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Kilobit-Scale — Analytical TVD Forecast

1,000-Qubit TVD Forecast — Certified < 0.001

25×40 grid  ·  Depth 2  ·  KLT phase Z2  ·  No exact reference possible  ·  Analytically certified

TVD < 0.001
Certified bound
180
Saturation depth dsat
22 s
CPU time

What this benchmark tests

At 1,000 qubits the Hilbert space contains 21000 ≈ 10301 basis states — a number so vast it exceeds the count of atoms in the observable universe by nearly 230 orders of magnitude. No classical computer can store, let alone compute, the exact quantum state.

Yet for a depth-2 circuit on this 25×40 grid, Qumulator's KLT engine returns a certified TVD bound of less than 0.001 in 22 seconds on a single CPU. This certification requires no reference simulation — it is derived analytically from the circuit's Lyapunov spectrum and entropy profile. At depth 2, the circuit sits deep in the Z2 phase (natural depth τ = 0.011 — barely 1% of the way to scrambling), where the entanglement structure is sparse and provably controllable.

Result: 1,000-qubit 25×40 grid, depth 2. KLT phase: Z2. Smean = 0.16 bits/qubit. Cumulant K = 2. Certified TVD < 0.001. Natural depth τ = 0.011. Saturation depth dsat = 180. Wall-clock: 22 s on CPU. No exponential reference possible.

How this scales across qubit counts

Qubits Depth Phase τ dsat TVD bound Time
20 20 Z5 (Haar) 1.000 20 — (exact SV) 44.7 s
105 2 Z3→Z4 0.300 52 < 0.002 3.4 s
1,000 2 Z2 0.011 180 < 0.001 22 s

Highlighted row = this benchmark. τ = d / dsat.

Full result summary

Grid25×40 = 1,000 qubits
Circuit depth2 layers
Gate setCX entangling + single-qubit Haar (Ry, Rz)
KLT phaseZ2 (low entanglement: 0.10–0.40 bits/qubit)
Mean entropy Smean0.16 bits/qubit
Cumulant level K2 (2nd-order correction — sufficient for Z2)
Certified TVD bound< 0.001  ✓
Exact reference needed?No — analytical (no reference possible anyway)  ✓
Lyapunov exponent λL0.038 bits/layer (slow scrambling at 1,000q)
Natural depth τ0.011 (= 2 / 180)
Saturation depth dsat180 layers
Bond dimension χ4 (ultra-low entanglement at depth 2)
Hilbert space dimension21000 ≈ 10301
Hardware4 vCPU / 16 GB RAM — CPU only
Wall-clock time22 s

Reproduce this result

The analytical TVD certificate is always returned in result.tvd_bound. At depth 2 on a 1,000-qubit grid, bond dimension χ = 4 is sufficient — the KLT preflight step selects this automatically based on the Z2 phase classification. 

pip install qumulator-sdk
import os
import numpy as np
from qumulator import QumulatorClient

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

# 25×40 grid, depth 2 — 1,000 qubits
nrows, ncols, depth = 25, 40, 2
N = nrows * ncols  # 1,000 qubits
rng = np.random.default_rng(99)

instructions = []
for d in range(depth):
    # Single-qubit Haar layer
    for q in range(N):
        instructions.append({"gate": "ry", "qubits": q,
                              "params": [rng.uniform(0, np.pi)]})
        instructions.append({"gate": "rz", "qubits": q,
                              "params": [rng.uniform(0, 2*np.pi)]})
    # CX entangling layer (horizontal bonds on even columns)
    for r in range(nrows):
        for c in range(0, ncols - 1, 2):
            q0 = r * ncols + c
            instructions.append({"gate": "cx", "qubits": [q0, q0 + 1]})

result = client.circuit.run(
    n_qubits=N,
    instructions=instructions,
    mode="klt_mps",
    bond_dim=4,           # auto-selected for Z2 phase; set explicitly here
    shots=2048,
    seed=99,
    return_entropy_map=True,
)

print(f"KLT phase        : {result.klt_phase}")          # → Z2
print(f"Mean entropy     : {result.mean_entropy:.3f} bits/qubit")  # → 0.160
print(f"Certified TVD    : < {result.tvd_bound:.4f}")    # → < 0.0010
print(f"Natural depth τ  : {result.natural_depth:.3f}")   # → 0.011
print(f"Saturation depth : {result.saturation_depth}")   # → 180
print(f"Bond dimension   : {result.bond_dim_used}")       # → 4
print(f"Exact ref needed : No — 2^1000 ≈ 10^301 dimensions")
Why does TVD tighten at scale? Counterintuitively, the TVD bound improves as qubit count grows at fixed depth. For a given depth d, the entanglement entropy per qubit Smean is roughly constant regardless of N — but the saturation depth dsat ≈ log(N) / λL grows logarithmically with N. As a result, τ = d / dsat shrinks at larger N, and the Pinsker-type TVD bound ε ≈ eλL × d − 1 is smaller. At 1,000 qubits depth 2, the circuit is so far from saturation (τ = 0.011) that the KLT MPS with χ = 4 captures it essentially exactly.

Context: what 21000 means

The number 21000 ≈ 10301 dwarfs the estimated number of atoms in the observable universe (1080) by 221 orders of magnitude. Storing a single complex amplitude per basis state at 64-bit precision would require ≈ 10292 GB of memory — more than all storage media ever manufactured by a similar margin. Classical brute-force simulation of this system is not merely impractical: it is physically impossible.

The KLT analytical certificate sidesteps this entirely. The TVD bound follows from the circuit's entropy profile and Lyapunov exponent alone — both quantities computable in O(N × χ² × d) time, polynomial in all parameters. For this benchmark, that computation completes in 22 seconds.

Tier 4 comparison: The Tier 4 depth-3 benchmark on the same 1,000-qubit 25×40 grid runs in 112 s with χ = 16. At depth 2, the Z2 phase requires only χ = 4 — a 16× reduction in bond dimension — cutting runtime to 22 s while delivering a tighter TVD certificate.