Exact finite checks and a schedule explorer

Copy the following standard-library Python source to 02_b_checks.py and run:

python en02_b_checks.py

The code uses Gaussian integers, represented by pairs of integers. Hadamards are applied without their $1/\sqrt{2}$ factors, and those factors are accounted for explicitly in the checked norms. There is no floating-point state simulation.

The checks cover the supplied code projector, all physical Pauli operators of weight at most two, the batch and streaming branch factorizations, and a finite two-level schedule class. The tree channel theorem itself follows by composition in the main text; no dense twenty-five-qubit simulation is needed or claimed.

"""En02-B: exact finite checks, not a universal proof certificate.

Standard library only. Gaussian integers are pairs (real, imag). No execution log is supplied by the manuscript. """

from itertools import combinations, product from functools import lru_cache from dataclasses import dataclass

class CheckFailure(Exception): pass

def need(condition, message): if not condition: raise CheckFailure(message)

ZERO = (0, 0) ONE = (1, 0) II = (0, 1)

def add(a, b): return (a[0] + b[0], a[1] + b[1])

def neg(a): return (-a[0], -a[1])

def mul(a, b): return (a[0] * b[0] - a[1] * b[1], a[0] * b[1] + a[1] * b[0])

def conj(a): return (a[0], -a[1])

def norm_sq(a): return a[0] * a[0] + a[1] * a[1]

def inner(v, w): out = ZERO for a, b in zip(v, w): out = add(out, mul(conj(a), b)) return out

def vadd(v, w): return [add(a, b) for a, b in zip(v, w)]

def bits(index, n): return tuple((index >> (n - 1 - j)) & 1 for j in range(n))

def index_of(bs): out = 0 for b in bs: out = 2 * out + b return out

def nqubits(v): size = len(v) need(size >= 1 and (size & (size - 1)) == 0, "state size is not a power of two") return size.bit_length() - 1

def pauli_action(v, word, phase=0): """Apply i**phase times a Pauli word exactly.""" n = nqubits(v) need(len(word) == n, "wrong Pauli arity") units = (ONE, II, (-1, 0), (0, -1)) out = [ZERO] * len(v) for k, amp in enumerate(v): bs = list(bits(k, n)) factor = units[phase for j, letter in enumerate(word): b = bs[j] if letter == "X": bs[j] ^= 1 elif letter == "Z": if b: factor = neg(factor) elif letter == "Y": factor = mul(factor, II if b == 0 else (0, -1)) bs[j] ^= 1 elif letter != "I": raise CheckFailure("invalid Pauli letter") target = index_of(bs) out[target] = add(out[target], mul(factor, amp)) return out

LOCAL_PRODUCT = ("X", "Y"): (1, "Z"), ("Y", "Z"): (1, "X"), ("Z", "X"): (1, "Y"), ("Y", "X"): (3, "Z"), ("Z", "Y"): (3, "X"), ("X", "Z"): (3, "Y"),