# US 10,320,360 B2 · Quantum algorithms for arithmetic and function synthesis

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- Number: US 10,320,360 B2
- Document ID: US10320360B2
- Status: Granted
- Kind: B2
- Application number: 15/316,500
- Filed: 2015-06-05
- Published: 2017-07-06
- Granted: 2019-06-11
- Earliest claimed: 2014-06-06
- Assignees: Microsoft Technology Licensing, LLC
- Applicants: Microsoft Technology Licensing, LLC
- Inventors: Nathan Wiebe; Martin Roetteler
- Web page: https://patentgrep.com/patent/US10320360B2

## CPC

- H03H 7/21: providing two or more phase shifted output signals, e.g. n-phase output
- B82Y 10/00: Nanotechnology for information processing, storage or transmission, e.g. quantum computing or single electron logic
- G06F 7/523: Multiplying only
- G06F 7/548: Trigonometric functions; Co-ordinate transformations
- G06F 17/17: Function evaluation by approximation methods, e.g. inter- or extrapolation, smoothing, least mean square method ({G06F17/18 takes precedence } ; interpolation for numerical control G05B19/18)
- G06N 10/20: Models of quantum computing, e.g. quantum circuits or universal quantum computers
- G06N 10/60: Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms
- H03H: IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS (waveguides, resonators, lines or other devices of the waveguide type H01P)
- B82Y: SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS

## Abstract

Quantum circuits and associated methods use Repeat-Until-Success (RUS) circuits to perform approximate multiplication and approximate squaring of input values supplied as rotations encoded on ancilla qubits. So-called gearbox and programmable ancilla circuits are coupled to encode even or odd products of input values as a rotation of a target qubit. In other examples, quantum RUS circuits provide target qubit rotations that are associated with reciprocals using series expansion representations.

## Claims

### Claim 1

A quantum circuit, comprising:
a repeat-until-success (RUS) multiplication circuit coupled to a first ancilla qubit and a target qubit at respective RUS multiplication circuit inputs so as to produce, at a target qubit output, a target qubit phase based on a product of a phase coupled to the RUS multiplication circuit at the first ancilla qubit;
a measurement circuit coupled to the first ancilla qubit at an RUS multiplication circuit output; and
a correction circuit coupled to the measurement circuit to apply a correction to the target qubit if the measurement circuit indicates failure.

### Claim 2 (depends on 1)

The quantum circuit of claim 1, wherein the multiplication circuit is a gearbox circuit.

### Claim 3 (depends on 1)

The quantum circuit of claim 1, wherein the multiplication circuit is a PAR circuit.

### Claim 4 (depends on 1)

The quantum circuit of claim 1, wherein a plurality of ancilla qubits are coupled to the RUS multiplication circuit, and the target qubit phase is based on a product of phases supplied to the RUS multiplication circuit associated with the plurality of ancilla qubits.

### Claim 5 (depends on 4)

The quantum circuit of claim 4, wherein the RUS multiplication circuit is a gearbox circuit, and the target qubit phase corresponds to an even power of a product of the phases supplied to the RUS multiplication circuit with the plurality of ancilla qubits.

### Claim 6 (depends on 4)

The quantum circuit of claim 4, wherein the RUS multiplication circuit is a gearbox circuit, and the target qubit phase corresponds to an odd power of a product of the phases supplied to the RUS multiplication circuit with the plurality of ancilla qubits.

### Claim 7 (depends on 1)

The quantum circuit of claim 1, wherein the target qubit phase is associated with a reciprocal.

### Claim 8 (depends on 7)

The quantum circuit of claim 7, wherein the correction circuit includes at least one Clifford gate.

### Claim 9

At least one computer readable storage device comprising computer executable instructions for a method, comprising:
obtaining a representation of a function of a plurality of variables;
expressing at least a portion of the representation based on at least one product of the plurality of variables; and
assigning at least one repeat-until-success quantum circuit so as to produce a qubit phase corresponding to the at least one product upon success.

### Claim 10 (depends on 9)

The at least one computer readable storage device of claim 9, wherein the method further comprises identifying odd and even powers of the plurality of variables in the portion of the representation, and assigning the at least one repeat-until-success quantum circuit based on the identification.

### Claim 11 (depends on 9)

The at least one computer readable storage device of claim 9, wherein the at least one quantum circuit is a gearbox circuit if an even power is identified or a programmable ancilla rotation (PAR) circuit if an odd power is identified.

### Claim 12 (depends on 9)

The at least one computer readable storage device of claim 9, wherein the at least one quantum circuit is a gearbox circuit or a PAR circuit.

### Claim 13 (depends on 9)

The at least one computer readable storage device of claim 9, wherein method further comprises obtaining an error limit associated with the product, and assigning a plurality of quantum circuits based on the error limit so as to compensate product errors.

### Claim 14 (depends on 9)

The at least one computer readable storage device of claim 9, wherein method further comprises arranging the at least one selected quantum circuit so that the variables of the plurality of variables are coupled to respective ancilla qubits, the target qubit receives the qubit phase based on an error limit associated with the product, and a plurality of quantum circuits is assigned based on the error limit.

### Claim 15 (depends on 9)

The at least one computer readable storage device of claim 9, wherein the method further comprises representing the function as a series of products of the variables.

### Claim 16 (depends on 15)

The at least one computer readable storage device of claim 15, further comprising defining a quantum correction circuit that is coupled to the target qubit and returns the target qubit to an initial state upon failure.

### Claim 17 (depends on 15)

The at least one computer readable storage device of claim 15, wherein values of the variables of the plurality of variables are represented as rotations on respective qubits.

### Claim 18 (depends on 9)

The at least one computer readable storage device of claim 9, wherein the representation of the function includes a least one reciprocal of a variable that is obtained based on a series representation.

### Claim 19 (depends on 9)

The at least one computer readable storage device of claim 9, wherein at least one of the variables is offset so as change an even or odd power term in a series expansion to an odd or even power term, respectively.

### Claim 20

A method of obtaining a value of a function in quantum computer, comprising:
representing values of a plurality of variables on respective ancilla qubits;
coupling the ancilla qubits to a quantum repeat-until-success circuit that includes at least one gearbox circuit or PAR circuit;
obtaining a corresponding value of the function as a phase on a target qubit if output states of the ancilla qubit correspond to success; and
correcting the phase of the output target qubit upon failure, and coupling the corrected target qubit state to a target qubit input of the quantum repeat-until-success multiplication circuit.

### Claim 21 (depends on 1)

The quantum circuit of claim 1, wherein the product of the phase coupled to the first ancilla qubit is a power of the phase coupled to the first ancilla qubit or a product of the phase coupled to the first ancilla qubit with a second phase that is coupled to a second ancilla qubit.