WO2022045938A1 - Résolution d'un système d'équations linéaires - Google Patents

Résolution d'un système d'équations linéaires Download PDF

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WO2022045938A1
WO2022045938A1 PCT/SE2020/050825 SE2020050825W WO2022045938A1 WO 2022045938 A1 WO2022045938 A1 WO 2022045938A1 SE 2020050825 W SE2020050825 W SE 2020050825W WO 2022045938 A1 WO2022045938 A1 WO 2022045938A1
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qubit
eigenvalue
state
rotation operation
quantum
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PCT/SE2020/050825
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Ahsan Javed AWAN
Markus TOLLET
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Telefonaktiebolaget Lm Ericsson (Publ)
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N20/00—Machine learning
    • G06N20/10—Machine learning using kernel methods, e.g. support vector machines [SVM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/12—Simultaneous equations, e.g. systems of linear equations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/20—Models of quantum computing, e.g. quantum circuits or universal quantum computers

Definitions

  • Examples of the present disclosure relate to a method of solving a system of linear equations using a quantum computing device, and to a quantum computing device configured to solve a system of linear equations.
  • the original SVM technique may be described as a dual problem, in which a first problem may be solved in order to then solve a second problem.
  • This problem may be converted into a Linear System Problem (LSP) via the Least Square formulation of SVM.
  • LSP Linear System Problem
  • the SVM is a supervised machine learning technique for solving classification or regression problems.
  • the computational complexity of SVM is O[poly(NM)], and it is proportional to the polynomial in NM, where N is the number of dimensions of the data, and M is the number of training data.
  • HHL Harrow-Hassidim-Lloyd
  • QLSP Quantum Linear System Problem
  • HHL algorithm uses n register qubits to encode the eigenvalues of the linear system as bit-strings.
  • the HHL algorithm also includes a Phase Estimation procedure, which involves performing a Hamiltonian simulation.
  • quantum algorithms proposed for simulating Hamiltonian, H such as the divide and conquer approach, the quantum walk, and quantum signal processing.
  • the query and gate complexity of these major approaches are complex enough to run inefficiently on the near-term noisy intermediate scale quantum (NISQ) computers.
  • NISQ near-term noisy intermediate scale quantum
  • quantum computing devices are performance-limited by quantum noise and coherence time of the qubits. This, in turn, limits the size of quantum circuits that can be reliably executed. Furthermore, there is a limit on the number of qubits that are available to use in these near-term quantum machines. This implies that existing solutions would not be applicable on near-term quantum machines, and would require customized solutions that take into account the limitations of near-term quantum machines. Thus, there is a need to reduce the complexity of quantum algorithms to observe expected results on NISQ computers.
  • the quantum circuit 100 consists of three parts as follows, a phase estimation part 102, a controlled rotation part 104, and an un-computation part 106.
  • the quantum circuit 400 consists of four qubits: the first qubit
  • the phase estimation part 102 initially consists of a Hadamard gate and two controlled- NOT gates. Following the execution of the Hadamard gate and the two controlled-NOT gates, the first, second and third qubits become a 3-qubit Greenberger-Horne-Zeilinger state (GHZ) state, 1/ 2(
  • 111 >), and the first and second qubit encode the eigenvalues of the matrix 1 which are Ai
  • 01> and A 2
  • the Control-X gate is then followed by an X gate and a Hadamard gate.
  • the controlled rotation part 104 of the quantum circuit 100 contains an X gate and two H(0) rotations.
  • the X gate function is to find the reciprocals
  • 1/Aj>, i 1, 2 from the eigenvalues
  • Aj>, i 1, 2.
  • the H(0) gate is defined as _ CO s(20) ⁇
  • the un-computation part 106 of the quantum circuit 100 is composed of two Hadamard gates for computing the inverse phase estimation. A measurement of the quantum state of the third qubit may then be obtained, and the solution of the system of linear equations may be obtained from this measurement.
  • phase estimation part 102 only features a two-qubit register (
  • One aspect of the present disclosure provides a method of solving a system of linear equations using a quantum computing device.
  • the method comprises applying a first state based on b to a first qubit of the quantum computing device, performing a first rotation operation on a second qubit, wherein the first rotation operation is controlled by a state of the first qubit, and performing a second rotation operation on the second qubit, wherein the second rotation operation is controlled by the state of the first qubit, and wherein the first rotation operation and the second rotation operation are based on eigenvalues of A.
  • the method also comprises performing a measurement of a state of the first qubit, and solving the system of linear equations based on the measurement.
  • Another aspect of the present disclosure provides a method of encoding, on a qubit of a quantum computing device, a first eigenvalue and a second eigenvalue of a matrix A corresponding to a system of linear equations.
  • the method comprises performing a first rotation operation on an initial quantum state of the qubit to obtain a second quantum state of the qubit that corresponds to the first eigenvalue and the second eigenvalue, wherein the first rotation operation is based on an angle relating to the first eigenvalue and the second eigenvalue.
  • a further aspect of the present disclosure provides a quantum computing device configured to perform the method of any of the above aspects.
  • Figure 1 shows an example of a quantum circuit
  • Figure 2 is a flow chart for encoding a first eigenvalue and a second eigenvalue of a matrix on a qubit of a quantum computing device
  • Figure 3 is a flow chart for solving a system of linear equations using a quantum computing device
  • Figure 4 shows a further example of a quantum circuit
  • Figures 5a and 5b show a comparison of the norm of the solution difference, and the Jensen-Shannon divergence, and the final classification result based on the execution of two quantum circuits;
  • Figure 6 is a schematic of an example of apparatus for encoding a first eigenvalue and a second eigenvalue of a matrix on a qubit of a quantum computing device
  • Figure 7 is a schematic of an example of apparatus for solving a system of linear equations using a quantum computing device.
  • Hardware implementation may include or encompass, without limitation, digital signal processor (DSP) hardware, a reduced instruction set processor, hardware (e.g., digital or analogue) circuitry including but not limited to application specific integrated circuit(s) (ASIC) and/or field programmable gate array(s) (FPGA(s)), and (where appropriate) state machines capable of performing such functions.
  • DSP digital signal processor
  • ASIC application specific integrated circuit
  • FPGA field programmable gate array
  • Methods and apparatus disclosed herein provide an amplitude encoding procedure as opposed to a binary encoding procedure in order encode the eigenvalues of a linear system on a qubit of a quantum computing device. Furthermore, improved methods and apparatus for solving a system of linear equations using a quantum computing device are provided, wherein the quantum phase estimation procedure in a HHL algorithm may be replaced with the aforementioned encoding based procedure.
  • the presented amplitude encoding procedure may use a rotation operator in order to encode the eigenvalues in the amplitudes of a qubit, rather than estimating eigenvalues in bit-string encoded states.
  • the method 200 comprises, in step 202, performing a first rotation operation on an initial quantum state of the qubit to obtain a second quantum state of the qubit that corresponds to the first eigenvalue and the second eigenvalue, wherein the first rotation operation is based on an angle relating to the first eigenvalue and the second eigenvalue.
  • the angle will be suitable for encoding the first eigenvalue and the second eigenvalue on a qubit of a quantum computing device.
  • the angle may be suitable for amplitude encoding a first eigenvalue on a first state (e.g.
  • a vector b may be encoded as a quantum state
  • i) corresponds to the computational basis. In the case that N 2, this corresponds to the quantum states
  • the 2-dimensional vector b can be represented as the quantum state
  • the initial state of the qubit may be
  • the first eigenvalue and the second eigenvalue may be L 2 normalized.
  • the method 200 may further comprise the steps of performing a measurement on the second quantum state of the qubit, and determining the first eigenvalue and the second eigenvalue based on the measurement.
  • the method 200 may be executed a plurality of times, and a plurality of measurements may be performed on the second quantum state of the qubit.
  • the first eigenvalue and the second eigenvalue may be determined based on the plurality of measurements.
  • a probability distribution may be obtained from the plurality of measurements, and the first eigenvalue and the second eigenvalue may be obtained from this obtained probability distribution.
  • the plurality of times that the method 200 may be executed, and the plurality of measurements may be performed on the second quantum state of the qubit may be adjusted depending on the desired statistical error of the determined first eigenvalue and the second eigenvalue.
  • the first and second eigenvalues may be encoded in terms of amplitudes relating to quantum states (such as in the method 200)
  • ceil(N/2) qubits e.g. the next integer higher than N/2
  • the binary limitation for storing eigenvalues is equivalent to that of a classical system. This therefore yields better flexibility for solving linear systems.
  • this approach may allow for a set of QLSP problems to be solved using a quantum computing device
  • A P o b ] e R L aJ 2
  • a,b are generic real numbers.
  • the method 300 comprises, in step 302, applying a first state based on b to a first qubit of the quantum computing device.
  • Step 304 of the method 300 comprises performing a first rotation operation on a second qubit, wherein the first rotation operation is controlled by a state of the first qubit.
  • the first eigenvalue and a second eigenvalue may be L 2 normalized.
  • the first rotation operation may comprises a Hadamard operation.
  • Step 306 of the method 300 comprises performing a second rotation operation on the second qubit, wherein the second rotation operation is controlled by the state of the first qubit, and wherein the first rotation operation and the second rotation operation are based on eigenvalues of A.
  • the second rotation operation may comprise a Hadamard operation.
  • an initial state of the second qubit may be
  • the difference between the first rotation operation and the second rotation operation may be based on one or more eigenvalues of A.
  • the method 300 may further comprise performing a third rotation operation on a third qubit, wherein the third rotation operation is based on one or more eigenvalues of A. In some embodiments, the method 300 may further comprise performing a fourth rotation operation on the third qubit, wherein the third rotation operation is the inverse of the third rotation operation.
  • an initial state of the third qubit may be
  • the third rotation operation may be based on an angle relating to a first eigenvalue and a second eigenvalue of A.
  • Step 308 of the method 300 comprises performing a measurement of a state of the first qubit.
  • the step of performing a measurement of a state of the first qubit comprises performing the method a plurality of times to obtain a plurality of measurements of the state of the first qubit.
  • the state(s) of one or more other qubits may also be measured. An example is given below.
  • Step 310 of the method 300 comprises solving the system of linear equations based on the measurement.
  • the method 300 may be executed a plurality of times, and a measurement of a state of the first qubit performed a plurality of times.
  • solving the system of linear equations may be based on the plurality of obtained measurements.
  • a probability distribution may be obtained from the plurality of obtained measurements, and the system of linear equations may be solved based on the plurality of obtained probability distribution.
  • the plurality of times that the method 300 may be executed, and the plurality of measurements may be performed on the first qubit may be adjusted depending on the desired statistical error for the solution of the system of linear equations.
  • the method 300 may comprise determining x from the measured state of the first qubit. In some embodiments, determining x from the measured state of the first qubit may comprise determining a value for each element of x from probabilities of states of the first qubit. In some embodiments, determining a value for each element of x from probabilities of states of the first qubit may comprise determining a first value for a first element of x from a probability of a state of the first qubit, and determining a second value for a second element of x from a probability of the state of the first qubit.
  • FIG. 4 shows an example of a quantum circuit 400.
  • the quantum circuit 400 may implement the methods 200 or 300 described above. It will be appreciated that the quantum circuit 400 may be considered equivalent to a quantum computing device.
  • the quantum circuit 400 shown in Figure 4 consists of three parts as follows, a phase estimation part 402, a controlled rotation part 404, and an un-computation part 406.
  • the quantum circuit 400 consists of three qubits: the first qubit Iq- is the input qubit, and is initialized as the quantum state ⁇ b) that represents the vector b.
  • the quantum state ⁇ b) may be the quantum state
  • q 2 ) is the reduced clock register, or the encoding qubit, and is initialized as
  • q 3 is the ancilla register, and has the final the value of
  • a rotation operator R y (q>) is executed on the second qubit
  • a Hadamard operator is also executed on the first qubit iQi) in the phase estimation part, which expresses the input state Iq- in terms of the eigenvectors, A 1; A 2 , of the linear system.
  • a controlled rotation rotates the ancilla qubit,
  • the controlled rotation is based on the first and second eigenvalues.
  • the first controlled rotation is based on a normalization constant (in other words, is arbitrary), and the second controlled rotation is based on the first and second eigenvalues.
  • the first controlled rotation may be based on the first eigenvalue
  • the second controlled rotation may be based on the second eigenvalue
  • the second controlled rotation may be an arbitrary value (e.g. a normalization constant) and the first controlled rotation may be based on the first and second eigenvalues
  • both controlled rotations may be based on the first and second eigenvalues; or any other suitable arrangement.
  • the controlled rotations may be C/A ⁇ and C/A 2 .
  • the first controlled rotation is controlled on the
  • the second controlled rotation is controlled on the
  • the ancilla qubit is also rotated to have amplitude C/Aj for the
  • the quantum circuit 100 described above and shown in Figure 1 it will be appreciated that two qubits are used to control the rotation. Hence, the number of qubits required to solve the same 2x2 linear system is reduced in the quantum circuit 400 when compared to the quantum circuit 100.
  • the un-computation part 406 a rotation operator R y ' 1 ((p) is executed on the second qubit
  • the un-computation part 406 effectively undoes the phase estimation part 402. It will be appreciated that the un-computation part 406 maximizes the probability of the states of the first qubit Iq- that correspond to the solution of the linear system.
  • the uncomputation part 106 of the quantum circuit 100 constitutes executing two Hadamard gates on the clock register (iQi) and
  • the un-computation part 406 is considerably computationally cheaper in terms of gate sequences to inverse the amplitude encoding version of the phase estimation (in order to maximize the probability of quantum states that correspond to the solution of the linear system).
  • q 3 ) may be performed.
  • x> may be obtained from the measurement of the first qubit
  • 0> may be obtained from the measurement of the second qubit
  • 1> may be obtained from the measurement of the third qubit if the execution of the quantum circuit 400 has resulted in a solution.
  • q 3 ) may indicate that a solution was not obtained, e.g. due to noise.
  • the rotations may be for example p and ⁇ p 2 .
  • the quantum circuit 400 may be executed a plurality of times, and a measurement of a state of the first qubit performed a plurality of times.
  • solving the system of linear equations may be based on the plurality of obtained measurements.
  • a probability distribution may be obtained from the plurality of obtained measurements, and the system of linear equations may be solved based on the plurality of obtained probability distribution.
  • the plurality of times that the quantum circuit 400 may be executed, and the plurality of measurements may be performed on the first qubit may be adjusted depending on the desired statistical error for the solution of the system of linear equations.
  • b>, representing different input vectors b, and different matrices A) and four corresponding classical solutions x are presented.
  • x> have been obtained using a local noise/error-free simulator (Pyquil), and the classical solutions have been found with numpy.linalg.solve(A,b).
  • the results obtained from the execution of the quantum circuit 400 are close to the classically obtained results.
  • a further norm of the solution difference may also be obtained for results obtained from the execution of the quantum circuit 100 and the corresponding classical solutions, in that the error between the classically obtained results and results obtained using embodiments of this disclosure is very small.
  • This further norm of the solution difference may be compared with the previously discussed norm of the solution difference to provide a metric for determining improvements of the quantum circuit 400 (implementing the methods 200 and 300 described above), over the quantum circuit 100 described above.
  • the Jensen-Shannon divergence is a metric that determines distinguishability of mixed quantum states represented by two probability distributions Pi , P2:
  • the QSVM may be implemented on the quantum circuit 400 to classify numerical digits for the OCR MNIST dataset.
  • a linear 2 x 2 kernel matrix K may be generated by running the training data oracle according to reference [2], to create the state K then may be calculated via
  • K was found to be:
  • the corresponding QLSP is: r 1 0.481 r «ii _ r— 11
  • Figure 5a shows the output probability distribution and the classification result obtained from executing the quantum circuit 400 according to the QLSP described above.
  • Figure 5b shows the output probability distribution and the classification result obtained from executing the quantum circuit 100 according to the QLSP described above (in this illustrated example, the quantum circuit 100 has been executed on Rigetti machine ASPEN-7-4Q-D).
  • Figures 5A and 5B show a comparison the norm of the solution difference, and the Jensen-Shannon divergence, and the final classification result based on the execution of the quantum circuit 400 (Figure 5A) and the quantum circuit 100 (Figure 5B) running on the Rigetti quantum system ASPEN-7-4Q-D, respectively.
  • Figures 5A(i) and Figure 5B(i) respectively show the probability distribution of executing the quantum circuit 400 (Figure 5a(i)), and executing the quantum circuit 100 ( Figure 5B(i)), on the local noise/error-free simulator (Pyquil) (as shown in blue) and the Rigetti quantum system ASPEN-7-4Q-D (as shown in red).
  • Figure 5A(ii) and Figure 5B(ii) respectively show example results of a classification process for a plurality of test vectors, the classification boundary (straight line) from solving the QLSP, and the overall classification success ratio for executing the quantum circuit 400 (Figure 5A(ii)), and the quantum circuit 100 ( Figure 5B(ii)) on the Rigetti quantum system ASPEN-7-4Q-D.
  • the training process e.g. matrix inversion followed by training data oracle
  • test vectors falling on one side of the respective lines are classified in a first classification (e.g. +1), whereas test vectors falling on the other side of the respective lines are classified in a second classification (e.g. -1).
  • the quantum circuit 400 reduces the circuit depth by approximately 2.4 time compared to the quantum circuit 100. Furthermore, the difference between the probability distributions obtained for the quantum circuit 400 (for both the local noise/error-free simulator (Pyquil) and the Rigetti quantum system ASPEN-7-4Q-D) is reduced by approximately 2 times compared to the corresponding probability distributions obtained for the quantum circuit 100. Furthermore, in one example, the classification accuracy obtained when executing the quantum circuit 400 using a particular dataset is equal to 98.5%. This is an improvement over the classification accuracy obtained when executing the quantum circuit 100 using the particular dataset, which is equal to 97.5%.
  • the apparatus 600 comprises processing circuitry 602 (e.g. one or more processors) and a memory 604 in communication with the processing circuitry 602.
  • the memory 604 contains instructions executable by the processing circuitry 602.
  • the memory 604 contains instructions executable by the processing circuitry 602 such that the apparatus 600 is operable to performing a first rotation operation on an initial quantum state of the qubit to obtain a second quantum state of the qubit that corresponds to the first eigenvalue and the second eigenvalue, wherein the first rotation operation is based on an angle relating to the first eigenvalue and the second eigenvalue.
  • the memory 604 contains instructions executable by the processing circuitry 602 such that the apparatus 600 is operable to carry out the method 200 shown in Figure 2 described above.
  • the apparatus 700 comprises processing circuitry 702 (e.g. one or more processors) and a memory 704 in communication with the processing circuitry 702.
  • the memory 704 contains instructions executable by the processing circuitry 702.
  • the memory 704 contains instructions executable by the processing circuitry 702 such that the apparatus 700 is operable to apply a first state based on b to a first qubit of the quantum computing device, perform a first rotation operation on a second qubit, wherein the first rotation operation is controlled by a state of the first qubit, perform a second rotation operation on the second qubit, wherein the second rotation operation is controlled by the state of the first qubit, and wherein the first rotation operation and the second rotation operation are based on eigenvalues of A, perform a measurement of a state of the first qubit, and solve the system of linear equations based on the measurement.
  • the memory 704 contains instructions executable by the processing circuitry 702 such that the apparatus 700 is operable to carry out the method 300 shown in Figure 3 described above.
  • Methods and apparatus disclosed herein provide an amplitude encoding procedure as opposed to a binary encoding procedure in order encode the eigenvalues of a linear system on a qubit of a quantum computing device. Furthermore, improved methods and apparatus for solving a system of linear equations using a quantum computing device are provided, wherein the quantum phase estimation procedure in a HHL algorithm may be replaced with the aforementioned encoding based procedure. These methods and apparatus provide a way to reduce the number of qubits and the depth of the circuit required to solve the QLSP. It will also be appreciated that the amplitude encoding based approach does not require the Hamiltonian simulation present in the original HHL algorithm, thus reducing the circuit depth.
  • the proposed methods and apparatus allow greater flexibility in the linear system that may be solved, as the number of qubits needed to encode the eigenvalues are no longer limited by binary encoding. This allows the amplitude encoding based approach to scale well with the size of the linear system.

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Abstract

Est divulgué un procédé de résolution d'un système d'équations linéaires à l'aide d'un dispositif informatique quantique, le système d'équations linéaires étant équivalent à Ax = b, où A est une matrice n x n, x est un vecteur de colonne avec n entrées, et b est un vecteur de colonne avec n entrées. Le procédé consiste à appliquer un premier état sur la base d'un premier bit quantique du dispositif informatique quantique, à effectuer une première opération de rotation sur un second bit quantique, la première opération de rotation étant commandée par un état du premier bit quantique, à effectuer une seconde opération de rotation sur le second bit quantique, la seconde opération de rotation étant commandée par l'état du premier bit quantique, et la première opération de rotation et la seconde opération de rotation étant basées sur des valeurs propres de A, à réaliser une mesure d'un état du premier bit quantique, et à résoudre le système d'équations linéaires sur la base de la mesure.
PCT/SE2020/050825 2020-08-27 2020-08-27 Résolution d'un système d'équations linéaires WO2022045938A1 (fr)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
EP4283535A1 (fr) * 2022-05-25 2023-11-29 Rolls-Royce plc Détermination de solutions à un certain nombre d'équations matricielles linéaires

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