Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators
Abstract
:1. Introduction
2. Materials and Methods
2.1. Difference Scheme of the Poisson Equation
2.2. Discretization of the Equation
- Representation of the input vector b:|x〉|0〉 → |x〉|b〉
- Conversion of the matrix A into a quantum operation:|x〉|y〉|0〉 → |x〉|y〉|A(x,y)〉
- Calculation of the solution vector x:|x〉|y〉|A(x,y)〉|b〉 → |x〉|y〉|A(x,y)〉|b〉 − A(x,y)|x〉
- Measurement of the states of the qubits representing the solution vector x:|x〉|y〉|A(x,y)〉|b〉 → |x〉|y〉|A(x,y)〉|b〉 − A(x,y)|x〉
2.3. HHL Algorithm
2.4. Using the HHL Algorithm on the Poisson Equation
- The right-hand-side vector b is represented as a superposition of two states |0〉 and |1〉 proportional to it, i.e., |b〉 = sqrt(p0)|0〉 + sqrt(p1)|1〉;
- An operator is applied that effectively implements the action of the matrix A on the state |1〉, i.e., |1〉 → |A〉|1〉, where |A〉 is some state that encodes the matrix A;
- The quantum Fourier transform (QFT) is performed on the qubits corresponding to the right-hand-side vector and matrix A;
- The qubit corresponding to the right-side vector is measured. In this case, it is the probability of getting 0 or 1;
- Inverse QFT (IQFT) is applied on all qubits to obtain the inverse Fourier transform for the state |y〉;
- The first N qubits are measured on the standard basis and the output results are obtained. Then, using the results of measurements and the IQFT, an estimate of the solution x is obtained;
- To estimate the accuracy of the solution, the vector is calculated, where is the estimate of the solution x;
- The norm is evaluated to obtain an estimate of the error of the solution.
3. Results
3.1. Quantum Scheme and Probabilities for States
- A quantum circuit is created with N × N qubits and one classical bit.
- The gates associated with matrix A are applied. Each non-zero element of matrix A leads to the application of the corresponding gates. If the element is on the main diagonal (i = j), the rz gate is used. Otherwise, when the element is off the main diagonal (i! = j), the CRZ gate is used.
- The Hadamard h gate is applied to all qubits.
- QFT quantum Fourier transform is applied on all qubits.
- cswap gate is applied between certain qubits.
- gates rz and rx with defined angles are applied on each qubit.
- The last rz valve with an angle of 0 can be omitted because it is equivalent to the identity valve.
- The last qubit is measured, and the result is stored in the classic bit.
3.2. Error Calculations in Different Quantum Simulators and in Different Qubits
- Problem size: The table shows that the HHL algorithm uses only 5 qubits on real quantum computers and 63 qubits on a quantum simulator. If the problem scales with increasing numbers of qubits, real quantum computers may begin to show their advantages.
- Noise and decoherence: Real quantum computers are subject to noise and decoherence, which can slow down the execution of algorithms. Simulators can bypass this limitation because they do not encounter the real physics and noise associated with real quantum systems. If the matrix size is increased, the number of qubits may not be sufficient to run it on real quantum computers. Therefore, the results of work in quantum simulators are obtained faster than on real quantum computers.
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Quantum Simulators | Type | Time (s) |
---|---|---|
qasm_simulator | General | 51.353 |
ibmq_qasm_simulator | General, context aware | 24.219 |
statevector_simulator | Schrödinger wavefunction | 5.745 |
matrix_product_state | Extended Clifford | 0.6654 |
Quantum Resources | Number of Qubits | Time (s) |
---|---|---|
simulator_extended_stabilizer | 63 | 0.7246 |
ibmq_quito | 5 | 4.7431 |
ibmq_belem | 5 | 5.0169 |
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Daribayev, B.; Mukhanbet, A.; Imankulov, T. Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators. Appl. Sci. 2023, 13, 11491. https://doi.org/10.3390/app132011491
Daribayev B, Mukhanbet A, Imankulov T. Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators. Applied Sciences. 2023; 13(20):11491. https://doi.org/10.3390/app132011491
Chicago/Turabian StyleDaribayev, Beimbet, Aksultan Mukhanbet, and Timur Imankulov. 2023. "Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators" Applied Sciences 13, no. 20: 11491. https://doi.org/10.3390/app132011491