# US 12,118,432 B2 · Method for synthesizing product of Pauli rotations in a quantum circuit and process for synthesizing quantum circuits for Trotter-Suzuki n-order expansion

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- Number: US 12,118,432 B2
- Document ID: US12118432B2
- Status: Granted
- Kind: B2
- Application number: 17/458,758
- Filed: 2021-08-27
- Published: 2022-03-03
- Granted: 2024-10-15
- Earliest claimed: 2020-09-02
- Assignees: Bull SAS
- Applicants: Bull SAS
- Inventors: Simon Martiel; Timothée Goubault de Brugière
- Web page: https://patentgrep.com/patent/US12118432B2

## CPC

- G06N 10/00: Quantum computing, i.e. information processing based on quantum-mechanical phenomena
- G06F 11/3442: {for planning or managing the needed capacity}
- G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

A method for generalizing an algorithm configured to synthesize a diagonal product of Pauli rotations to synthesize a product of Pauli rotations comprising X, Y and Z rotations, the method comprising: Providing a table of p number of rows and m number of columns, where p is a number of qubits and m a number of rotations in the quantum circuit, and where the table comprises X, Y, Z or I entry corresponding to the respective rotations of the qbits; Determining a pivot row, and recursively, until all rotations of the product of Pauli rotations are 1-qubit rotations: Determine a target row, Conjugate the target row with the pivot row by insertion of predetermined quantum gates on the qubits corresponding to the target row and/or pivot row by calling, at each recursive call, entries of the same type of the pivot row and by always calling first the identity entry.

## Claims

### Claim 1

Method for synthesizing product of Pauli rotations, wherein said product of Pauli rotations comprises at least two different types of X, Y, and/or Z n-qubits Pauli rotations and identities (I), said method comprising:
providing a table of p number of rows and m number of columns, where p is a number of qubits and m a number of rotations in a quantum circuit, and where the table comprises X, Y, Z or I entry corresponding to the respective rotations of the qubits;
determining a pivot row of said table, the pivot row being a first row comprising a maximum number of a same type of X, Y, Z or I entry;
performing a subset of steps recursively on said table, until all qubits perform only 1-qubit-rotation, the subset of steps comprising:
determining if the pivot row comprises at least one identity entry;
determining a target row among remaining rows, said target row containing a maximum of non-identity entries interacting with the entries of the pivot row;
conjugating said pivot row and said target row by inserting a predetermined type of quantum gates on their respective qubits in order to bring the X, Y or Z entry of at least the pivot row to I entry; and
rewriting the table according to a result of the conjugation, by replacing the X, Y and/or Z entry of at least the pivot row by an identity entry; and
returning the rewritten table,
wherein if the pivot row is determined to comprise at least one identity entry, the pivot row is determined as being a second row having the maximum number of the same type of X, Y, Z, or I entry, and being different from the first row, and
wherein if the pivot row is determined to not comprise at least one identity entry, the determination of a target row is performed directly after.

### Claim 2 (depends on 1)

Method according to claim 1, comprising, before determining the pivot row:
suppressing one or more columns of the table that contain only one rotation; and
and inserting a corresponding phase gate in the quantum circuit.

### Claim 3 (depends on 1)

Method according to claim 1, wherein the table is provided after:
providing the quantum circuit corresponding to the product of Pauli rotations; and
rewriting the quantum circuits in a form of a table.

### Claim 4 (depends on 3)

Method according to claim 3, wherein the conjugating step comprises:
at least one insertion of Clifford quantum gates between the pivot row and the target row in the quantum circuit.

### Claim 5 (depends on 4)

Method according to claim 4, wherein the Clifford quantum gates comprise:
RX(π/2) quantum rotation gates and Hadamard quantum gates,
CNOT quantum gates, or
a combination thereof.

### Claim 6 (depends on 1)

Method according to claim 1, further comprising:
at each conjugation step, inserting the predetermined type of quantum gates on their respective qubits in the quantum circuit;
according to a result of a conjugation, creating a synthesized quantum circuit based on the quantum circuit, by replacing the n-qubits rotation of at least the pivot row by a 1-qubit rotation; and
returning the synthesized quantum circuit.

### Claim 7 (depends on 6)

Method according to claim 6, wherein the conjugation step further comprises:
inserting the predetermined type of quantum gates on their respective qubits in the quantum circuit before the X, Y and/or Z n-qubits Pauli operators; and
inserting reversed quantum gates on the same qubits in the quantum circuit, after the X, Y and/or Z n-qubits Pauli operators.

### Claim 8 (depends on 6)

Method according to claim 6, further comprising an optimizing step wherein the returned synthesized quantum circuit is split to obtain:
a first sub-quantum circuit comprising only Clifford quantum gates and 1-qubit rotations; and
a second sub-quantum circuit comprising only reversed Clifford gates,
wherein only the first sub-circuit is executed.

### Claim 9 (depends on 8)

Method according to claim 8, wherein a Clifford correction corresponding to the second sub-circuit is applied to the first sub-circuit.

### Claim 10 (depends on 1)

Method according to claim 1, wherein the product of Pauli rotations corresponds to a Hamiltonian simulation.

### Claim 11 (depends on 1)

Process for synthesizing quantum circuits for Trotter-Suzuki n-order expansion comprising:
applying iteratively to each layer of the quantum circuit the method for synthesizing Pauli rotations in a quantum circuit according claim 1.