sympauli

Symbolic Pauli Heisenberg Evolution Engine

Evolves quantum observables through parameterized circuits in the Heisenberg picture, keeping all coefficients as exact SymPy expressions. Vibe coded using Claude AI in one afternoon.

Install dependencies

pip install sympy numpy

Usage

from sympauli import PauliSum, evolve, gradient, expectation_value
from sympauli.gates import gate_Ry, gate_CNOT
import sympy as sp
import numpy as np

theta = sp.Symbol('theta', real=True)
n = 2

# Define observable
H = PauliSum.from_dict({'ZZ': 1, 'XI': 1, 'IX': 1}, n=n)

# Define circuit
circuit = [gate_Ry(theta, target=0), gate_Ry(theta, target=1), gate_CNOT(0, 1)]

# Symbolically evolve in the Heisenberg picture: U†·H·U
H_evolved = evolve(H, circuit, n_qubits=n)

# Symbolic gradient w.r.t. theta
dH = gradient(H, circuit, n_qubits=n, param_symbol=theta)

# Expectation value on a state
state = np.array([1, 0, 0, 0], dtype=complex)
ev = expectation_value(H_evolved, state, {theta: 0.5})

Approximate simulation

Exact evolution grows the number of Pauli terms with circuit depth, in the worst case until it saturates the 4ⁿ available strings. evolve_truncated bounds the working set instead, discarding terms above a chosen Pauli weight after every gate — truncating once at the end would save nothing, since the intermediate blow-up has already been paid for. This is approximate Pauli-path simulation, and it is lossy by construction: the result equals evolve's only when the cutoff cannot bite.

from sympauli import evolve_truncated

H_approx = evolve_truncated(H, circuit, n_qubits=n, max_weight=2)

A coefficient cutoff is available alongside the weight cutoff via min_magnitude, which needs a subs dict since coefficients are symbolic; a term whose value is still unknown after substitution is kept rather than guessed at.

Truncated evolution routes each conjugation through conjugate_by_gate_fast, which recognizes a single-generator rotation exp(-iθ/2·Q) and applies the closed form G†PG = P when [P,Q] = 0 and cos(θ)·P − i·sin(θ)·PQ when {P,Q} = 0, instead of building the triple product and simplifying away the terms that cancel. Anything else falls back to the exact path, so it is a drop-in replacement for conjugate_by_gate.

Package structure

Qubit convention

• PauliString.from_string('AB'): A acts on qubit 1 (MSB), B acts on qubit 0 (LSB).
• gate_CNOT(control=0, target=1): control is qubit 0, target is qubit 1.
• to_matrix() uses big-endian ordering: qubit n-1 is the most significant bit.

Citation

If you found this useful, please consider citing us.

@misc{shirgure2026challengesbarrenplateaumitigation,
      title={Challenges in Barren Plateau Mitigation with Dynamic Parameterized Quantum Circuits}, 
      author={Sumeet Shirgure and Efekan Kökcü and Siyuan Niu},
      year={2026},
      eprint={2606.23751},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2606.23751}, 
}