Quantum approximate optimization algorithm

Tinatayang paggamit: 22 minuto sa Heron r3 processor (TANDAAN: Ito ay isang tantiya lamang. Maaaring mag-iba ang inyong runtime.)

Background

Ipinakikita ng tutorial na ito kung paano ipatupad ang Quantum Approximate Optimization Algorithm (QAOA) – isang hybrid (quantum-classical) iterative method – sa konteksto ng Qiskit patterns. Lulutasin mo muna ang Maximum-Cut (o Max-Cut) problem para sa isang maliit na graph at pagkatapos ay matututo kung paano ito isagawa sa utility scale. Lahat ng hardware executions sa tutorial ay dapat tumakbo sa loob ng time limit para sa libreng-maaccessible na Open Plan.

Ang Max-Cut problem ay isang optimization problem na mahirap lutasin (mas tiyak, ito ay isang NP-hard problem) na may ilang iba't ibang aplikasyon sa clustering, network science, at statistical physics. Isinasaalang-alang ng tutorial na ito ang isang graph ng mga node na konektado ng mga edge, at naglalayong hatiin ang mga node sa dalawang set nang sa gayon ay ang bilang ng mga edge na tinatahak ng cut na ito ay pinakamataas.

Requirements

Bago magsimula ng tutorial na ito, siguraduhing mayroon kayong sumusunod na naka-install:

• Qiskit SDK v1.0 o mas bago, na may visualization support
• Qiskit Runtime v0.22 o mas bago (pip install qiskit-ibm-runtime)

Bilang karagdagan, kakailanganin ninyo ng access sa isang instance sa IBM Quantum Platform. Tandaan na ang tutorial na ito ay hindi maaaring isagawa sa Open Plan, dahil nagpapatakbo ito ng workloads gamit ang sessions, na available lamang sa Premium Plan access.

Setup

# Added by doQumentation — required packages for this notebook
!pip install -q matplotlib numpy qiskit qiskit-ibm-runtime rustworkx scipy

import matplotlib
import matplotlib.pyplot as plt
import rustworkx as rx
from rustworkx.visualization import mpl_draw as draw_graph
import numpy as np
from scipy.optimize import minimize
from collections import defaultdict
from typing import Sequence

from qiskit.quantum_info import SparsePauliOp
from qiskit.circuit.library import QAOAAnsatz
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager

from qiskit_ibm_runtime import QiskitRuntimeService
from qiskit_ibm_runtime import Session, EstimatorV2 as Estimator
from qiskit_ibm_runtime import SamplerV2 as Sampler

Part I. Small-scale QAOA

Ang unang bahagi ng tutorial na ito ay gumagamit ng isang small-scale Max-Cut problem upang ilarawan ang mga hakbang upang malutas ang isang optimization problem gamit ang quantum computer.

Upang magbigay ng konteksto bago i-map ang problemang ito sa isang quantum algorithm, mas mauunawaan ninyo kung paano ang Max-Cut problem ay nagiging isang classical combinatorial optimization problem sa pamamagitan ng pag-consider muna sa minimization ng isang function na $f(x)$

$$\min_{x\in \{0, 1\}^n}f(x),$$

kung saan ang input na $x$ ay isang vector na ang mga component ay tumutugma sa bawat node ng isang graph. Pagkatapos, limitahan ang bawat isa sa mga component na ito upang maging alinman sa $0$ o $1$ (na kumakatawan sa pagsasama o hindi pagsasama sa cut). Ang small-scale example case na ito ay gumagamit ng isang graph na may $n=5$ nodes.

Maaari kayong sumulat ng function ng isang pares ng nodes na $i,j$ na nagsasaad kung ang kaukulang edge na $(i,j)$ ay nasa cut. Halimbawa, ang function na $x_i + x_j - 2 x_i x_j$ ay 1 lamang kung isa sa alinman sa $x_i$ o $x_j$ ay 1 (na nangangahulugang ang edge ay nasa cut) at zero kung hindi. Ang problema ng pag-maximize sa mga edge sa cut ay maaaring iformula bilang

$$\max_{x\in \{0, 1\}^n} \sum_{(i,j)} x_i + x_j - 2 x_i x_j,$$

na maaaring isulat muli bilang minimization sa anyo ng

$$\min_{x\in \{0, 1\}^n} \sum_{(i,j)} 2 x_i x_j - x_i - x_j.$$

Ang minimum ng $f(x)$ sa kasong ito ay kapag ang bilang ng mga edge na tinatahak ng cut ay pinakamataas. Tulad ng inyong makikita, walang kaugnayan sa quantum computing pa. Kailangan ninyong i-reformulate ang problemang ito sa isang bagay na mauunawaan ng quantum computer. Simulan ang inyong problema sa pamamagitan ng paglikha ng isang graph na may $n=5$ nodes.

n = 5

graph = rx.PyGraph()
graph.add_nodes_from(np.arange(0, n, 1))
edge_list = [
    (0, 1, 1.0),
    (0, 2, 1.0),
    (0, 4, 1.0),
    (1, 2, 1.0),
    (2, 3, 1.0),
    (3, 4, 1.0),
]
graph.add_edges_from(edge_list)
draw_graph(graph, node_size=600, with_labels=True)

Step 1: Map classical inputs to a quantum problem

Ang unang hakbang ng pattern ay ang pag-map ng classical problem (graph) sa mga quantum circuits at operators. Upang gawin ito, may tatlong pangunahing hakbang na dapat gawin:

1. Gumamit ng serye ng mathematical reformulations, upang katawanin ang problemang ito gamit ang Quadratic Unconstrained Binary Optimization (QUBO) problems notation.
2. Isulat muli ang optimization problem bilang isang Hamiltonian kung saan ang ground state ay tumutugma sa solusyon na nagpapaliit sa cost function.
3. Lumikha ng quantum circuit na maghahanda sa ground state ng Hamiltonian na ito sa pamamagitan ng prosesong katulad ng quantum annealing.

Tandaan: Sa QAOA methodology, sa huli ay gusto ninyong magkaroon ng operator (Hamiltonian) na kumakatawan sa cost function ng ating hybrid algorithm, pati na rin ang isang parametrized circuit (Ansatz) na kumakatawan sa mga quantum states na may mga kandidatong solusyon sa problema. Maaari kayong mag-sample mula sa mga kandidatong estado na ito at pagkatapos ay suriin ang mga ito gamit ang cost function.

Graph → optimization problem

Ang unang hakbang ng mapping ay isang notation change. Ang sumusunod ay nagpapahayag ng problema sa QUBO notation:

$$\min_{x\in \{0, 1\}^n}x^T Q x,$$

kung saan ang $Q$ ay isang $n\times n$ matrix ng mga real numbers, ang $n$ ay tumutugma sa bilang ng mga node sa inyong graph, ang $x$ ay ang vector ng mga binary variables na ipinakilala sa itaas, at ang $x^T$ ay nagsasaad ng transpose ng vector na $x$.

Maximize
 -2*x_0*x_1 - 2*x_0*x_2 - 2*x_0*x_4 - 2*x_1*x_2 - 2*x_2*x_3 - 2*x_3*x_4 + 3*x_0
 + 2*x_1 + 3*x_2 + 2*x_3 + 2*x_4

Subject to
  No constraints

  Binary variables (5)
    x_0 x_1 x_2 x_3 x_4

Optimization problem → Hamiltonian

Maaari ninyong i-reformulate ang QUBO problem bilang isang Hamiltonian (dito, isang matrix na kumakatawan sa enerhiya ng isang sistema):

$$H_C=\sum_{ij}Q_{ij}Z_iZ_j + \sum_i b_iZ_i.$$

Mga hakbang sa reformulation mula sa QAOA problem hanggang sa Hamiltonian

Upang ipakita kung paano ang QAOA problem ay maaaring isulat muli sa paraang ito, palitan muna ang mga binary variables na $x_i$ sa isang bagong set ng mga variables na $z_i\in\{-1, 1\}$ sa pamamagitan ng

$$x_i = \frac{1-z_i}{2}.$$

Dito makikita ninyo na kung ang $x_i$ ay $0$, kung gayon ang $z_i$ ay dapat na $1$. Kapag ang mga $x_i$ ay pinalitan para sa mga $z_i$ sa optimization problem ($x^TQx$), maaaring makuha ang isang equivalent formulation.

$$x^TQx=\sum_{ij}Q_{ij}x_ix_j \\ =\frac{1}{4}\sum_{ij}Q_{ij}(1-z_i)(1-z_j) \\=\frac{1}{4}\sum_{ij}Q_{ij}z_iz_j-\frac{1}{4}\sum_{ij}(Q_{ij}+Q_{ji})z_i + \frac{n^2}{4}.$$

Ngayon kung tayo ay magdepina ng $b_i=-\sum_{j}(Q_{ij}+Q_{ji})$, alisin ang prefactor, at ang constant na $n^2$ term, darating tayo sa dalawang equivalent formulations ng parehong optimization problem.

$$\min_{x\in\{0,1\}^n} x^TQx\Longleftrightarrow \min_{z\in\{-1,1\}^n}z^TQz + b^Tz$$

Dito, ang $b$ ay nakadepende sa $Q$. Tandaan na upang makuha ang $z^TQz + b^Tz$ ay inalis natin ang factor na 1/4 at isang constant offset na $n^2$ na hindi gumaganap ng papel sa optimization.

Ngayon, upang makakuha ng quantum formulation ng problema, itaas ang mga $z_i$ variables sa isang Pauli $Z$ matrix, tulad ng isang $2\times 2$ matrix sa anyo ng

$$Z_i = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}.$$

Kapag pinalitan ninyo ang mga matrix na ito sa optimization problem sa itaas, makukuha ninyo ang sumusunod na Hamiltonian

$$H_C=\sum_{ij}Q_{ij}Z_iZ_j + \sum_i b_iZ_i.$$

Tandaan din na ang mga $Z$ matrices ay naka-embed sa computational space ng quantum computer, iyon ay, isang Hilbert space na may laki na $2^n\times 2^n$. Samakatuwid, dapat ninyong unawain ang mga terminong tulad ng $Z_iZ_j$ bilang tensor product na $Z_i\otimes Z_j$ na naka-embed sa $2^n\times 2^n$ Hilbert space. Halimbawa, sa isang problemang may limang decision variables, ang termino na $Z_1Z_3$ ay nauunawaan na nangangahulugang $I\otimes Z_3\otimes I\otimes Z_1\otimes I$ kung saan ang $I$ ay ang $2\times 2$ identity matrix.

Ang Hamiltonian na ito ay tinatawag na cost function Hamiltonian. Mayroon itong property na ang ground state nito ay tumutugma sa solusyon na nagpapaliit sa cost function $f(x)$. Samakatuwid, upang malutas ang inyong optimization problem, kailangan na ninyong ihanda ngayon ang ground state ng $H_C$ (o isang estado na may mataas na overlap dito) sa quantum computer. Pagkatapos, ang pag-sample mula sa estadong ito ay, na may mataas na probabilidad, ay magbubunga ng solusyon sa $min~f(x)$. Isaalang-alang natin ngayon ang Hamiltonian na $H_C$ para sa Max-Cut problem. Hayaang ang bawat vertex ng graph ay maiugnay sa isang qubit sa estado na $|0\rangle$ o $|1\rangle$, kung saan ang halaga ay nagsasaad ng set kung nasaan ang vertex. Ang layunin ng problema ay i-maximize ang bilang ng mga edge na $(v_1, v_2)$ kung saan ang $v_1 = |0\rangle$ at $v_2 = |1\rangle$, o vice-versa. Kung iuugnay natin ang $Z$ operator sa bawat qubit, kung saan

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

kung gayon ang isang edge na $(v_1, v_2)$ ay kabilang sa cut kung ang eigenvalue ng $(Z_1|v_1\rangle) \cdot (Z_2|v_2\rangle) = -1$; sa madaling salita, ang mga qubit na nauugnay sa $v_1$ at $v_2$ ay magkaiba. Katulad nito, ang $(v_1, v_2)$ ay hindi kabilang sa cut kung ang eigenvalue ng $(Z_1|v_1\rangle) \cdot (Z_2|v_2\rangle) = 1$. Tandaan na, hindi tayo interesado sa eksaktong qubit state na nauugnay sa bawat vertex, sa halip ang ating pakialam lamang ay kung pareho sila o hindi sa kabuuan ng isang edge. Ang Max-Cut problem ay nangangailangan sa atin na makahanap ng assignment ng mga qubit sa mga vertices nang sa gayon ay ang eigenvalue ng sumusunod na Hamiltonian ay minimized

$$H_C = \sum_{(i,j) \in e} Q_{ij} \cdot Z_i Z_j.$$

Sa madaling salita, ang $b_i = 0$ para sa lahat ng $i$ sa Max-Cut problem. Ang halaga ng $Q_{ij}$ ay nagsasaad ng timbang ng edge. Sa tutorial na ito isinasaalang-alang natin ang isang unweighted graph, iyon ay, $Q_{ij} = 1.0$ para sa lahat ng $i, j$.

def build_max_cut_paulis(
    graph: rx.PyGraph,
) -> list[tuple[str, list[int], 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()):
        weight = graph.get_edge_data(edge[0], edge[1])
        pauli_list.append(("ZZ", [edge[0], edge[1]], weight))
    return pauli_list

max_cut_paulis = build_max_cut_paulis(graph)
cost_hamiltonian = SparsePauliOp.from_sparse_list(max_cut_paulis, n)
print("Cost Function Hamiltonian:", cost_hamiltonian)

Cost Function Hamiltonian: SparsePauliOp(['IIIZZ', 'IIZIZ', 'ZIIIZ', 'IIZZI', 'IZZII', 'ZZIII'],
              coeffs=[1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j])

Hamiltonian → quantum circuit

Ang Hamiltonian na $H_C$ ay naglalaman ng quantum definition ng inyong problema. Ngayon maaari na kayong lumikha ng quantum circuit na tutulong sa pag-sample ng mga magagandang solusyon mula sa quantum computer. Ang QAOA ay inspirado ng quantum annealing at nag-apply ng alternating layers ng mga operators sa quantum circuit.

Ang pangkalahatang ideya ay magsimula sa ground state ng isang kilalang sistema, na $H^{\otimes n}|0\rangle$ sa itaas, at pagkatapos ay gabayan ang sistema tungo sa ground state ng cost operator na inyong interesado. Ginagawa ito sa pamamagitan ng pag-apply ng mga operators na $\exp\{-i\gamma_k H_C\}$ at $\exp\{-i\beta_k H_m\}$ na may mga angles na $\gamma_1,...,\gamma_p$ at $\beta_1,...,\beta_p~$.

Ang quantum circuit na inyong ginagawa ay parametrized ng $\gamma_i$ at $\beta_i$, kaya maaari ninyong subukan ang iba't ibang mga halaga ng $\gamma_i$ at $\beta_i$ at mag-sample mula sa resultang estado.

Sa kasong ito, susubukan ninyo ang isang halimbawa na may isang QAOA layer na naglalaman ng dalawang parameters: $\gamma_1$ at $\beta_1$.

circuit = QAOAAnsatz(cost_operator=cost_hamiltonian, reps=2)
circuit.measure_all()

circuit.draw("mpl")

circuit.parameters

ParameterView([ParameterVectorElement(β[0]), ParameterVectorElement(β[1]), ParameterVectorElement(γ[0]), ParameterVectorElement(γ[1])])

Step 2: Optimize problem for quantum hardware execution

Ang circuit sa itaas ay naglalaman ng serye ng mga abstractions na kapaki-pakinabang na isipin tungkol sa quantum algorithms, ngunit hindi posibleng patakbuhin sa hardware. Upang makapagsagawa sa isang QPU, ang circuit ay kailangang dumaan sa serye ng mga operasyon na bumubuo sa transpilation o circuit optimization step ng pattern.

Ang Qiskit library ay nag-aalok ng serye ng transpilation passes na tumutugon sa malawak na hanay ng circuit transformations. Kailangan ninyong siguraduhing ang inyong circuit ay naka-optimize para sa inyong layunin.

Ang transpilation ay maaaring magsangkot ng ilang hakbang, tulad ng:

• Initial mapping ng mga qubit sa circuit (tulad ng decision variables) sa physical qubits sa device.
• Unrolling ng mga instructions sa quantum circuit sa hardware-native instructions na nauunawaan ng backend.
• Routing ng anumang mga qubit sa circuit na nakikipag-ugnayan sa physical qubits na katabi ng isa't isa.
• Error suppression sa pamamagitan ng pagdaragdag ng single-qubit gates upang pigilan ang ingay sa dynamical decoupling.

Mas maraming impormasyon tungkol sa transpilation ay available sa aming documentation.

Ang sumusunod na code ay nag-transform at nag-optimize ng abstract circuit sa isang format na handa na para sa execution sa isa sa mga devices na accessible sa pamamagitan ng cloud gamit ang Qiskit IBM Runtime service.

service = QiskitRuntimeService()
backend = service.least_busy(
    operational=True, simulator=False, min_num_qubits=127
)
print(backend)

# Create pass manager for transpilation
pm = generate_preset_pass_manager(optimization_level=3, backend=backend)

candidate_circuit = pm.run(circuit)
candidate_circuit.draw("mpl", fold=False, idle_wires=False)

<IBMBackend('test_heron_pok_1')>

Step 3: Execute using Qiskit primitives

Sa QAOA workflow, ang optimal QAOA parameters ay nahanap sa isang iterative optimization loop, na nagpapatakbo ng serye ng circuit evaluations at gumagamit ng classical optimizer upang makahanap ng optimal na $\beta_k$ at $\gamma_k$ parameters. Ang execution loop na ito ay isinasagawa sa pamamagitan ng sumusunod na mga hakbang:

1. I-define ang mga initial parameters
2. Gumawa ng bagong Session na naglalaman ng optimization loop at ang primitive na ginagamit upang mag-sample ng circuit
3. Kapag natagpuan na ang optimal set ng parameters, isagawa ang circuit sa huling pagkakataon upang makakuha ng huling distribution na gagamitin sa post-process step.

Define circuit with initial parameters

Nagsisimula tayo sa arbitrarily chosen parameters.

initial_gamma = np.pi
initial_beta = np.pi / 2
init_params = [initial_beta, initial_beta, initial_gamma, initial_gamma]

Define backend and execution primitive

Gamitin ang Qiskit Runtime primitives upang makipag-ugnayan sa IBM® backends. Ang dalawang primitives ay Sampler at Estimator, at ang pagpili ng primitive ay nakadepende sa uri ng measurement na gusto ninyong isagawa sa quantum computer. Para sa minimization ng $H_C$, gamitin ang Estimator dahil ang measurement ng cost function ay simpleng expectation value ng $\langle H_C \rangle$.

Run

Ang primitives ay nag-aalok ng iba't ibang execution modes upang mag-schedule ng workloads sa quantum devices, at ang QAOA workflow ay tumatakbo nang iteratively sa isang session.

Maaari ninyong i-plug ang sampler-based cost function sa SciPy minimizing routine upang makahanap ng optimal parameters.

def cost_func_estimator(params, ansatz, hamiltonian, estimator):
    # transform the observable defined on virtual qubits to
    # an observable defined on all physical qubits
    isa_hamiltonian = hamiltonian.apply_layout(ansatz.layout)

    pub = (ansatz, isa_hamiltonian, params)
    job = estimator.run([pub])

    results = job.result()[0]
    cost = results.data.evs

    objective_func_vals.append(cost)

    return cost

objective_func_vals = []  # Global variable
with Session(backend=backend) as session:
    # If using qiskit-ibm-runtime<0.24.0, change `mode=` to `session=`
    estimator = Estimator(mode=session)
    estimator.options.default_shots = 1000

    # Set simple error suppression/mitigation options
    estimator.options.dynamical_decoupling.enable = True
    estimator.options.dynamical_decoupling.sequence_type = "XY4"
    estimator.options.twirling.enable_gates = True
    estimator.options.twirling.num_randomizations = "auto"

    result = minimize(
        cost_func_estimator,
        init_params,
        args=(candidate_circuit, cost_hamiltonian, estimator),
        method="COBYLA",
        tol=1e-2,
    )
    print(result)

message: Return from COBYLA because the trust region radius reaches its lower bound.
 success: True
  status: 0
     fun: -1.6295230263157894
       x: [ 1.530e+00  1.439e+00  4.071e+00  4.434e+00]
    nfev: 26
   maxcv: 0.0

Ang optimizer ay nakabawas ng cost at nakahanap ng mas mahusay na parameters para sa circuit.

plt.figure(figsize=(12, 6))
plt.plot(objective_func_vals)
plt.xlabel("Iteration")
plt.ylabel("Cost")
plt.show()

Kapag natagpuan na ninyo ang optimal parameters para sa circuit, maaari na ninyong i-assign ang mga parameters na ito at mag-sample ng huling distribution na nakuha gamit ang mga naka-optimize na parameters. Dito dapat gamitin ang Sampler primitive dahil ito ang probability distribution ng bitstring measurements na tumutugma sa optimal cut ng graph.

Tandaan: Nangangahulugan ito ng paghahanda ng quantum state na $\psi$ sa computer at pagkatapos ay pagsusukat nito. Ang isang measurement ay magko-collapse ng estado sa isang computational basis state - halimbawa, 010101110000... - na tumutugma sa isang kandidatong solusyon na $x$ sa ating initial optimization problem ($\max f(x)$ o $\min f(x)$ depende sa gawain).

optimized_circuit = candidate_circuit.assign_parameters(result.x)
optimized_circuit.draw("mpl", fold=False, idle_wires=False)

# If using qiskit-ibm-runtime<0.24.0, change `mode=` to `backend=`
sampler = Sampler(mode=backend)
sampler.options.default_shots = 10000

# Set simple error suppression/mitigation options
sampler.options.dynamical_decoupling.enable = True
sampler.options.dynamical_decoupling.sequence_type = "XY4"
sampler.options.twirling.enable_gates = True
sampler.options.twirling.num_randomizations = "auto"

pub = (optimized_circuit,)
job = sampler.run([pub], shots=int(1e4))
counts_int = job.result()[0].data.meas.get_int_counts()
counts_bin = job.result()[0].data.meas.get_counts()
shots = sum(counts_int.values())
final_distribution_int = {key: val / shots for key, val in counts_int.items()}
final_distribution_bin = {key: val / shots for key, val in counts_bin.items()}
print(final_distribution_int)

{28: 0.0328, 11: 0.0343, 2: 0.0296, 25: 0.0308, 16: 0.0303, 27: 0.0302, 13: 0.0323, 7: 0.0312, 4: 0.0296, 9: 0.0295, 26: 0.0321, 30: 0.031, 23: 0.0324, 31: 0.0303, 21: 0.0335, 15: 0.0317, 12: 0.0309, 29: 0.0297, 3: 0.0313, 5: 0.0312, 6: 0.0274, 10: 0.0329, 22: 0.0353, 0: 0.0315, 20: 0.0326, 8: 0.0322, 14: 0.0306, 17: 0.0295, 18: 0.0279, 1: 0.0325, 24: 0.0334, 19: 0.0295}

Step 4: Post-process and return result in desired classical format

Ang post-processing step ay nag-interpret ng sampling output upang magbalik ng solusyon para sa inyong orihinal na problema. Sa kasong ito, interesado kayo sa bitstring na may pinakamataas na probabilidad dahil ito ang tumutukoy sa optimal cut. Ang mga symmetries sa problema ay nagbibigay-daan para sa apat na posibleng solusyon, at ang sampling process ay magbabalik ng isa sa kanila na may bahagyang mas mataas na probabilidad, ngunit makikita ninyo sa plotted distribution sa ibaba na apat sa mga bitstrings ay kapansin-pansing mas malamang kaysa sa iba.

# auxiliary functions to sample most likely bitstring
def to_bitstring(integer, num_bits):
    result = np.binary_repr(integer, width=num_bits)
    return [int(digit) for digit in result]

keys = list(final_distribution_int.keys())
values = list(final_distribution_int.values())
most_likely = keys[np.argmax(np.abs(values))]
most_likely_bitstring = to_bitstring(most_likely, len(graph))
most_likely_bitstring.reverse()

print("Result bitstring:", most_likely_bitstring)

Result bitstring: [0, 1, 1, 0, 1]

matplotlib.rcParams.update({"font.size": 10})
final_bits = final_distribution_bin
values = np.abs(list(final_bits.values()))
top_4_values = sorted(values, reverse=True)[:4]
positions = []
for value in top_4_values:
    positions.append(np.where(values == value)[0])
fig = plt.figure(figsize=(11, 6))
ax = fig.add_subplot(1, 1, 1)
plt.xticks(rotation=45)
plt.title("Result Distribution")
plt.xlabel("Bitstrings (reversed)")
plt.ylabel("Probability")
ax.bar(list(final_bits.keys()), list(final_bits.values()), color="tab:grey")
for p in positions:
    ax.get_children()[int(p[0])].set_color("tab:purple")
plt.show()

Visualize best cut

Mula sa optimal bit string, maaari na ninyong i-visualize ang cut na ito sa orihinal na graph.

# auxiliary function to plot graphs
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=100, alpha=0.8, pos=pos
    )

plot_result(graph, most_likely_bitstring)

At kalkulahin ang halaga ng cut:

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])
        for u, v in list(graph.edge_list())
    )

cut_value = evaluate_sample(most_likely_bitstring, graph)
print("The value of the cut is:", cut_value)

The value of the cut is: 5

Part II. Scale it up!

Mayroon kayong access sa maraming devices na may higit sa 100 qubits sa IBM Quantum® Platform. Pumili ng isa kung saan lulutasin ang Max-Cut sa isang 100-node weighted graph. Ito ay isang "utility-scale" problem. Ang mga hakbang upang bumuo ng workflow ay sinusunod na pareho tulad sa itaas, ngunit may mas malaking graph.

n = 100  # Number of nodes in graph
graph_100 = rx.PyGraph()
graph_100.add_nodes_from(np.arange(0, n, 1))
elist = []
for edge in backend.coupling_map:
    if edge[0] < n and edge[1] < n:
        elist.append((edge[0], edge[1], 1.0))
graph_100.add_edges_from(elist)
draw_graph(graph_100, node_size=200, with_labels=True, width=1)

Step 1: Map classical inputs to a quantum problem

Graph → Hamiltonian

Una, i-convert ang graph na gusto ninyong lutasin direkta sa isang Hamiltonian na angkop para sa QAOA.

max_cut_paulis_100 = build_max_cut_paulis(graph_100)

cost_hamiltonian_100 = SparsePauliOp.from_sparse_list(max_cut_paulis_100, 100)
print("Cost Function Hamiltonian:", cost_hamiltonian_100)

Cost Function Hamiltonian: SparsePauliOp(['IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIZIIIIIIIIIII', 'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 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              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, 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])

Hamiltonian → quantum circuit

circuit_100 = QAOAAnsatz(cost_operator=cost_hamiltonian_100, reps=1)
circuit_100.measure_all()

circuit_100.draw("mpl", fold=False, scale=0.2, idle_wires=False)

Step 2: Optimize problem for quantum execution

Upang i-scale ang circuit optimization step sa utility-scale problems, maaari ninyong samantalahin ang high performance transpilation strategies na ipinakilala sa Qiskit SDK v1.0. Ang iba pang tools ay kinabibilangan ng bagong transpiler service na may AI enhanced transpiler passes.

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

candidate_circuit_100 = pm.run(circuit_100)
candidate_circuit_100.draw("mpl", fold=False, scale=0.1, idle_wires=False)

Step 3: Execute using Qiskit primitives

Upang patakbuhin ang QAOA, kailangan ninyong malaman ang optimal parameters na $\gamma_k$ at $\beta_k$ na ilalagay sa variational circuit. I-optimize ang mga parameters na ito sa pamamagitan ng pagpapatakbo ng optimization loop sa device. Ang cell ay nagsusumite ng mga jobs hanggang sa ang cost function value ay nag-converge at ang optimal parameters para sa $\gamma_k$ at $\beta_k$ ay natukoy.

Find candidate solution by running the optimization on the device

Una, patakbuhin ang optimization loop para sa circuit parameters sa isang device.

initial_gamma = np.pi
initial_beta = np.pi / 2
init_params = [initial_beta, initial_gamma]

objective_func_vals = []  # Global variable
with Session(backend=backend) as session:
    # If using qiskit-ibm-runtime<0.24.0, change `mode=` to `session=`
    estimator = Estimator(mode=session)

    estimator.options.default_shots = 1000

    # Set simple error suppression/mitigation options
    estimator.options.dynamical_decoupling.enable = True
    estimator.options.dynamical_decoupling.sequence_type = "XY4"
    estimator.options.twirling.enable_gates = True
    estimator.options.twirling.num_randomizations = "auto"

    result = minimize(
        cost_func_estimator,
        init_params,
        args=(candidate_circuit_100, cost_hamiltonian_100, estimator),
        method="COBYLA",
    )
    print(result)

message: Return from COBYLA because the trust region radius reaches its lower bound.
 success: True
  status: 0
     fun: -3.9939191365979383
       x: [ 1.571e+00  3.142e+00]
    nfev: 29
   maxcv: 0.0

Kapag natagpuan na ang optimal parameters mula sa pagpapatakbo ng QAOA sa device, i-assign ang mga parameters sa circuit.

optimized_circuit_100 = candidate_circuit_100.assign_parameters(result.x)
optimized_circuit_100.draw("mpl", fold=False, idle_wires=False)

Sa wakas, isagawa ang circuit na may optimal parameters upang mag-sample mula sa kaukulang distribution.

# If using qiskit-ibm-runtime<0.24.0, change `mode=` to `backend=`
sampler = Sampler(mode=backend)
sampler.options.default_shots = 10000

# Set simple error suppression/mitigation options
sampler.options.dynamical_decoupling.enable = True
sampler.options.dynamical_decoupling.sequence_type = "XY4"
sampler.options.twirling.enable_gates = True
sampler.options.twirling.num_randomizations = "auto"

pub = (optimized_circuit_100,)
job = sampler.run([pub], shots=int(1e4))

counts_int = job.result()[0].data.meas.get_int_counts()
counts_bin = job.result()[0].data.meas.get_counts()
shots = sum(counts_int.values())
final_distribution_100_int = {
    key: val / shots for key, val in counts_int.items()
}

Suriin na ang cost na na-minimize sa optimization loop ay nag-converge sa isang tiyak na halaga.

plt.figure(figsize=(12, 6))
plt.plot(objective_func_vals)
plt.xlabel("Iteration")
plt.ylabel("Cost")
plt.show()

Step 4: Post-process and return result in desired classical format

Dahil ang likelihood ng bawat solusyon ay mababa, kunin ang solusyon na tumutugma sa pinakamababang cost.

_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."""
    packed_uint8 = np.packbits(observable.paulis.z, axis=1, bitorder="little")
    state_bytes = np.frombuffer(
        state.to_bytes(packed_uint8.shape[1], "little"), dtype=np.uint8
    )
    reduced = np.bitwise_xor.reduce(packed_uint8 & state_bytes, axis=1)
    return np.sum(observable.coeffs * _PARITY[reduced])

def best_solution(samples, hamiltonian):
    """Find solution with lowest cost"""
    min_cost = 1000
    min_sol = None
    for bit_str in samples.keys():
        # Qiskit use little endian hence the [::-1]
        candidate_sol = int(bit_str)
        # fval = qp.objective.evaluate(candidate_sol)
        fval = evaluate_sparse_pauli(candidate_sol, hamiltonian).real
        if fval <= min_cost:
            min_sol = candidate_sol

    return min_sol

best_sol_100 = best_solution(final_distribution_100_int, cost_hamiltonian_100)
best_sol_bitstring_100 = to_bitstring(int(best_sol_100), len(graph_100))
best_sol_bitstring_100.reverse()

print("Result bitstring:", best_sol_bitstring_100)

Result bitstring: [1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1]

Susunod, i-visualize ang cut. Ang mga node na may parehong kulay ay kabilang sa parehong grupo.

plot_result(graph_100, best_sol_bitstring_100)

Kalkulahin ang halaga ng cut.

cut_value_100 = evaluate_sample(best_sol_bitstring_100, graph_100)
print("The value of the cut is:", cut_value_100)

The value of the cut is: 124

Ngayon kailangan ninyong kalkulahin ang objective value ng bawat sample na sinukat ninyo sa quantum computer. Ang sample na may pinakamababang objective value ay ang solusyon na ibinalik ng quantum computer.

# auxiliary function to help plot cumulative distribution functions
def _plot_cdf(objective_values: dict, ax, color):
    x_vals = sorted(objective_values.keys(), reverse=True)
    y_vals = np.cumsum([objective_values[x] for x in x_vals])
    ax.plot(x_vals, y_vals, color=color)

def plot_cdf(dist, ax, title):
    _plot_cdf(
        dist,
        ax,
        "C1",
    )
    ax.vlines(min(list(dist.keys())), 0, 1, "C1", linestyle="--")

    ax.set_title(title)
    ax.set_xlabel("Objective function value")
    ax.set_ylabel("Cumulative distribution function")
    ax.grid(alpha=0.3)

# auxiliary function to convert bit-strings to objective values
def samples_to_objective_values(samples, hamiltonian):
    """Convert the samples to values of the objective function."""

    objective_values = defaultdict(float)
    for bit_str, prob in samples.items():
        candidate_sol = int(bit_str)
        fval = evaluate_sparse_pauli(candidate_sol, hamiltonian).real
        objective_values[fval] += prob

    return objective_values

result_dist = samples_to_objective_values(
    final_distribution_100_int, cost_hamiltonian_100
)

Sa wakas, maaari ninyong i-plot ang cumulative distribution function upang i-visualize kung paano ang bawat sample ay nag-aambag sa kabuuang probability distribution at ang kaukulang objective value. Ang horizontal spread ay nagpapakita ng saklaw ng objective values ng mga samples sa huling distribution. Sa ideal, makikita ninyo na ang cumulative distribution function ay may mga "tumalon" sa mas mababang dulo ng objective function value axis. Nangangahulugan ito na kakaunting solusyon na may mababang cost ay may mataas na probabilidad na ma-sample. Ang smooth, malawak na kurba ay nagsasaad na ang bawat sample ay pareho ang posibilidad, at maaari silang magkaroon ng napakaiiba ng objective values, mababa o mataas.

fig, ax = plt.subplots(1, 1, figsize=(8, 6))
plot_cdf(result_dist, ax, "Eagle device")

Conclusion

Ipinakita ng tutorial na ito kung paano lutasin ang isang optimization problem gamit ang quantum computer sa pamamagitan ng paggamit ng Qiskit patterns framework. Ang demonstrasyon ay may kasamang utility-scale example, na may mga circuit sizes na hindi maaaring eksakto na i-simulate nang classical. Sa kasalukuyan, ang mga quantum computers ay hindi lumalampas sa classical computers para sa combinatorial optimization dahil sa ingay. Gayunpaman, ang hardware ay patuloy na umuunlad, at ang mga bagong algorithms para sa quantum computers ay patuloy na binubuo. Tunay nga, karamihan sa pananaliksik na gumagawa sa quantum heuristics para sa combinatorial optimization ay sinusubukan gamit ang classical simulations na nagbibigay-daan lamang sa maliit na bilang ng qubits, karaniwang mga 20 qubits. Ngayon, na may mas malalaking qubit counts at devices na may mas kaunting ingay, makakapagsimula nang mag-benchmark ang mga mananaliksik ng mga quantum heuristics na ito sa malalaking problem sizes sa quantum hardware.

Tutorial survey

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Link to survey