HHL算法的QISKit实现
本实现基于IBM QISKit 0.7.0版本，python 3.7版本。
HHL Experiment (Quantum Algorithm for linear systems of equations)
This experiment is implemented based on IBM QISKit version 0.7.0. It is a quantum implementation of finding x⃗\vec{x}x satisfying Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b, and only for the case when AAA is a 2 by 2 matrx, while b⃗\vec{b}b is a 2 by 1 vector.
The whole process can be divided into several steps:
- Quantum phase estimation
- Controlled rotation
- Inverse quantum phase estimation
The quantum circuit for this experiment is (from reference3):
The first qubit ∣x1⟩|x_1 \rangle∣x1⟩ is the ancilla register for controlled rotation. The second and third qubit ∣x2x3⟩|x_2x_3\rangle∣x2x3⟩ (the register C) will save the superposition of the eigenvalues of A, after the quantum phase estimation. The forth qubit ∣x4⟩|x_4 \rangle∣x4⟩ is used to save ∣b⟩|b \rangle∣b⟩, and after the whole process of HHL, it will save the approximate value of x⃗\vec{x}x.
# Import packages
%matplotlib inline
from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister
from qiskit import execute
from qiskit import BasicAer
from math import pi
from qiskit.tools.visualization import plot_histogramimport warnings # Ignore the warning message
warnings.filterwarnings('ignore')
The known AAA and b⃗\vec{b}b are set like this in this experiment:
A=12(3113)A=\frac{1}{2}\begin{pmatrix} 3 & 1\\ 1 & 3 \end{pmatrix}A=21(3113) and b⃗=(b1b2)\vec{b}=\begin{pmatrix} b_1\\ b_2 \end{pmatrix}b=(b1b2).
b⃗\vec{b}b can be encoded in a quantum state that ∣b⟩=b1∣0⟩+b2∣1⟩|b \rangle = b_1|0 \rangle+b_2|1 \rangle∣b⟩=b1∣0⟩+b2∣1⟩, which fulfills b12+b22=1b_1^2+b_2^2=1b12+b22=1.
And the eigenvalues of A are λ1=1\lambda_1=1λ1=1 and λ2=2\lambda_2=2λ2=2 with corresponding eigenvectors ∣u1⟩|u_1 \rangle∣u1⟩ and ∣u2⟩|u_2 \rangle∣u2⟩. λ1\lambda_1λ1 and λ2\lambda_2λ2 can be encoded by ∣x2x3⟩=∣01⟩|x_2x_3 \rangle=|01 \rangle∣x2x3⟩=∣01⟩ and ∣x2x3⟩=∣10⟩|x_2x_3 \rangle=|10 \rangle∣x2x3⟩=∣10⟩ respectively. (Actually all 2 by 2 matrics with eigenvalues 1 and 2 can be realized by this circuit.)
Step 1. Apply quantum phase estimation
This step is to determine the eigenvalues of A.
1.1 Initialization
Here we use ∣b⟩=∣1⟩|b \rangle =|1 \rangle∣b⟩=∣1⟩ as an example, and the initial value for ∣x1x2x3x4⟩|x_1x_2x_3x_4 \rangle∣x1x2x3x4⟩ is ∣0001⟩|0001 \rangle∣0001⟩.
1.2 Create superposition
Apply Hadamard gate to the ancilla register and the register C.
1.3 Apply controlled-U gate
The controlled-U gates in this process can be implemented by phase shift gates, with phases exp(iAt0/4)exp(iAt_0/4)exp(iAt0/4) and exp(iAt0/2)exp(iAt_0/2)exp(iAt0/2) (here we let t0=2πt_0=2\pit0=2π):
The picture above is from https://www.youtube.com/watch?v=v1AUILJz3RU.
1.4 Inverse Fourier transform
Apply inverse Fourier transform to the register C.
## 1.1 Initialization
n = 4 # Set the number of qubits in the register
q = QuantumRegister(n, 'q') # Create a Quantum Register with n qubits.
register = QuantumCircuit(q) # Create a Quantum Circuit acting on the q register
register.x(q[3]) ## 1.2 Create superposition
register.h(q[1])
register.h(q[2])## 1.3 Apply controlled-U gate
register.u1(pi/2, q[2])
register.u1(pi, q[1])
register.cx(q[2], q[3]) #additional, still need to figure out why## 1.4 Inverse quantum Fourier transform
register.swap(q[1], q[2])
register.h(q[2]) #iQFT
register.cu1(-pi/2, q[1], q[2])
register.h(q[1])
<qiskit.extensions.standard.h.HGate at 0x222397f0400>
After the phase estimation the state of ∣x1x2x3⟩|x_1x_2x_3 \rangle∣x1x2x3⟩ becomes β1∣01⟩∣u1⟩+β2∣10⟩∣u2⟩\beta_1|01 \rangle|u_1 \rangle+\beta_2|10 \rangle|u_2 \rangleβ1∣01⟩∣u1⟩+β2∣10⟩∣u2⟩, where β1\beta_1β1 and β2\beta_2β2 are expansion coefficients of ∣b⟩|b \rangle∣b⟩ in AAA's eigenbasis.
Step 2. Controlled rotation
You can find more mathematical derivations about this step here.
2.1 Get λ1−1\lambda_1^{-1}λ1−1 and λ2−1\lambda_2^{-1}λ2−1 using a SWAP gate
After the SWAP gate, the state of ∣x1x2x3⟩|x_1x_2x_3 \rangle∣x1x2x3⟩ becomes β1∣10⟩∣u1⟩+β2∣01⟩∣u2⟩\beta_1|10 \rangle|u_1 \rangle+\beta_2|01 \rangle|u_2 \rangleβ1∣10⟩∣u1⟩+β2∣01⟩∣u2⟩. We can now interpret ∣x2x3⟩=∣10⟩|x_2x_3 \rangle=|10 \rangle∣x2x3⟩=∣10⟩ as the state encoding the inverted eigenvalue 2λ1−1=22\lambda_1^{-1}=22λ1−1=2 and ∣x2x3⟩=∣01⟩|x_2x_3 \rangle=|01 \rangle∣x2x3⟩=∣01⟩ as that encoding 2λ2−1=12\lambda_2^{-1}=12λ2−1=1.
Here in our implementation, the matix A have eignvalues 1 and 2. I also give an example of getting λ1−1\lambda_1^{-1}λ1−1 and λ2−1\lambda_2^{-1}λ2−1 when the eigenvalues are 1 and 3. Thus we use A=(2112)A=\begin{pmatrix} 2 & 1\\ 1 & 2 \end{pmatrix}A=(2112) as the input matrix instead. In this example, we need to use a X gate on ∣x2⟩|x_2 \rangle∣x2⟩ to instead the SWAP gate between ∣x2⟩|x_2 \rangle∣x2⟩ and ∣x3⟩|x_3\rangle∣x3⟩.
2.2 Apply rotation RyR_yRy
This process is to save λj−1\lambda_j^{-1}λj−1 into the amplitudes of the ancilla register.
Ry(θ)=(cos(θ2)−sin(θ2)sin(θ2)cos(θ2))R_y(\theta)=\begin{pmatrix} cos(\frac{\theta}{2}) & -sin(\frac{\theta}{2})\\ sin(\frac{\theta}{2})& cos(\frac{\theta}{2}) \end{pmatrix}Ry(θ)=(cos(2θ)sin(2θ)−sin(2θ)cos(2θ)), and the rrr in the RyR_yRy gate is a parameter that ranges between log22πlog_22\pilog22π and log22π/ωlog_22\pi/\omegalog22π/ω with ω\omegaω being the minimum angle that can be resolved. Here we choose r=4, because when x=4, the system has a good performance on both the fidelity and the probability, as is shown in the picture below (from reference 3):
A more detailed explanation for rrr can be found in reference 3.
## 2.1 SWAP
# register.swap(q[1], q[2]) # for eigenvalues 1 and 2
register.x(q[1]) # for eigen values 1 and 3## 2.2 rotation R_y
register.cu3(pi/8, 0, 0, q[1], q[0])
register.cu3(pi/16, 0, 0, q[2], q[0])
<qiskit.extensions.standard.cu3.Cu3Gate at 0x22239f26278>
Step 3. Uncompute
This process is to do the inverse of all operations before RyR_yRy, which is denoted by U†U^\daggerU† in the circuit.
# register.swap(q[1], q[2]) # for eigenvalues 1 and 2
register.x(q[1]) # for eigenvalues 1 and 3
register.h(q[1]) #iQFT
register.cu1(pi/2, q[1], q[2])
register.h(q[2])
register.swap(q[1], q[2])register.cx(q[2], q[3]) #additional, still need to figure out why
register.u1(-pi, q[1])
register.u1(-pi/2, q[2])register.h(q[2])
register.h(q[1])
<qiskit.extensions.standard.h.HGate at 0x222398e60b8>
Step 4. Measurement
4.1 Quantum circuit
# Create a Classical Register with n bits.
c = ClassicalRegister(n, 'c')
# Create a Quantum Circuit
measure = QuantumCircuit(q, c)
measure.barrier(q)
# map the quantum measurement to the classical bits
measure.measure(q,c)# The Qiskit circuit object supports composition using
# the addition operator.
qc = register + measure#drawing the circuit
qc.draw(output="mpl")
4.2 Run on local simulator
The final state for the ∣x1x2x3x4⟩|x_1x_2x_3x_4 \rangle∣x1x2x3x4⟩ should be a∣0000⟩+b∣0001⟩+c∣1000⟩+d∣1001⟩a|0000\rangle+b|0001\rangle+c|1000\rangle+d|1001\ranglea∣0000⟩+b∣0001⟩+c∣1000⟩+d∣1001⟩ (The ∣x2x3⟩|x_2x_3 \rangle∣x2x3⟩ is always ∣00⟩|00\rangle∣00⟩). Because we can only get the ∣x⟩|x \rangle∣x⟩ when the ancilla qubit, ∣x1⟩=∣1⟩|x_1 \rangle=|1\rangle∣x1⟩=∣1⟩, so we only care about the last two possible states of ∣x1x2x3x4⟩|x_1x_2x_3x_4 \rangle∣x1x2x3x4⟩.
Note that the rightmost qubit in the result is q0 (∣x1⟩|x_1 \rangle∣x1⟩), the leftmost is q3 (∣x4⟩|x_4 \rangle∣x4⟩). Thus in this implementation the two states that we need to pay attention to is ‘0001’ and ‘1001’. The ratio of these two probabilities, which is ∣c/d∣2|c/d|^2∣c/d∣2, is close to ∣x1/x2∣2|x_1/x_2|^2∣x1/x2∣2 (x⃗=(x1  x2)T\vec{x} = (x_1\; x_2)^Tx=(x1x2)T).
# Use Aer's qasm_simulator
backend_sim = BasicAer.get_backend('qasm_simulator')# Execute the circuit on the qasm simulator.
# We've set the number of repeats of the circuit to be 1024, which is the default.
job_sim = execute(qc, backend_sim, shots=8192)# Grab the results from the job.
result_sim = job_sim.result()counts = result_sim.get_counts(qc)
print(counts)
x1_square = counts['0001']
x2_square = counts['1001']
p = x1_square/x2_square
print('probability of state 0001:', x1_square)
print('probability of state 1001:', x2_square)
print('the ratio of two probabilities', p)
plot_histogram(result_sim.get_counts())
{'0000': 3, '0001': 85, '1000': 7809, '1001': 295}
probability of state 0001: 85
probability of state 1001: 295
the ratio of two probabilities 0.288135593220339
Reference:
- Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems ofequations. Physical review letters, 103(15):150502, 2009.
- Yudong Cao, Anmer Daskin, Steven Frankel, and Sabre Kais. Quantum circuit design forsolving linear systems of equations.Molecular Physics, 110(15-16):1675–1680, 2012.
- Yudong Cao, Anmer Daskin, Steven Frankel, and Sabre Kais. Quantum circuits for solvinglinear systems of equations.arXiv preprint arXiv:1110.2232v3, 2013.
- Jian Pan, Yudong Cao, Xiwei Yao, Zhaokai Li, Chenyong Ju, Hongwei Chen, Xinhua Peng,Sabre Kais, and Jiangfeng Du. Experimental realization of quantum algorithm for solving linearsystems of equations.Physical Review A, 89(2):022313, 2014.
- X-D Cai, Christian Weedbrook, Z-E Su, M-C Chen, Mile Gu, M-J Zhu, Li Li, Nai-Le Liu,Chao-Yang Lu, and Jian-Wei Pan. Experimental quantum computing to solve systems of linearequations.Physical review letters, 110(23):230501, 2013.
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