Quantum-like models as circuits

The order-effect, Bayesian-network, walk and belief models as quantum circuits: compare circuit output with the analytic model on the ideal Aer simulator, on Aer with the FakeTorino (IBM Heron) noise model, and on the Braket local simulator.

import numpy as np
from quantum_mind.core import tvd
from quantum_mind.circuits import (run, order_effects_circuit, qlbn_circuit, walk_circuit, belief_circuit)
from quantum_mind.families.order_effects import QuantumOrderModel
from quantum_mind.families.qlbn import BayesNet, quantum_like_marginal
from quantum_mind.families.dynamics import QuantumWalk, OpenSystemBelief, final_yes

Build the circuits

Each builder returns a circuit and a decoder that turns counts into the quantity the model predicts.

ql = QuantumOrderModel(a=2.3543, b=0.9676, g=0.5846, ranks=(1, 2))
cases = []
qc, dec = order_effects_circuit(ql, 'AB', 'deferred'); cases.append(('order effects (AB)', qc, dec, ql.predict()['AB']))
net = BayesNet({'O': ['w', 'l'], 'P': ['y', 'n']}, {'P': ['O']}, {'O': {(): {'w': .5, 'l': .5}}, 'P': {('w',): {'y': .69, 'n': .31}, ('l',): {'y': .59, 'n': .41}}})
qc, dec = qlbn_circuit(net, 'P', None, [0, 2.2]); m = quantum_like_marginal(net, 'P', None, [0, 2.2]); cases.append(('QLBN', qc, dec, np.array([m['y'], m['n']])))
w = QuantumWalk(mu=3, sigma=3, n_states=16); qc, dec = walk_circuit(w, 1.0); cases.append(('quantum walk', qc, dec, w.predict({'a': ('single', 1.0)})['a']))

Run on three back ends

Ideal Aer, Aer with the IBM FakeTorino noise model, and the Amazon Braket local simulator. The total variation distance (TVD) to the exact prediction measures the sampling and noise error.

for name, qc, dec, exact in cases:
    row = [name]
    for be in ('aer', 'aer:FakeTorino', 'braket_local'):
        try:
            row.append('%s TVD %.3f' % (be, tvd(dec(run(qc, be, 8000)), exact)))
        except ValueError as e:
            row.append('%s n/a (%s)' % (be, str(e)[:40]))
    print(' | '.join(row))

order effects (AB) | aer TVD 0.003 | aer:FakeTorino TVD 0.069 | braket_local TVD 0.009
QLBN | aer TVD 0.006 | aer:FakeTorino TVD 0.019 | braket_local TVD 0.006
quantum walk | aer TVD 0.002 | aer:FakeTorino TVD 0.187 | braket_local TVD 0.005

Belief dynamics with dephasing

Dephasing is implemented by an ancilla that triggers a Z gate with the right probability.

b = OpenSystemBelief(); qc, dec = belief_circuit(b, (1, 1, 0, 1, 0, 1), (2, 5))
print('belief dynamics: circuit %.3f, exact %.3f' % (dec(run(qc, 'aer', 20000)), final_yes(b, (1, 1, 0, 1, 0, 1), (2, 5))))

belief dynamics: circuit 0.651, exact 0.649