105-Qubit MPS RCS

7×15 grid  ·  Depth 5  ·  MPS(χ=16)  ·  Tier 3 limit (N 55–105, d≤5)

What this benchmark tests

This is the Tier 3 maximum: 105 qubits at depth 5 using the KLT MPS engine. At 105 qubits, the Hilbert space has 2¹⁰⁵ ≈ 4 × 10³¹ dimensions — a number that exceeds the total number of atoms in the Earth by 10 orders of magnitude. Exact statevector simulation of this system is completely infeasible; MPS is the only practical approach.

The circuit uses a 7×15 nearest-neighbour grid with alternating horizontal and vertical CX layers (Sycamore-style topology). At depth 5, the mean entropy per qubit is 0.979 bits — very close to full scrambling. All 2,048 shots produced distinct bitstrings, confirming the distribution is spread across an astronomically large output space.

Result summary

Reproduce this result

Install the Qumulator SDK and run the following. Use mode='tensor' with bond_dim=16.

pip install qumulator

import os, time, math, random
from qumulator import QumulatorClient

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

# Tier 3 max: 105-qubit depth-5 RCS on a 7x15 nearest-neighbour grid
N, ROWS, COLS, DEPTH = 105, 7, 15, 5
rng = random.Random(3)

eng = client.circuit.engine(n_qubits=N, mode='tensor', bond_dim=16)

h_even = [(r*COLS+c, r*COLS+c+1) for r in range(ROWS) for c in range(0, COLS-1, 2)]
h_odd  = [(r*COLS+c, r*COLS+c+1) for r in range(ROWS) for c in range(1, COLS-1, 2)]
v_even = [(r*COLS+c, (r+1)*COLS+c) for r in range(0, ROWS-1, 2) for c in range(COLS)]
v_odd  = [(r*COLS+c, (r+1)*COLS+c) for r in range(1, ROWS-1, 2) for c in range(COLS)]
layers = [h_even, v_even, h_odd, v_odd]

for d in range(DEPTH):
    for q in range(N):
        eng.apply('rz', q, params=[rng.uniform(0, 2 * math.pi)])
        eng.apply('rx', q, params=[rng.uniform(0, 2 * math.pi)])
    for q0, q1 in layers[d % 4]:
        eng.apply('cx', [q0, q1])

t0 = time.time()
result = eng.run(shots=2048, seed=3, return_entropy_map=True)
elapsed = time.time() - t0

print(f"Elapsed       : {elapsed:.1f}s")
print(f"Trunc error   : {result.trunc_error:.4f}")
print(f"Mean S        : {sum(result.entropy_map)/N:.3f} bits/qubit")
print(f"Distinct      : {len(result.counts)} / {result.shots}")

Willow comparison

Google's Willow chip has 105 superconducting qubits — the same count as this benchmark. Willow runs at depth ≈ 20+ for its RCS experiments; Tier 3 caps at depth 5. At depth 5, Willow-scale circuits are within the tractable MPS regime on commodity CPU hardware. Deeper circuits (beyond depth 5 at 105 qubits) enter a regime where classical simulation cost grows exponentially and exact results become infeasible without specialised hardware.

Qumulator's Tier 3 limit is set precisely at this boundary — the maximum depth where the KLT MPS engine can reliably execute at 105 qubits within the platform's CPU resource budget.