Willow-Scale Analytical Forecast — TVD < 0.002

105 qubits  ·  Depth 2  ·  No exact reference needed  ·  KLT phase Z3→Z4  ·  Analytically certified

What this benchmark tests

At 105 qubits, no classical computer can produce an exact statevector to verify against — the Hilbert space has 2¹⁰⁵ ≈ 4×10³¹ dimensions. Yet Qumulator certifies the accuracy of its simulation analytically, without any exponential reference, using the KLT Lyapunov-exponent framework.

This benchmark shows the full analytical certificate for a 105-qubit 7×15 Willow-scale SYC circuit at depth 2: TVD < 0.002. The bound comes from the measured entropy profile and the KLT scrambling-time prediction — not from comparing against a brute-force result. It is a rigorous consequence of the circuit's entanglement geometry.

Depth-to-saturation timeline

The circuit depth relative to the saturation depth determines which phase the circuit is in and what accuracy guarantee applies:

Full result summary

Reproduce this result

Submit the circuit with return_entropy_map=True. The analytical TVD bound is in result.tvd_bound; the Lyapunov exponent and saturation depth are in result.lyapunov_exponent and result.saturation_depth.

pip install qumulator-sdk

import os, math
import numpy as np
from qumulator import QumulatorClient

client = QumulatorClient(
    api_url=os.environ["QUMULATOR_API_URL"],
    api_key=os.environ["QUMULATOR_API_KEY"],
)

# 7×15 Willow-scale SYC circuit, depth 2
nrows, ncols, depth = 7, 15, 2
N = nrows * ncols  # 105 qubits
rng = np.random.default_rng(42)

# SYC gate: fSim(θ=π/2, φ=π/6)
def syc():
    th, ph = math.pi/2, math.pi/6
    c, s = math.cos(th), math.sin(th)
    U = [[1,0,0,0],[0,c,-1j*s,0],[0,-1j*s,c,0],[0,0,0,np.exp(-1j*ph)]]
    M = np.array(U, dtype=complex)
    return M.real.tolist(), M.imag.tolist()

syc_r, syc_i = syc()

instructions = []
for d in range(depth):
    for q in range(N):
        instructions.append({"gate": "ry", "qubits": q, "params": [rng.uniform(0, math.pi)]})
        instructions.append({"gate": "rz", "qubits": q, "params": [rng.uniform(0, 2*math.pi)]})
    for r in range(nrows):
        for c in range(ncols - 1):
            q0, q1 = r*ncols+c, r*ncols+c+1
            instructions.append({
                "gate": "unitary", "qubits": [q0, q1],
                "matrix_real": syc_r, "matrix_imag": syc_i,
            })

result = client.circuit.run(
    n_qubits=N,
    instructions=instructions,
    mode="klt_mps",
    bond_dim=8,           # shallow circuit → low entanglement → χ=8 sufficient
    shots=2048,
    seed=42,
    return_entropy_map=True,
)

print(f"KLT phase          : {result.klt_phase}")          # → Z3 or Z4
print(f"Certified TVD      : < {result.tvd_bound:.4f}")    # → < 0.0020
print(f"Natural depth τ    : {result.natural_depth:.3f}")   # → 0.300
print(f"Saturation depth   : {result.saturation_depth}")   # → 52
print(f"Lyapunov λ         : {result.lyapunov_exponent:.2f}") # → 0.15–0.80
print(f"Exact ref needed?  : No — analytical certificate")

Why this matters at scale

Beyond ~50 qubits, no classical computer can store the exact quantum state. The ability to certify accuracy analytically — from the circuit's structure and the KLT entropy geometry, without a ground-truth reference — is what makes Qumulator useful at Willow scale and beyond.

The saturation depth d_sat = 52 means that for this 105-qubit circuit, exact simulation remains feasible (via KLT MPS with small bond dimension χ = 8) for all circuits up to depth ≈ 16. At depths 16–52, the KLT cumulant expansion (K=4) provides certified accuracy with moderate MPS bond dimensions. Only at depths beyond 52 does full χ=2^N/2 MPS become necessary.