QASM circuit can also be applied directly to an MPS state. That is a different task than TN evolution under the full LSH-MPO. The local Hilbert space now consists of qubits and the gates are exactly those of the downloaded circuit.

For the step-5 pair of SCV and meson we found:

engine	total time	set max. bond	meaning
Qiskit Aer MPS	34.281 s	64	compiled QASM, 512 shot route
ITensor QASM-MPS	174.611s	64	compiled QASM, expectation values

The ITensor run reached a maximum link dimension 50 and returned \(Q=60.000000824\) for SCV and \(Q=60.000000671\) for meson. This confirms that the ideal circuit preserves the global sector very accurately. The different runtimes mainly measure engine, implementation and measurement strategy; they are not a universal complexity equation.

These local MPS runs are useful for the decoder and circuit. They cannot replace the paper-TN route because both sides then simulate the same approximate QASM and the step back to the full Hamiltonian remains uncontrolled.

Why Majorana is not the quick way out here

Quadratic fermion Hamiltonians and matchgate circuits can be simulated exactly via the linear evolution of Majorana operators. That is polynomial and can reach much larger systems than a general state simulator.

However, the downloaded hadron QASM is not a purely fermionic Gaussian circuit under the standard encoding. The local gate inventory over the examined circuits includes 59,200 CNOTs, 23,600 Hadamards and 11,560 regular SWAPs. CNOT and Hadamard do not generally lie in the fermionic Gaussian group; an ordinary qubit SWAP also lacks the parity sign of a fermionic SWAP.

A Majorana simulator can therefore at most provide a separate free fermion surrogate or lower bound model. It is not an exact replacement for PP or the full LSH-TN calculation.

Which classical era is fair?

A timing number can only be interpreted when five things are certain:

1. Is the full Hamiltonian or compiled circuit simulated?
2. Is the full state or just a local observable calculated?
3. Which cutoff, bond dimension or term threshold is used?
4. Is it one time point or the entire time series?
5. What hardware and what fault tolerance are included in the run?

The paper TN and PP times do not answer exactly the same algorithmic question, but they do provide the two necessary layers of validation for the same reported physical observables. Our local times add implementation checks. Part 7 will therefore not lump them all together.

What has been demonstrated here?

The hadron simulation has a layered classical control. Small exact evolution checks the construction, TN checks the full LSH physics, PP checks the ideal circuit and two local MPS engines check the downloaded QASM and the decoder.

What has not yet been proven?

None of the local classical routes are a converged, paper-scale reproduction of the full \(N=60\)-LSH-TDVP calculation. The Majorana route is not exactly applicable and a single runtime without an accuracy target does not prove an asymptotic classical impossibility.

In part 7 we combine the physical validation and the times for the first time. The question then does not become is quantum faster?, but: for which exactly defined observable-estimation task, with which error and which clock?

Sources

1. Fran Ilčić et al., paper v3, TN and PP methods.
2. U. Schollwöck, MPS and DMRG.
3. M. C. Bañuls, Tensor Network Algorithms: A Route Map.
4. M. S. Rudolph et al., Pauli Propagation: A Computational Framework for Simulating Quantum Systems.
5. Local Quantum Win Claim Boundary.
6. Local Majorana Feasibility.
7. Local LSH-MPO Summary.