Abstract: Methods and apparatus for performing surface code computations using Auto-CCZ states. In one aspect, a method for implementing a delayed choice CZ operation on a first and second data qubit using a quantum computer includes: preparing a first and second routing qubit in a magic state; interacting the first data qubit with the first routing qubit and the second data qubit with the second routing qubit using a first and second CNOT operation, where the first and second data qubits act as controls for the CNOT operations; if a received first classical bit represents an off state: applying a first and second Hadamard gate to the first and second routing qubit; measuring the first and second routing qubit using Z basis measurements to obtain a second and third classical bit; and performing classically controlled fixup operations on the first and second data qubit using the second and third classical bits.

Abstract: In some aspects, a heterogeneous computing system includes a quantum processor unit and a classical processor unit. In some instances, variables defined by a computer program are stored in a classical memory in the heterogeneous computing system. The computer program is executed in the heterogeneous computing system by operation of the quantum processor unit and the classical processor unit. Instructions are generated for the quantum processor by a host processor unit based on values of the variables stored in the classical memory. The instructions are configured to cause the quantum processor unit to perform a data processing task defined by the computer program. The values of the variables are updated in the classical memory based on output values generated by the quantum processor unit. The classical processor unit processes the updated values of the variables.

Abstract: Systems and methods for performing signed matrix operations using a linear photonic processor are provided. The linear photonic processor is formed as an array of first amplitude modulators and second amplitude modulators, the first amplitude modulators configured to encode elements of a vector into first optical signals and the second amplitude modulators configured to encode a product between the vector elements and matrix elements into second optical signals. An apparatus may be used to implement a signed value of an output of the linear processor. The linear photonic processor may be configured to perform matrix-vector and/or matrix-matrix operations.

Abstract: An integrated photonics computing system implements a residue number system (RNS) to achieve orders of magnitude improvements in computational speed per watt over the current state-of-the-art. RNS and nanophotonics have a natural affinity where most operations can be achieved as spatial routing using electrically controlled directional coupler switches, thereby giving rise to an innovative processing-in-network (PIN) paradigm. The system provides a path for attojoule-per-bit efficient and fast electro-optic switching devices, and uses them to develop optical compute engines based on residue arithmetic leading to multi-purpose nanophotonic computing.

Abstract: Techniques for creating a replacement for optical elements with diffractive planar components based on metasurfaces are provided. In one example, a substantially flat optical component for lensing incoming electromagnetic radiation having at least one wavelength and a first phase into outgoing electromagnetic radiation having a second phase is provided.

Abstract: A method for realizing a Hadamard product, a device and a storage medium, includes: acquiring a plurality of to-be-treated optical signals with unequal wavelengths; inputting the to-be-treated optical signals into a wavelength division multiplexer; by using the wavelength division multiplexer, feeding the to-be-treated optical signals to a micro-ring-resonator component, wherein the micro-ring-resonator component includes a plurality of micro-ring-resonator groups each of which is formed by two micro-ring resonators with equal radii; and applying a corresponding electric current to the micro-ring-resonator component, to obtain a result of the Hadamard product according to an outputted light intensity.

Abstract: A gas analyzer that easily facilitates alignment is provided. The gas analyzer is a gas analyzer for measuring a predetermined component in a measurement gas by irradiating light on the measurement gas from a light emitting element and receiving light that passes through the measurement gas. The gas analyzer includes a base member configured to be adjustable in position along at least one axis that is not parallel to the optical axis of the light emitting element, and a holding member configured to hold the light emitting element and to be held to the base member in an angularly adjustable manner around at least one axis that is not parallel to the optical axis.

Abstract: The disclosure describes various aspects of techniques for cooling a chain of ions to near the combined ground state that does not grow with the number of ions in the chain. By addressing each ion individually and using each ion to cool a different motional mode, it is possible to cool the motional modes concurrently. In an example, a third of the total motional modes can be cooled at the same time. In an aspect, the techniques include generating a sideband cooling laser beam for each ion in the ion chain, concurrently cooling two or more motional modes associated with the ions in the ion chain using the respective sideband cooling laser beam until each of the two or more motional modes reaches a motional ground state, and performing a quantum computation using the ion chain after the two or more motional modes have reached the motional ground state.

Abstract: Systems and methods for performing matrix operations using a path-number balanced optical network are provided. The optical network is formed as an array including active optical components and passive optical components arranged at a substantially central location of the array. The optical network includes at least NM active optical components which are used to implement a first matrix of any size N×M by embedding the first matrix in a second matrix of a larger size. The optical network performs matrix-vector and matrix-matrix operations by propagating one or more pluralities of optical signals corresponding to an input vector through the optical network.

Abstract: Systems and methods for performing signed matrix operations using a linear photonic processor are provided. The linear photonic processor is formed as an array of first amplitude modulators and second amplitude modulators, the first amplitude modulators configured to encode elements of a vector into first optical signals and the second amplitude modulators configured to encode a product between the vector elements and matrix elements into second optical signals. An apparatus may be used to implement a signed value of an output of the linear processor. The linear photonic processor may be configured to perform matrix-vector and/or matrix-matrix operations.

Abstract: An efficient search includes: inputting data comprising a vector that requires a first amount of memory; compressing the vector into a compressed representation while preserving information content of the vector, including: encoding, using one or more non-quantum processors, at least a portion of the vector to implement a quantum gate matrix; and modulating a reference vector using the quantum gate matrix to generate the compressed representation; searching the compressed vector in a database; and outputting a search result to be displayed, stored, and/or further processed.

Abstract: Data to be processed includes vector element values of an input vector and matrix element values of a model matrix associated with a neural network model. A vector-matrix multiplication module receives a set of matrix element values for performing a vector-matrix multiplication operation. Processing the data includes computing a plurality of intermediate vectors based on element-wise vector multiplication between different subsets of the vector element values and different respective pre-processing vectors. The vector-matrix multiplication module is loaded with a core matrix, and the input vector is multiplied by the model matrix based on separately multiplying each of the intermediate vectors by the loaded core matrix.

Abstract: A method of performing a multiplication operation in the optical domain using a device (100) comprising: an optical waveguide (101), and a modulating element (102) that is optically coupled to the optical waveguide (101), the modulating element (102) modifying a transmission, reflection or absorption characteristic of the waveguide (101) dependant on its state, wherein the state of the modulating element (102) is adjustable by a write signal (103). The method comprises: encoding a first value to the write signal (103), using the write signal (103) to map the first value to a state of the modulating element (102); encoding a second value to a read signal (104); producing an output signal intensity as the transmitted or reflected read signal, wherein the product of the first value and the second value is encoded in the output signal intensity.

Abstract: Aspects relate to a photonic processing system, a photonic processor, and a method of performing matrix-vector multiplication. An optical encoder may encode an input vector into a first plurality of optical signals. A photonic processor may receive the first plurality of optical signals; perform a plurality of operations on the first plurality of optical signals, the plurality of operations implementing a matrix multiplication of the input vector by a matrix; and output a second plurality of optical signals representing an output vector. An optical receiver may detect the second plurality of optical signals and output an electrical digital representation of the output vector.

Abstract: A quantum communications system includes a first quantum repeater and a second quantum repeater each positioned at a repeater node and each having a first quantum memory and a second quantum memory. A first channel switch is optically coupled to the first quantum repeater and a second channel switch is optically coupled to the second quantum repeater. Further, a first sub-channel extends between and optically couples the first channel switch and the first quantum memory of the first quantum repeater, a second sub-channel extends between and optically couples the first channel switch and the first quantum memory of the second quantum repeater, a third sub-channel extends between and optically couples the second channel switch and the second quantum memory of the first quantum repeater, and a fourth sub-channel extends between and optically couples the second channel switch and the second quantum memory of the second quantum repeater.

Abstract: Methods, systems and apparatus for performing windowed quantum arithmetic. In one aspect, a method for performing a product addition operation includes: determining multiple entries of a lookup table, comprising, for each index in a first set of indices, multiplying the index value by a scalar for the product addition operation; for each index in a second set of indices, determining multiple address values, comprising extracting source register values corresponding to indices between i) the index in the second set of indices, and ii) the index in the second set of indices plus the predetermined window size; and adjusting values of a target quantum register based on the determined multiple entries of the lookup table and the determined multiple address values.

Abstract: Data compression includes: inputting data comprising a vector that requires a first amount of memory; compressing the vector into a compressed representation while preserving information content of the vector, including: encoding, using one or more non-quantum processors, at least a portion of the vector to implement a quantum gate matrix; and modulating a reference vector using the quantum gate matrix to generate the compressed representation, wherein the compressed representation requires a second amount of memory that is less than the first amount of memory; and outputting the compressed representation to be displayed, stored, and/or further processed.

Abstract: A method of RF signal processing comprises receiving an incoming RF signal at each of a plurality of antenna elements that are arranged in a first pattern. The received RF signals from each of the plurality of antenna elements are modulated onto an optical carrier to generate a plurality of modulated signals that each have at least one sideband. The modulated signals are directed along a corresponding plurality of optical channels with outputs arranged in a second pattern corresponding to the first pattern. A composite optical signal is formed using light emanating from the outputs of the plurality of optical channels. Non-spatial information contained in at least one of the received RF signals is extracted from the composite signal.

Abstract: A system includes a first unit configured to generate a plurality of modulator control signals, and a processor unit. The processor unit includes: a light source or port configured to provide a plurality of light outputs, and a first set of optical modulators coupled to the light source or port and the first unit. The optical modulators in the first set are configured to generate an optical input vector by modulating the plurality of light outputs provided by the light source or port based on digital input values corresponding to a first set of modulator control signals in the plurality of modulator control signals, the optical input vector comprising a plurality of optical signals. The processor unit also includes a matrix multiplication unit that includes a second set of optical modulators.

Abstract: A method for performing a matrix multiplication operation being secure against side-channel attacks according to one embodiment, which is performed by a computing device comprising one or more processors and a memory storing one or more programs to be executed by the one or more processors, includes shuffling an order of execution of multiplication operations between elements of a first matrix and elements of a second matrix for a matrix multiplication operation between the first matrix and the second matrix; and performing the matrix multiplication operation based on the shuffled order of execution.

Abstract: A system for parallel photonic computation, preferably including a source module, a plurality of input modulator units, an optical interference unit (OIU), and a plurality of detector banks. An OIU, preferably including one or more unitary matrix modules and optionally including a diagonal matrix module. An input modulator, which can include one or more waveguides, couplers, and/or modulator banks. A method for parallel photonic computing, preferably including encoding input vectors, performing a desired matrix operation, and receiving output values, and optionally including performing electronic computations and/or performing further optical computations based on the outputs, which can function to compute the results of a matrix operation on many different input vectors in parallel.

Abstract: A method of error correction for a quantum computer includes identifying each of a plurality of physical qubits arranged in a lattice pattern over a surface in a quantum processor of the quantum computer as a one of a data qubit, an ancilla qubit or a flag qubit to define a plurality of data qubits, ancilla qubits and flag qubits. Each pair of interacting data qubits interact with a flag qubit and adjacent flag qubits both interact with a common ancilla qubit. The method further includes performing measurements of weight-four stabilizers, weight-two stabilizers, or both of a surface code formed using at least a sub-plurality of the plurality of physical qubits, or performing measurements of weight-four Bacon-Shor type gauge operators; and correcting fault-tolerantly quantum errors in one or more of the at least sub-plurality of physical qubits based on a measurement from at least one flag qubit.

Abstract: An imaging system includes a first camera and a second camera. A first antenna arrangement collects image light from a first scene as seen by the first camera, and a second antenna arrangement collects image light from a second, different scene as seen by the second camera. The first antenna arrangement includes a first polarized dish antenna and the second antenna arrangement includes a second polarized dish antenna. The first camera and the second camera are supported with a first polarization of the first polarized dish antenna orthogonal to a second polarization of the second polarized dish antenna such that at least some of the image light from the first scene travels through the second polarized dish antenna to reach the first camera and at least some of the image light from the second scene travels through the first polarized dish antenna to reach the second camera.

Abstract: Systems and methods for performing signed matrix operations using a linear photonic processor are provided. The linear photonic processor is formed as an array of first amplitude modulators and second amplitude modulators, the first amplitude modulators configured to encode elements of a vector into first optical signals and the second amplitude modulators configured to encode a product between the vector elements and matrix elements into second optical signals. An apparatus may be used to implement a signed value of an output of the linear processor. The linear photonic processor may be configured to perform matrix-vector and/or matrix-matrix operations.

Abstract: An all-optical Diffractive Deep Neural Network (D2NN) architecture learns to implement various functions or tasks after deep learning-based design of the passive diffractive or reflective substrate layers that work collectively to perform the desired function or task. This architecture was successfully confirmed experimentally by creating 3D-printed D2NNs that learned to implement handwritten classifications and lens function at the terahertz spectrum. This all-optical deep learning framework can perform, at the speed of light, various complex functions and tasks that computer-based neural networks can implement, and will find applications in all-optical image analysis, feature detection and object classification, also enabling new camera designs and optical components that can learn to perform unique tasks using D2NNs. In alternative embodiments, the all-optical D2NN is used as a front-end in conjunction with a trained, digital neural network back-end.

Abstract: The disclosure describes various aspects of techniques for cooling a chain of ions to near the combined ground state that does not grow with the number of ions in the chain. By addressing each ion individually and using each ion to cool a different motional mode, it is possible to cool the motional modes concurrently. In an example, a third of the total motional modes can be cooled at the same time. In an aspect, the techniques include generating a sideband cooling laser beam for each ion in the ion chain, concurrently cooling two or more motional modes associated with the ions in the ion chain using the respective sideband cooling laser beam until each of the two or more motional modes reaches a motional ground state, and performing a quantum computation using the ion chain after the two or more motional modes have reached the motional ground state.

Abstract: Novel and useful electronic and magnetic control of several quantum structures that provide various control functions. An electric field provides control and is created by a voltage applied to a control terminal. Alternatively, an inductor or resonator provides control. An electric field functions as the main control and an auxiliary magnetic field provides additional control on the control gate. The magnetic field is used to control different aspects of the quantum structure. The magnetic field impacts the spin of the electron by tending to align to the magnetic field. The Bloch sphere is a geometrical representation of the state of a two-level quantum system and defined by a vector in x, y, z spherical coordinates.

Abstract: Aspects of the disclosure provide for mechanisms for providing optimization recommends for quantum computing. A method of the disclosure includes: receiving a first file including a first plurality of quantum instructions for implementing an algorithm; receiving hardware information of a plurality of quantum computer systems, wherein the hardware information comprises information about hardware capacities of the quantum computer systems; and generating, by a processing device, one or more optimization recommendations for implementing the algorithm in view of the first plurality of instructions and the hardware information. In some embodiments, the one or more optimization recommendations include an estimated qubit size required to implement the algorithm in at least one of the plurality of quantum computer systems.

Abstract: A system can include a filter assembly with a filter and a substance in the filter assembly, and at least one optical computing device having an integrated computational element which receives electromagnetic radiation from the substance. A method can include receiving electromagnetic radiation from a substance in a filter assembly, the electromagnetic radiation from the substance being received by at least one optical computing device having an integrated computational element, and the receiving being performed while a filter is positioned in the filter assembly. A detector may receive electromagnetic radiation from the integrated computational element and produce an output correlated to a characteristic of the substance. A mitigation technique may be selected, based on the detector output.

Abstract: Systems and methods for performing matrix operations using a path-number balanced optical network are provided. The optical network is formed as an array including active optical components and passive optical components arranged at a substantially central location of the array. The optical network includes at least NM active optical components which are used to implement a first matrix of any size N×M by embedding the first matrix in a second matrix of a larger size. The optical network performs matrix-vector and matrix-matrix operations by propagating one or more pluralities of optical signals corresponding to an input vector through the optical network.

Abstract: Systems and methods for an optical ternary content addressable memory (TCAM) are provided. The optical TCAM implements a time-division multiplexing (TDM) based encoding scheme to encode each bit position of a search word in the time domain. Each bit position is associated with at least two time slots. The encoded optical signal comprising the search word is routed through one or more modulators configured to represent a respective TCAM stored word. If a mismatch between at least one bit position of the search word and at least one TCAM stored word occurs, a photodetector or photodetector array will detect light.

Abstract: A transmitter comprising a plurality of modulator and multiplexer (Mod-MUX) units, each Mod-MUX unit operating at an optical wavelength different from the other Mod-MUX units. The transmitter can additional include in each Mod-MUX unit two optical taps and three photodetectors that are configured to allow the respective Mod-MUX unit to be tuned to achieve thermal stabilization and achieve effective modulation and WDM operation across a range of temperatures. The Mod-MUX transmitter avoids the use of a frequency comb. The Mod-MUX transmitter avoids cross-modulation between different modulators for different laser signals.

Abstract: An optical numerical computation device relates light from a plurality of light sources to calculate an arithmetic solution. The optical numerical computation device includes input circuitry, pre-calculation circuitry, calculation circuitry, a light collection cavity, and a plurality of light computation components. The pre-calculation circuitry and calculation circuitry cause light sources to emit light representing the values of input operands, which is subsequently related within the light collection cavity. Sensors then generate resultant outputs at values indicative of the sensed light value. The respective wavelengths of light used may be associated with an operand arithmetic sign or an order of magnitude.

Abstract: A method and associated systems for using direct sums and invariance groups to optimize the testing of partially symmetric quantum-logic circuits is disclosed. A test system receives information that describes the architecture of a quantum-logic circuit to be tested. The system uses this information to organize the circuit's inputs into two or more mutually exclusive subsets of inputs. The system computes a direct sum of a set of groups associated with the subsets in order to generate an invariance group that contains one or more invariant permutations of the circuit's inputs. These invariant permutations can be used to reduce the number of tests required to fully verify the circuit for all possible input vectors. Once one specific input vector has been verified, there is no need to test other vectors that can be generated by performing any one of the invariant permutations upon the previously verified vector.

Abstract: High-performance ultra-wideband Phased Array Antennas (PAA) are disclosed, having unique capabilities, enabled through photonic integrated circuits and novel optical architectures. Unique capabilities for PAA systems are enabled by photonic integration and ultra-low-loss waveguides. Novel aspects include optical multiplexing combining wavelength division multiplexing and/or a novel extension to array photodetectors, providing the capability to combine many RF photonic signals with very low loss. Architectures include tunable optical up-conversion and down-conversion systems, moving a chosen frequency band between baseband and a high RF frequency band with high dynamic range. Simultaneous multi-channel RF beamforming is achieved through power combining/splitting of optical signals.

Abstract: A system for parallel photonic computation, preferably including one or more source modules, a plurality of multiplication modules, and a plurality of summation modules. In one embodiment, each multiplication module can include a set of input modulators, a splitter, and a plurality of multiplication banks. Each summation module can include one or more detectors. Each summation module preferably receives an output from multiple multiplication modules and computes the sum of all channels of all the received outputs. A method for parallel photonic computation, preferably including generating input signals, computing products, and computing sums.

Abstract: A method for directing non-modulated light from a Spatial Light Modulator (SLM) and allowing through modulated light for producing an interference based holographic image, the method including illuminating the SLM with coherent light, thereby producing a mix of light modulated by the SLM and light not modulated by the SLM, and projecting the mix of the modulated light and the not modulated light along an optical axis onto a volume grating, wherein the volume grating directs the not modulated light away from the holographic image and allows through modulated light for producing the holographic image. Related apparatus and methods are also described.

Abstract: A method and apparatus for processing an image interaction are provided. The apparatus extracts, using an encoder, an input feature from an input image, converts the input feature to a second feature based on an interaction for an application to the input image, and generates, using a decoder, a result image from the second feature.

Abstract: Example photoconductive devices and example methods for using photoconductive devices are described. An example method may include providing a photoconductive device having a metal-semiconductor-metal structure. The method may also include controlling, based on a first input state, illumination of the photoconductive device by a first optical beam during a time period, and controlling, based on a second input state, illumination of the photoconductive device by a second optical beam during the time period. Further, the method may include detecting an amount of current produced by the photoconductive device during the time period, and based on the detected amount of current, providing an output indicative of the first input state and the second input state. The example devices can be used individually as discrete components or in integrated circuits for memory or logic applications.

Abstract: Systems and methods are provided in which a direction of arrival of a radio frequency (RF) signal received by a plurality of antennas is determined. A plurality of first converter receives RF signals from the plurality of antennas and outputs a minimum of a first optical signal and a second optical signal each modulated by their corresponding RF signal. A plurality of second converters receives a minimum of the first optical signal via a first optical channel that introduces a first delay and the second optical signal via a second optical channel that introduces a second delay. The second converter outputs a first RF signal that corresponds to the RF modulation on the first optical signal and a second RF signal that corresponds to the RF modulation on the second optical signal. A switch serially receives, from the second converter outputs, the first RF signal and the second RF signal.

Abstract: Metasurfaces comprise an array of pillars in a lattice. The dimensions of the pillars and the spacing are varied to obtain desired optical properties. The dispersionless metasurfaces can focus optical light over a broad wavelength range. Specific dispersion profiles for the metasurfaces can be designed. Gratings can be fabricated having similar properties as the array of pillars. Pillars in the metasurfaces can have different cross-section profiles.

Abstract: The present disclosure provides methods and systems for performing non-classical computations. The methods and systems generally use a plurality of spatially distinct optical trapping sites to trap a plurality of atoms, one or more electromagnetic delivery units to apply electromagnetic energy to one or more atoms of the plurality to induce the atoms to adopt one or more superposition states of a first atomic state and a second atomic state, one or more entanglement units to quantum mechanically entangle at least a subset of the one or more atoms in the one or more superposition states with at least another atom of the plurality, and one or more readout optical units to perform measurements of the superposition states to obtain the non-classical computation.

Abstract: A laser system may include a laser source configured to generate a first laser light beam, a beam stabilizer downstream from the laser source and configured to stabilize the first laser light beam, and a beamsplitter downstream from the beam stabilizer and configured to split the stabilized first laser light beam into a plurality of second laser light beams. The system may further include a multi-channel acousto-optic modulator (AOM) including a common acousto-optic medium configured to receive the plurality of second laser light beams, and a respective phase array transducer comprising a plurality of electrodes coupled to the common acousto-optic medium for each of the second laser light beams. The system may further include a plurality of radio frequency (RF) drivers each configured to generate respective RF drive signals for the phased array transducer electrodes.

Abstract: Embodiments of methods, apparatuses, devices and/or systems for manipulating character expressions to determine relationships among such character expressions.

Abstract: A technique relates to a circuit for a sum frequency generator. A first resonator is connected to a Josephson ring modulator (JRM), and the first resonator is configured to receive a first photon at a first frequency. A second resonator is connected to the JRM, and the second resonator is configured to have a first harmonic and no second harmonic. The second resonator is configured to receive a second photon at a second frequency, and the first resonator is configured to output an up-converted photon. The up-converted photon has an up-converted frequency that is a sum of the first frequency and the second frequency.

Abstract: Optical systems for performing matrix-matrix multiplication in real time utilizing spatially coherent input light and wavelength multiplexing.

Abstract: There is provided a computer implemented data driven classification and troubleshooting system and method. The system has an interface application enabled to receive data. The system has an associative memory software in communication with the interface application via an API. The associative memory software has an associative memory and a machine learning algorithm. The system has one or more individual areas, within the associative memory, requiring one or more troubleshooting actions to improve accuracy of the individual areas. The system has at least one troubleshooting tool enabled by the interface application. The at least one troubleshooting tool enables or performs the troubleshooting actions. The system has a quality rating metric (QRM) that measures a strength and an assurance that one or more predictions of the associative memory are correct. The one or more troubleshooting actions results in improving the accuracy and the performance of the associative memory.

Abstract: In the Waveguide Array Modulator (WAM) a single electrical signal drives an array of waveguide optical modulators, creating multiple modulated output signals that can be combined to provide a higher output power than from a single waveguide based modulator, enabling a higher dynamic range system. Alternatively, using a WAM in which different waveguide optical modulators are designed for different dynamic ranges, e.g. one highly efficient modulator for low level signals and one low efficiency but linear modulator for high level signals, the WAM based system can provide a higher dynamic range than from a single waveguide based modulator. Various WAM based systems for different applications are included.

Abstract: A method and apparatus for measuring a quality of a holographic image are provided. An image quality measuring method may include acquiring a first image including a holographic image in a background, and a second image that does not include a holographic image, the first image and the second image being on the same background, extracting a gray level value and a dark-noise counted level value from the first image in a selected range of the first image, extracting a gray level value and a dark-noise counted level value from the second image in the same range as the selected range of the first image, and determining a holographic contrast ratio (HCR) of the holographic image based on the extracted gray level values and the extracted dark-noise counted level values.

Abstract: The inventive concept relates to a quantum circuit that shifts the phase of a target qubit by ?/2n?1 based on a control qubit. The quantum circuit may include a first auxiliary circuit, a rotation gate, and a second auxiliary circuit. The first auxiliary circuit converts the first qubit state into the second qubit state according to the entanglement of a control qubit, a target qubit, and an ancillary qubit having a |0> state. The rotation gate shifts the phase for some basis states of the second qubit state by ?/2n?1, and converts the second qubit state to the third qubit state. The second auxiliary circuit converts the third qubit state to the fourth qubit state such that the phase of the target qubit is shifted by ?/2n?1.