Fortgschritteni Technike für QAOA

Nutzungsschätzung: 3 Minute uf emm Heron r2 Prozessor (HINWEIS: Des isch nur e Schätzung. Ihri Laufzeit ka variiere.)

Hintergrund

Des Notebook stellt fortgschritteni Technike zur Verbesserung vo dr Leistung vom Quantum Approximate Optimization Algorithm (QAOA) mit ere große Anzahl vo Qubits vor. E Einführung in QAOA finde mir im Tutorial Utility-scale Quanteoptimierungsproblem löse.

D'in dem Notebook vorgstellte fortgschritteni Technike umfasse:

• SWAP-Strategie mit SAT-Initial-Mapping: Des isch e speziell für QAOA entwickelter Transpiler-Pass, wo e SWAP-Strategie un en SAT-Solver zämme verwendet, um d'Auswahl vo de physische Qubits uf dr QPU z'verbessere. D'SWAP-Strategie nutzt d'Kommutativität vo de QAOA-Operatore, um Gates so umz'ordne, dass Schichte vo SWAP-Gates gleichziitig usgführt werre kenne, wodurch d'Tiefe vom Circuit reduziert wird [1]. Dr SAT-Solver wird verwendet, um e Initial-Mapping z'finde, des d'Anzahl vo de SWAP-Operatione minimiert, wo bruucht werre, um d'Qubits im Circuit uf d'physische Qubits vom Gerät abz'bilde [2].
• CVaR-Kostenfunktion: Typischerwiis wird dr Erwartungswert vom Kosten-Hamiltonian als Kostenfunktion für QAOA verwendet, aber wie in [3] zeigt worre isch, ka d'Konzentration uf dr Tail vo dr Verteilung anstatt vom Erwartungswert d'Leistung vo QAOA für kombinatorischi Optimierungsproblem verbessere. Dr CVaR erreicht des. Für e gegebeni Menge vo Shots mit entsprechende Zielfunktionswerte vom betrachtete Optimierungsproblem isch dr Conditional Value at Risk (CVaR) mit Konfidenzniveau $\alpha \in [0, 1]$ als Durchschnitt vo de $\alpha$ beschte Shots definiert [3]. Somit entspricht $\alpha = 1$ dem Standard-Erwartungswert, während $\alpha=0$ dem Minimum vo de gegebene Shots entspricht, un $\alpha \in (0, 1)$ isch e Kompromiss zwische dr Konzentration uf besseri Shots un dr Anwendung vo ere gwisse Mittelung zur Glättung vo dr Optimierungslandschaft. Zusätzlich ka dr CVaR als Fehlerminderungstechnik verwendet werre, um d'Qualität vo dr Schätzung vom Zielfunktionswert z'verbessere [4].

Anforderungen

Schau vor em Aafange vo diesem Tutorial, dass mir des Folgende installiert hän:

• Qiskit SDK v2.0 oder höher, mit Unterstützung für Visualisierung
• Qiskit Runtime v0.43 oder höher (pip install qiskit-ibm-runtime)
• Rustworkx Graph-Bibliothek (pip install rustworkx)
• Python SAT (pip install python-sat)

Setup

# Added by doQumentation — installs packages not in the Binder environment
!pip install -q python-sat

from __future__ import annotations

import numpy as np
import rustworkx as rx
from dataclasses import dataclass
from itertools import combinations
from threading import Timer
from collections.abc import Callable, Iterable
from pysat.formula import CNF, IDPool
from pysat.solvers import Solver
from scipy.optimize import minimize
from rustworkx.visualization import mpl_draw as draw_graph

from qiskit.quantum_info import SparsePauliOp
from qiskit.circuit.library import QAOAAnsatz
from qiskit.circuit import QuantumCircuit, ParameterVector
from qiskit.transpiler import CouplingMap, PassManager
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager
from qiskit.transpiler.passes.routing.commuting_2q_gate_routing import (
    SwapStrategy,
    FindCommutingPauliEvolutions,
    Commuting2qGateRouter,
)

from qiskit_ibm_runtime import QiskitRuntimeService, Session
from qiskit_ibm_runtime import SamplerV2 as Sampler

Max-Cut-Problem

Betrachte mir d'Lösung vom Max-Cut-Problem uf emm Graphe mit 100 Knote mittels QAOA. Des Max-Cut-Problem isch e kombinatorisches Optimierungsproblem, des uf emm Graphe $G = (V, E)$ definiert isch, wobei $V$ d'Menge vo de Knote un $E$ d'Menge vo de Kante isch. Des Ziel isch, d'Knote in zwei Menge, $S$ un $V \setminus S$, z'partitioniere, so dass d'Anzahl vo de Kante zwische de beide Menge maximiert wird. In dem Beispiel verwende mir en Graphe mit 100 Knote, wo uf ere Hardware-Coupling-Map basiert.

Schritt 1: Klassischi Eingabe uf e Quanteproblem abbilden

Graph → Hamiltonian

Bilde mir zunächscht des Problem uf en Quantecircuit ab, wo für QAOA geeignet isch. Details zu dem Prozess finde mir im einführende QAOA-Tutorial.

# Instantiate runtime to access backend
service = QiskitRuntimeService()
backend = service.least_busy(
    min_num_qubits=100, operational=True, simulator=False
)
print(backend)

<IBMBackend('ibm_fez')>

backend.coupling_map.is_symmetric

True

n = 100
graph_100 = rx.PyGraph()
graph_100.add_nodes_from((np.arange(0, n, 1)))
w = 1.0
elist = []

for edge in backend.coupling_map:
    if (edge[0] < n) and (edge[1] < n):
        if (edge[1], edge[0], w) not in elist:
            elist.append((edge[0], edge[1], w))

graph_100.add_edges_from(elist)
draw_graph(graph_100, with_labels=True)

# Construct cost hamiltonian

def build_max_cut_paulis(graph: rx.PyGraph) -> list[tuple[str, float]]:
    """Convert the graph to Pauli list.

    This function does the inverse of `build_max_cut_graph`
    """
    pauli_list = []
    for edge in list(graph.edge_list()):
        paulis = ["I"] * len(graph)
        paulis[edge[0]], paulis[edge[1]] = "Z", "Z"

        weight = graph.get_edge_data(edge[0], edge[1])

        pauli_list.append(("".join(paulis)[::-1], weight))

    return pauli_list

max_cut_paulis = build_max_cut_paulis(graph_100)

cost_hamiltonian = SparsePauliOp.from_list(max_cut_paulis)
print("Cost Function Hamiltonian:", cost_hamiltonian)

Cost Function Hamiltonian: SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIZIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIZIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIZIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIZIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIZIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIZIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIZIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIZIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IZIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'ZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'],
              coeffs=[1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j])

Schritt 2: Problem für d'Ausführung uf Quantehardware optimiere

SWAP-Strategie mit SAT-Initial-Mapping

Mir werre demonstriere, wie QAOA-Circuits mit dr SWAP-Strategie mit SAT-Initial-Mapping erstellt un optimiert werre, emm speziell für QAOA entwickelte Transpiler-Pass für quadratischi Problem.

In dem Beispiel wähle mir e SWAP-Einfügungsstrategie für Blöck vo kommutierend Zwei-Qubit-Gates, wo Schichte vo SWAP-Gates aawendet, wo gleichziitig uf dr Coupling-Map usgführt werre kenne. Die Strategie wird in [1] vorgstellt un an Commuting2qGateRouter übergebe, des als standardisierter Qiskit-Transpiler-Pass verfügbar isch (lueg Commuting2qGateRouter). In dem Beispiel verwende mir e Line-Swap-Strategie.

# Extract longest path with no repeated nodes
nodes = rx.longest_simple_path(graph_100)

# Collect even edges and odd edges
even_edges = [
    (nodes[i], nodes[i + 1])
    if nodes[i] < nodes[i + 1]
    else (nodes[i + 1], nodes[i])
    for i in range(0, len(nodes) - 1, 2)
]
odd_edges = [
    (nodes[i], nodes[i + 1])
    if nodes[i] < nodes[i + 1]
    else (nodes[i + 1], nodes[i])
    for i in range(1, len(nodes) - 1, 2)
]
edge_list = [
    (edge[0], edge[1]) if edge[0] < edge[1] else (edge[1], edge[0])
    for edge in graph_100.edge_list()
]

swap_strategy = SwapStrategy(CouplingMap(edge_list), (even_edges, odd_edges))

Dr Graphe mit emm SAT-Mapper neu abbilde

Selbscht wenn e Circuit us kommutierend Gates beschtoht (des isch dr Fall für dr QAOA-Circuit, aber au für Trotterisierti Simulatione vo Ising-Hamiltonians), isch des Finde vo emm guete Initial-Mapping e herausfordernd Ufgab. Dr in [2] vorgstellte SAT-basierte Ansatz ermöglicht d'Entdeckung vo effektive Initial-Mappings für Circuits mit kommutierend Gates, was zu ere signifikante Reduzierung vo dr Anzahl erforderlicher SWAP-Schichte führt. Der Ansatz isch uf bis zu 500 Qubits skaliert worre, wie im Paper dargestellt.

Dr folgende Code demonstriert, wie dr SATMapper vo Matsuo et al. verwendet wird, um dr Graphe neu abz'bilde. Der Prozess ermöglicht's, des Problem uf en optimalere Usgangszustand für e bestimmti SWAP-Strategie abz'bilde, was zu ere signifikante Reduzierung vo dr Anzahl vo de SWAP-Schichte führt, wo zur Usführung vom Circuit erforderlich sin.

Im SATMapper wird des Problem, e guets Initial-Mapping z'finde, als SAT-Problem formuliert. E SAT-Solver wird verwendet, um e solches Initial-Mapping für dr QAOA-Circuit z'finde. python-sat (pysat kurz) isch e Python-Bibliothek für en SAT-Solver, un mir werre sie verwende, um des SAT-Problem in dem Beispiel z'löse.

"""A class to solve the SWAP gate insertion initial mapping problem
using the SAT approach from https://arxiv.org/abs/2212.05666.
"""

@dataclass
class SATResult:
    """A data class to hold the result of a SAT solver."""

    satisfiable: bool  # Satisfiable is True if the SAT model could be solved
    # in a given time.
    solution: dict  # The solution to the SAT problem if it is satisfiable.
    mapping: list  # The mapping of nodes in the pattern graph to nodes in the
    # target graph.
    elapsed_time: float  # The time it took to solve the SAT model.

class SATMapper:
    r"""A class to introduce a SAT-approach to solve
    the initial mapping problem in SWAP gate insertion for commuting gates.

    When this pass is run on a DAG it will look for the first instance of
    :class:`.Commuting2qBlock` and use the program graph :math:`P` of this block
    of gates to find a layout for a given swap strategy. This layout is found
    with a binary search over the layers :math:`l` of the swap strategy. At each
    considered layer a subgraph isomorphism problem formulated as a SAT is solved
    by a SAT solver. Each instance is whether it is possible to embed the program
    graph :math:`P` into the effective connectivity graph :math:`C_l` that is
    achieved by applying :math:`l` layers of the swap strategy to the coupling map
    :math:`C_0` of the backend. Since solving SAT problems can be hard, a
    ``time_out`` fixes the maximum time allotted to the SAT solver for each
    instance. If this time is exceeded the considered problem is deemed
    unsatisfiable and the binary search proceeds to the next number of swap
    layers :math:``l``.
    """

    def __init__(self, timeout: int = 60):
        """Initialize the SATMapping.

        Args:
            timeout: The allowed time in seconds for each iteration of the SAT
                solver. This variable defaults to 60 seconds.
        """
        self.timeout = timeout

    def find_initial_mappings(
        self,
        program_graph: rx.Graph,
        swap_strategy: SwapStrategy,
        min_layers: int | None = None,
        max_layers: int | None = None,
    ) -> dict[int, SATResult]:
        r"""Find an initial mapping for a given swap strategy. Perform a
        binary search over the number of swap layers, and for each number
        of swap layers solve a subgraph isomorphism problem formulated as
        a SAT problem.

        Args:
            program_graph (rx.Graph): The program graph with commuting gates, where
                                        each edge represents a two-qubit gate.
            swap_strategy (SwapStrategy): The swap strategy to use to find the
                                        initial mapping.
            min_layers (int): The minimum number of swap layers to consider.
                                        Defaults to the maximum degree of the
                                        program graph - 2.
            max_layers (int): The maximum number of swap layers to consider.
                                        Defaults to the number of qubits in the
                                        swap strategy - 2.

        Returns:
            dict[int, SATResult]: A dictionary containing the results of the SAT
                                    solver for each number of swap layers.
        """
        num_nodes_g1 = len(program_graph.nodes())
        num_nodes_g2 = swap_strategy.distance_matrix.shape[0]
        if num_nodes_g1 > num_nodes_g2:
            return SATResult(False, [], [], 0)
        if min_layers is None:
            # use the maximum degree of the program graph - 2
            # as the lower bound.
            min_layers = max((d for _, d in program_graph.degree)) - 2
        if max_layers is None:
            max_layers = num_nodes_g2 - 1

        variable_pool = IDPool(start_from=1)
        variables = np.array(
            [
                [variable_pool.id(f"v_{i}_{j}") for j in range(num_nodes_g2)]
                for i in range(num_nodes_g1)
            ],
            dtype=int,
        )
        vid2mapping = {v: idx for idx, v in np.ndenumerate(variables)}
        binary_search_results = {}

        def interrupt(solver):
            # This function is called to interrupt the solver when the
            # timeout is reached.
            solver.interrupt()

        # Make a cnf (conjunctive normal form) for the one-to-one
        # mapping constraint
        cnf1 = []
        for i in range(num_nodes_g1):
            clause = variables[i, :].tolist()
            cnf1.append(clause)
            for k, m in combinations(clause, 2):
                cnf1.append([-1 * k, -1 * m])
        for j in range(num_nodes_g2):
            clause = variables[:, j].tolist()
            for k, m in combinations(clause, 2):
                cnf1.append([-1 * k, -1 * m])

        # Perform a binary search over the number of swap layers to find the
        # minimum number of swap layers that satisfies the subgraph isomorphism
        # problem.
        while min_layers < max_layers:
            num_layers = (min_layers + max_layers) // 2

            # Create the connectivity matrix. Note that if the swap strategy
            # cannot reach full connectivity then its distance matrix will have
            # entries with -1. These entries must be treated as False.
            d_matrix = swap_strategy.distance_matrix
            connectivity_matrix = (
                (-1 < d_matrix) & (d_matrix <= num_layers)
            ).astype(int)
            # Make a cnf for the adjacency constraint
            cnf2 = []
            for e_0, e_1 in list(program_graph.edge_list()):
                clause_matrix = np.multiply(
                    connectivity_matrix, variables[e_1, :]
                )
                clause = np.concatenate(
                    (
                        [[-variables[e_0, i]] for i in range(num_nodes_g2)],
                        clause_matrix,
                    ),
                    axis=1,
                )
                # Remove 0s from each clause
                cnf2.extend([c[c != 0].tolist() for c in clause])

            cnf = CNF(from_clauses=cnf1 + cnf2)

            with Solver(bootstrap_with=cnf, use_timer=True) as solver:
                # Solve the SAT problem with a timeout.
                # Timer is used to interrupt the solver when the
                # timeout is reached.
                timer = Timer(self.timeout, interrupt, [solver])
                timer.start()
                status = solver.solve_limited(expect_interrupt=True)
                timer.cancel()
                # Get the solution and the elapsed time.
                sol = solver.get_model()
                e_time = solver.time()

                print(
                    f"Layers: {num_layers}, Status: {status}, Time: {e_time}"
                )
                if status:
                    # If the SAT problem is satisfiable, convert the solution
                    # to a mapping.
                    mapping = [vid2mapping[idx] for idx in sol if idx > 0]
                    binary_search_results[num_layers] = SATResult(
                        status, sol, mapping, e_time
                    )
                    max_layers = num_layers
                else:
                    # If the SAT problem is unsatisfiable, return the last
                    # satisfiable solution.
                    binary_search_results[num_layers] = SATResult(
                        status, sol, [], e_time
                    )
                    min_layers = num_layers + 1

        return binary_search_results

    def remap_graph_with_sat(
        self, graph: rx.Graph, swap_strategy, max_layers
    ):
        """Applies the SAT mapping.

        Args:
            graph (nx.Graph): The graph to remap.
            swap_strategy (SwapStrategy): The swap strategy to use
                                            to find the initial mapping.

        Returns:
            tuple: A tuple containing the remapped graph, the edge map, and the
            number of layers of the swap strategy that was used to find the
            initial mapping. If no solution is found then the tuple contains
            None for each element. Note the returned edge map `{k: v}` means that
            node `k` in the original graph gets mapped to node `v` in the
            Pauli strings.
        """
        num_nodes = len(graph.nodes())
        results = self.find_initial_mappings(
            graph, swap_strategy, 0, max_layers
        )
        solutions = [k for k, v in results.items() if v.satisfiable]

        if len(solutions):
            min_k = min(solutions)
            edge_map = dict(results[min_k].mapping)
            # Create the remapped graph
            remapped_graph = rx.PyGraph()
            remapped_graph.add_nodes_from(range(num_nodes))
            mapping = dict(results[min_k].mapping)
            for i, graph_edge in enumerate(list(graph.edge_list())):
                remapped_edge = tuple(mapping[node] for node in graph_edge)
                remapped_graph.add_edge(*remapped_edge, graph.edges()[i])
            return remapped_graph, edge_map, min_k
        else:
            return None, None, None

sm = SATMapper(timeout=10)
remapped_graph, edge_map, min_swap_layers = sm.remap_graph_with_sat(
    graph=graph_100, swap_strategy=swap_strategy, max_layers=1
)
print("Map from old to new nodes: ", edge_map)
print("Min SWAP layers:", min_swap_layers)
draw_graph(remapped_graph, node_size=200, with_labels=True, width=1)

Layers: 0, Status: True, Time: 0.022812999999999306
Map from old to new nodes:  {0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 7: 7, 8: 8, 9: 9, 10: 10, 11: 11, 12: 12, 13: 13, 14: 14, 15: 15, 16: 16, 17: 17, 18: 18, 19: 19, 20: 20, 21: 21, 22: 22, 23: 23, 24: 24, 25: 25, 26: 26, 27: 27, 28: 28, 29: 29, 30: 30, 31: 31, 32: 32, 33: 33, 34: 34, 35: 35, 36: 36, 37: 37, 38: 38, 39: 39, 40: 40, 41: 41, 42: 42, 43: 43, 44: 44, 45: 45, 46: 46, 47: 47, 48: 48, 49: 49, 50: 50, 51: 51, 52: 52, 53: 53, 54: 54, 55: 55, 56: 56, 57: 57, 58: 58, 59: 59, 60: 60, 61: 61, 62: 62, 63: 63, 64: 64, 65: 65, 66: 66, 67: 67, 68: 68, 69: 69, 70: 70, 71: 71, 72: 72, 73: 73, 74: 74, 75: 75, 76: 76, 77: 77, 78: 78, 79: 79, 80: 80, 81: 81, 82: 82, 83: 83, 84: 84, 85: 85, 86: 86, 87: 87, 88: 88, 89: 89, 90: 90, 91: 91, 92: 92, 93: 93, 94: 94, 95: 95, 96: 96, 97: 97, 98: 98, 99: 99}
Min SWAP layers: 0

remapped_max_cut_paulis = build_max_cut_paulis(remapped_graph)
# define a qiskit SparsePauliOp from the list of paulis
remapped_cost_operator = SparsePauliOp.from_list(remapped_max_cut_paulis)
print(remapped_cost_operator)

SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIZIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 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'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIZIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIZIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIZIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIZIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIZIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIZIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIZIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IZIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'IIIIZZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII', 'ZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'],
              coeffs=[1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j,
 1.+0.j, 1.+0.j, 1.+0.j])

E QAOA-Circuit mit SWAP-Strategie un SAT-Mapping erstelle

Mir möchte d'SWAP-Strategie nur uf d'Kosten-Operator-Schicht aawende, deshalb erstelle mir zunächscht dr isolierte Block, wo mir später transformiere un an dr endgültige QAOA-Circuit aahänge werre.

Hierfür kenne mir d'Klasse QAOAAnsatz vo Qiskit verwende. Mir gebe en leere Circuit in d'Felder initial_state un mixer_operator ii, um sicherzstelle, dass mir e isolierti Kosten-Operator-Schicht erstelle. Mir definiere au d'edge_coloring-Map, damit RZZ-Gates nebe SWAP-Gates positioniert werre. Die strategischi Platzierung ermöglicht's uns, CX-Uslöschunge usz'nutze un dr Circuit für e besseri Leistung z'optimiere. Der Prozess wird innerhalb vo dr Funktion create_qaoa_swap_circuit usgführt.

def make_meas_map(circuit: QuantumCircuit) -> dict:
    """Return a mapping from qubit index (the key) to classical bit (the value).

    This allows us to account for the swapping order introduced by the SWAP strategy.
    """
    creg = circuit.cregs[0]
    qreg = circuit.qregs[0]

    meas_map = {}
    for inst in circuit.data:
        if inst.operation.name == "measure":
            meas_map[qreg.index(inst.qubits[0])] = creg.index(inst.clbits[0])

    return meas_map

def apply_swap_strategy(
    circuit: QuantumCircuit,
    swap_strategy: SwapStrategy,
    edge_coloring: dict[tuple[int, int], int] | None = None,
) -> QuantumCircuit:
    """Transpile with a SWAP strategy.

    Returns:
        A quantum circuit transpiled with the given swap strategy.
    """

    pm_pre = PassManager(
        [
            FindCommutingPauliEvolutions(),
            Commuting2qGateRouter(
                swap_strategy,
                edge_coloring,
            ),
        ]
    )
    return pm_pre.run(circuit)

def apply_qaoa_layers(
    cost_layer: QuantumCircuit,
    meas_map: dict,
    num_layers: int,
    gamma: list[float] | ParameterVector = None,
    beta: list[float] | ParameterVector = None,
    initial_state: QuantumCircuit = None,
    mixer: QuantumCircuit = None,
):
    """Applies QAOA layers to construct circuit.

    First, the initial state is applied. If `initial_state` is None, we begin in the
    initial superposition state. Next, we alternate between layers of the cost operator
    and the mixer. The cost operator is alternatively applied in order and in reverse
    instruction order. This allows us to apply the swap strategy on odd `p` layers
    and undo the swap strategy on even `p` layers.
    """

    num_qubits = cost_layer.num_qubits
    new_circuit = QuantumCircuit(num_qubits, num_qubits)

    if initial_state is not None:
        new_circuit.append(initial_state, range(num_qubits))
    else:
        # all h state by default
        new_circuit.h(range(num_qubits))

    if gamma is None or beta is None:
        gamma = ParameterVector("γ'", num_layers)
        if mixer is None or mixer.num_parameters == 0:
            beta = ParameterVector("β'", num_layers)
        else:
            beta = ParameterVector("β'", num_layers * mixer.num_parameters)

    if mixer is not None:
        mixer_layer = mixer
    else:
        mixer_layer = QuantumCircuit(num_qubits)
        mixer_layer.rx(-2 * beta[0], range(num_qubits))

    for layer in range(num_layers):
        bind_dict = {cost_layer.parameters[0]: gamma[layer]}
        cost_layer_ = cost_layer.assign_parameters(bind_dict)
        bind_dict = {
            mixer_layer.parameters[i]: beta[layer + i]
            for i in range(mixer_layer.num_parameters)
        }
        layer_mixer = mixer_layer.assign_parameters(bind_dict)

        if layer % 2 == 0:
            new_circuit.append(cost_layer_, range(num_qubits))
        else:
            new_circuit.append(cost_layer_.reverse_ops(), range(num_qubits))

        new_circuit.append(layer_mixer, range(num_qubits))

    for qidx, cidx in meas_map.items():
        new_circuit.measure(qidx, cidx)

    return new_circuit

def create_qaoa_swap_circuit(
    cost_operator: SparsePauliOp,
    swap_strategy: SwapStrategy,
    edge_coloring: dict = None,
    theta: list[float] = None,
    qaoa_layers: int = 1,
    initial_state: QuantumCircuit = None,
    mixer: QuantumCircuit = None,
):
    """Create the circuit for QAOA.

    Notes: This circuit construction for QAOA works for quadratic terms in `Z` and will be
    extended to first-order terms in `Z`. Higher-orders are not supported.

    Args:
        cost_operator: the cost operator.
        swap_strategy: selected swap strategy
        edge_coloring: A coloring of edges that should correspond to the coupling
            map of the hardware. It defines the order in which we apply the Rzz
            gates. This allows us to choose an ordering such that `Rzz` gates will
            immediately precede SWAP gates to leverage CNOT cancellation.
        theta: The QAOA angles.
        qaoa_layers: The number of layers of the cost operator and the mixer operator.
        initial_state: The initial state on which we apply layers of cost operator
            and mixer.
        mixer: The QAOA mixer. It will be applied as is onto the QAOA circuit. Therefore,
            its output must have the same ordering of qubits as its input.
    """

    num_qubits = cost_operator.num_qubits

    if theta is not None:
        gamma = theta[: len(theta) // 2]
        beta = theta[len(theta) // 2 :]
        qaoa_layers = len(theta) // 2
    else:
        gamma = beta = None

    # First, create the ansatz of one layer of QAOA without mixer
    cost_layer = QAOAAnsatz(
        cost_operator,
        reps=1,
        initial_state=QuantumCircuit(num_qubits),
        mixer_operator=QuantumCircuit(num_qubits),
    ).decompose()

    # This will allow us to recover the permutation of the measurements that the swaps introduce.
    cost_layer.measure_all()

    # Now, apply the swap strategy for commuting gates
    cost_layer = apply_swap_strategy(cost_layer, swap_strategy, edge_coloring)

    # Compute the measurement map (qubit to classical bit).
    # We will apply this for odd layers where the swaps were inserted.
    if qaoa_layers % 2 == 1:
        meas_map = make_meas_map(cost_layer)
    else:
        meas_map = {idx: idx for idx in range(num_qubits)}

    cost_layer.remove_final_measurements()

    # Finally, introduce the mixer circuit and add measurements following measurement map
    circuit = apply_qaoa_layers(
        cost_layer, meas_map, qaoa_layers, gamma, beta, initial_state, mixer
    )

    return circuit

# We can define the edge_coloring map so that RZZ gates are positioned right before SWAP gates to exploit CX cancellations
# We use greedy edge coloring from rustworkx to color the edges of the graph. This coloring is used to order the RZZ gates in the circuit.

edge_coloring_idx = rx.graph_greedy_edge_color(graph_100)
edge_coloring = {
    edge: edge_coloring_idx[idx]
    for idx, edge in enumerate(list(graph_100.edge_list()))
}
edge_coloring = {tuple(sorted(k)): v for k, v in edge_coloring.items()}

qaoa_circ = create_qaoa_swap_circuit(
    remapped_cost_operator,
    swap_strategy,
    edge_coloring=edge_coloring,
    qaoa_layers=1,
)
qaoa_circ.draw(output="mpl", fold=False)

/Users/mirko/Workspace/documentation/.venv/lib/python3.13/site-packages/qiskit/circuit/quantumcircuit.py:4625: UserWarning: Trying to add QuantumRegister to a QuantumCircuit having a layout
  circ.add_register(qreg)

Schritt 3: Usführung mit Qiskit Primitives

E CVaR-Kostenfunktion definiere

Des Beispiel zeigt, wie d'in [3] iigführte Conditional Value at Risk (CVaR)-Kostenfunktion innerhalb variationaler Quanteoptimierungsalgorithme verwendet wird.

Dr CVaR vo ere Zufallsvariable $X$ für e Konfidenzniveau $α ∈ (0, 1]$ isch definiert als $CVaR_{\alpha}(X) = \mathbb{E} \lbrack X | X \leq F_X^{-1}(\alpha) \rbrack$ wobei $F_X^{-1}(p)$ d'inverse kumulative Verteilungsfunktion vo $X$ isch. Mit andere Wort isch CVaR dr Erwartungswert vom undere $\alpha$-Tail vo dr Verteilung vo $X$.

pass_manager = generate_preset_pass_manager(
    backend=backend,
    optimization_level=3,
)

transpiled_qaoa_circ = pass_manager.run(qaoa_circ)

# Utility functions for the evaluation of the expectation value of a measured state
# In this code, for optimization, the measured state is converted into a bit string,
# and the sign of the value is determined by taking the exclusive OR of the bits
# corresponding to Pauli Z.

_PARITY = np.array(
    [-1 if bin(i).count("1") % 2 else 1 for i in range(256)],
    dtype=np.complex128,
)

def evaluate_sparse_pauli(state: int, observable: SparsePauliOp) -> complex:
    """Utility for the evaluation of the expectation value of a measured state.

    Args:
        state (int): The measured state.
        observable (SparsePauliOp): The observable to evaluate the expectation value for.

    Returns:
        complex: The expectation value of the measured state.
    """
    packed_uint8 = np.packbits(
        observable.paulis.z, axis=1, bitorder="little"
    )  # convert observable to array with 8 bit integer
    state_bytes = np.frombuffer(
        state.to_bytes(packed_uint8.shape[1], "little"),
        dtype=np.uint8,  # convert bitstring to array with 8 bit integer
    )
    reduced = np.bitwise_xor.reduce(
        packed_uint8 & state_bytes, axis=1
    )  # take bitwise xor of the result of 'and' conditional on the above two, return 0 or 1
    return np.sum(observable.coeffs * _PARITY[reduced])

def qaoa_sampler_cost_fun(
    params, ansatz, hamiltonian, sampler, aggregation=None
):
    """Standard sampler-based QAOA cost function to be plugged into optimizer routines.

    Args:
        params (np.ndarray): Parameters for the ansatz.
        ansatz (QuantumCircuit): Ansatz circuit.
        hamiltonian (SparsePauliOp): Hamiltonian to be minimized.
        sampler (QAOASampler): Sampler to be used.
        aggregation (Callable | float | None): Aggregation function to be applied to
            the sampled results. If None, the sum of the expectation values is returned.
            If float, the CVaR with the given alpha is used.
    """
    # Run the circuit
    job = sampler.run([(ansatz, params)])
    sampler_result = job.result()
    sampled_int_counts = sampler_result[
        0
    ].data.c.get_int_counts()  # bitstrings are stored as integers
    shots = sum(sampled_int_counts.values())
    int_count_distribution = {
        key: val / shots for key, val in sampled_int_counts.items()
    }

    # a dictionary containing: {state: (measurement probability, value)}
    evaluated = {
        state: (
            probability,
            np.real(evaluate_sparse_pauli(state, hamiltonian)),
        )
        for state, probability in int_count_distribution.items()
    }

    # If aggregation is None, return the sum of the expectation values.
    # If aggregation is a float, return the CVaR with the given alpha.
    # Otherwise, use the aggregation function.
    if aggregation is None:
        result = sum(
            probability * value for probability, value in evaluated.values()
        )
    elif isinstance(aggregation, float):
        cvar_aggregation = _get_cvar_aggregation(aggregation)
        result = cvar_aggregation(evaluated.values())
    else:
        result = aggregation(evaluated.values())

    global iter_counts, result_dict
    iter_counts += 1
    temp_dict = {}
    temp_dict["params"] = params.tolist()
    temp_dict["cvar_fval"] = result
    temp_dict["fval"] = sum(
        probability * value for probability, value in evaluated.values()
    )
    temp_dict["distribution"] = sampled_int_counts
    temp_dict["evaluated"] = evaluated
    result_dict[iter_counts] = temp_dict
    print(f"Iteration {iter_counts}: {result}")

    return result

def _get_cvar_aggregation(alpha: float | None) -> Callable:
    """Return the CVaR aggregation function with the given alpha.

    Args:
        alpha (float | None): Alpha value for the CVaR aggregation. If None, 1 is used
            by default.
    Raises:
        ValueError: If alpha is not in [0, 1].
    """
    if alpha is None:
        alpha = 1
    elif not 0 <= alpha <= 1:
        raise ValueError(f"alpha must be in [0, 1], but {alpha} was given.")

    def cvar_aggregation(
        objective_dict: Iterable[tuple[float, float]],
    ) -> float:
        """Return the CVaR of the given measurements.
        Args:
            objective_dict (Iterable[tuple[float, float]]): An iterable of tuples containing
                the measured bit string and the objective value based on the bit string.

        """
        sorted_measurements = sorted(objective_dict, key=lambda x: x[1])
        # accumulate the probabilities until alpha is reached
        accumulated_percent = 0.0
        cvar = 0.0
        for probability, value in sorted_measurements:
            cvar += value * min(probability, alpha - accumulated_percent)
            accumulated_percent += probability
            if accumulated_percent >= alpha:
                break
        return cvar / alpha

    return cvar_aggregation

Dr CVaR ka als Fehlerminderungstechnik verwendet werre, wie zuvor diskutiert [4]. In dem Beispiel bestimme mir $\alpha$ un d'Anzahl vo de Shots entsprechend dr Fehlerrate vom Circuit.

num_2q_ops = transpiled_qaoa_circ.count_ops()[
    "cz"
]  # the two qubit gates on our backend are cz's.

for el in backend.properties().general:
    if el.name[:2] == "lf" and el.name[3:] == str(
        n
    ):  # pick out lf_100, lf of the best 100q chain
        lf = el.value  # layer fidelity
        print("layer fidelity", lf)
        eplg = 1 - lf ** (1 / (n - 1))  # error per layered gate (EPLG)
        fid_cz = 1 - eplg
        gamma_cz = 1 / fid_cz**2
        gamma_circ = gamma_cz**num_2q_ops

cvar_aggregation = 1 / np.sqrt(gamma_circ)
print("")
print("The corresponding CVaR aggregation value is: ", cvar_aggregation)
print(
    "To mitigate the twirled noise, increase shots by a factor of",
    np.sqrt(gamma_circ),
)

layer fidelity 0.5454643821399414

The corresponding CVaR aggregation value is:  0.2568730767702702
To mitigate the twirled noise, increase shots by a factor of 3.8929731857197782

iter_counts = 0
result_dict = {}
init_params = [np.pi, np.pi / 2]

with Session(backend=backend) as session:
    sampler = Sampler(mode=session)
    sampler.options.default_shots = int(1000 / cvar_aggregation)
    sampler.options.dynamical_decoupling.enable = True
    sampler.options.dynamical_decoupling.sequence_type = "XY4"
    sampler.options.twirling.enable_gates = True
    sampler.options.twirling.enable_measure = True

    result = minimize(
        qaoa_sampler_cost_fun,
        init_params,
        args=(
            transpiled_qaoa_circ,
            remapped_cost_operator,
            sampler,
            cvar_aggregation,
        ),
        method="COBYLA",
        tol=1e-2,
    )
print(result)

Iteration 1: -13.227556797094595
Iteration 2: -13.181545294899571
Iteration 3: -13.149537293372594
Iteration 4: -3.305576300816324
Iteration 5: -12.647411769418035
Iteration 6: -13.443610807401718
Iteration 7: -12.475368761210511
Iteration 8: -15.905726329447413
Iteration 9: -18.011752834505565
Iteration 10: -14.125781339945583
Iteration 11: -19.693673319331744
Iteration 12: -21.175543794613695
Iteration 13: -21.805701324676196
Iteration 14: -22.121280244318488
Iteration 15: -20.02575633517435
Iteration 16: -22.399349757584158
Iteration 17: -22.569392265696226
Iteration 18: -21.877719328111898
Iteration 19: -22.79144777628963
Iteration 20: -22.437359259397432
Iteration 21: -23.021505287264777
Iteration 22: -22.69742427180412
Iteration 23: -23.12553129222746
Iteration 24: -22.893473281156922
 message: Return from COBYLA because the trust region radius reaches its lower bound.
 success: True
  status: 0
     fun: -23.12553129222746
       x: [ 2.766e+00  1.080e+00]
    nfev: 24
   maxcv: 0.0

Schritt 4: Nachbearbeitung un Rückgab vom Ergebnis im gwünschte klassische Format

from matplotlib import pyplot as plt

plt.figure(figsize=(12, 6))
plt.plot(
    [result_dict[i]["cvar_fval"] for i in range(1, iter_counts + 1)],
    label="CVaR",
)
plt.plot(
    [result_dict[i]["fval"] for i in range(1, iter_counts + 1)],
    label="Standard",
)
plt.legend()
plt.xlabel("Iteration")
plt.ylabel("Cost")
plt.show()

Des Folgende ruft d'beschti Lösung us de gesampelte Bitstrings ab

# sort the result_dict[iter_counts]['evaluated'] by the CVaR value
sorted_result_dict = [
    (k, v)
    for k, v in sorted(
        result_dict[iter_counts]["evaluated"].items(),
        key=lambda item: item[1][1],
    )
]
print(
    f"bitstring (int): {sorted_result_dict[0][0]}, probability: {sorted_result_dict[0][1][0]}, objective value: {sorted_result_dict[0][1][1]}"
)

bitstring (int): 283561207335785714592526814041, probability: 0.00025693730729701953, objective value: -43.0

Betrachte mir dr Hamiltonian $H_C$ für des Max-Cut-Problem. Sei jeder Knote vom Graphe emm Qubit im Zustand $|0\rangle$ oder $|1\rangle$ zugeordnet, wobei dr Wert d'Menge aagibt, in wo sich dr Knote befindet. Des Ziel vom Problem isch, d'Anzahl vo de Kante $(v_1, v_2)$ z'maximiere, für wo $v_1 = |0\rangle$ un $v_2 = |1\rangle$ isch, oder umgekehrt. Wenn mir dr $Z$-Operator jedem Qubit zuordne, wobei

$$Z|0\rangle = |0\rangle \qquad Z|1\rangle = -|1\rangle$$

dann ghört e Kante $(v_1, v_2)$ zum Cut, wenn dr Eigenwert vo $(Z_1|v_1\rangle) \cdot (Z_2|v_2\rangle) = -1$ isch; mit andere Wort, d'Qubits, wo mit $v_1$ un $v_2$ verbunde sin, sin unterschiedlich. Ebenso ghört $(v_1, v_2)$ nit zum Cut, wenn dr Eigenwert vo $(Z_1|v_1\rangle) \cdot (Z_2|v_2\rangle) = 1$ isch

from typing import Sequence

def to_bitstring(integer, num_bits):
    result = np.binary_repr(integer, width=num_bits)
    return [int(digit) for digit in result]

def evaluate_sample(x: Sequence[int], graph: rx.PyGraph) -> float:
    assert len(x) == len(
        list(graph.nodes())
    ), "The length of x must coincide with the number of nodes in the graph."
    return sum(
        x[u] * (1 - x[v])
        + x[v]
        * (
            1 - x[u]
        )  # x[u] = x[v] if same cut, x[u] \neq x[v] if different cuts
        for u, v in list(graph.edge_list())
    )

bitstring = to_bitstring(
    sorted_result_dict[0][0], len(list(remapped_graph.nodes()))
)
bitstring = bitstring[::-1]
print(f"Result bitstring (binary) : {bitstring}")

cut_value = evaluate_sample(bitstring, remapped_graph)
print(f"The value of the cut is: {cut_value}")

Result bitstring (binary) : [1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0]
The value of the cut is: 77

Zeichne mir abschließend en Graphe basierend uf em CVaR-Ergebnis. Mir teile d'Graphknote basierend uf em CVaR-Ergebnis in zwei Menge uf. D'Knote in dr erschte Menge sin grau gfärbt, un d'Knote in dr zweite Menge sin violett gfärbt. D'Kante zwische de beide Menge sin d'Kante, wo durch d'Partitionierung gschnitte werre.

def plot_result(G, x):
    colors = ["tab:grey" if i == 0 else "tab:purple" for i in x]
    pos, _default_axes = rx.spring_layout(G), plt.axes(frameon=True)
    rx.visualization.mpl_draw(
        G,
        node_color=colors,
        node_size=150,
        alpha=0.8,
        pos=pos,
        with_labels=True,
        width=1,
    )

plot_result(graph_100, to_bitstring(sorted_result_dict[0][0], 100)[::-1])

Referenze

[1] Weidenfeller, J., Valor, L. C., Gacon, J., Tornow, C., Bello, L., Woerner, S., & Egger, D. J. (2022). Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware. Quantum, 6, 870.

[2] Matsuo, A., Yamashita, S., & Egger, D. J. (2023). A SAT approach to the initial mapping problem in SWAP gate insertion for commuting gates. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 106(11), 1424-1431.

[3] Barkoutsos, P. K., Nannicini, G., Robert, A., Tavernelli, I., & Woerner, S. (2020). Improving variational quantum optimization using CVaR. Quantum, 4, 256.

[4] Barron, S. V., Egger, D. J., Pelofske, E., Bärtschi, A., Eidenbenz, S., Lehmkuehler, M., & Woerner, S. (2023). Provable bounds for noise-free expectation values computed from noisy samples. arXiv preprint arXiv:2312.00733.

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