Quantum channel phase estimation

Abstract

Various aspects of phase error estimation in a quantum communication system are provided. In some aspects, a semiclassical quantum fourier transform (SQFT), using Hadamard gates and rotation gates, may be applied to incoming quantum states. Measurements results from the SQFT may be used to estimate a phase error. Various SQFT circuitry is further described, including parallel SQFT circuitry and sequential SQFT circuitry.

Description

QUANTUM CHANNEL PHASE ESTIMATIONCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U. S. Provisional Application No. 63 / 592,779 filed on October 24, 2023, the contents of which are incorporated in its entirety.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under H9240522F0041 awarded by United States Special Operations Command. The government has certain rights in the invention.TECHNICAL FIELD

[0003] Various embodiments and implementations described herein relate generally to systems and methods for quantum communications. More specifically, embodiments and implementations hereof may involve quantum channel phase error estimation.BACKGROUND

[0004] The success of quantum key distribution (QKD), quantum computers, and Shor's algorithm opens the era of quantum supremacy. While most attention has been focused on quantum computing and quantum error correction, quantum signal processing can be useful in many military or civil application scenarios of interest such as quantum wireless communications, sensing, information processing, and distributed computing. Of ultimate interest in quantum wireless communication or sensing systems, we focus on an effective engineering design of channel estimation while transmitting classic information over a quantum channel. Although channel estimation or equivalent techniques have been well studied in classic systems, collapsing quantum state after the quantum measurement prohibits classic techniques such as sampling and filtering.

[0005] Parameter estimation of quantum channels has been studied and the theoretical foundation of quantum Fourier transform is known. However, existing solutions typically rely on the generation and transmission of entanglement through a quantum channel and / or the use of 2-qubit gates at the receiver, which implies circuit complexity in quantum optical implementation. To alleviate these requirements, low-complexity quantum channel phase estimation (LC-QCPE) has been proposed, which uses measurement statistics from the communication channel to estimate the quantum channel.BRIEF DESCRIPTION OF THE DRAWINGS

[0006] FIG. 1 illustrates an example quantum communication system.

[0007] FIG. 2 illustrates an example Quantum Fourier Transform (QFT) circuit.

[0008] FIG. 3A-C are graphs illustrating various simulation results with respect to various simulated quantum parameters.

[0009] FIG. 4 illustrates various examples of types of quantum communications and quantum communication processing.

[0010] FIG. 5 illustrates an example parallel and sequential QFT circuit.

[0011] FIG. 6 illustrates an example parallel and sequential semiclassical QFT (SQFT) circuit.

[0012] FIG. 7 illustrates an example Qiskit quantum circuit model.

[0013] FIG. 8 illustrates a second example Qiskit quantum circuit model.

[0014] FIG. 9A-B are graphs illustrating example simulation results comparing various quantum phase error estimations.

[0015] FIG. 10 illustrates example optical quantum state transmitter components.

[0016] FIG. 11 illustrates example optical quantum gate components.

[0017] FIG. 12 illustrates example optical quantum measurement gate components.

[0018] FIG. 13A illustrates an example 2-qubit SQFT circuit.

[0019] FIG. 13B illustrates example components of the example illustrated in FIG. 13A.

[0020] FIG. 14 illustrates an example optical 2-qubit SQFT circuit.

[0021] FIG. 15 illustrates an example 3-qubit SQFT circuit.

[0022] FIG. 16 illustrates an example parallel and sequential optical 3-qubit SQFT circuit.

[0023] FIG. 17 illustrates an example sequential optical 2-qubit SQFT circuit.

[0024] FIG. 18 illustrates an example sequential optical 3-qubit SQFT circuit.

[0025] FIG. 19 illustrates an example sequential optical N-qubit SQFT circuit.

[0026] FIG. 20 illustrates a block diagram of an example quantum communications receiver.DETAILED DESCRIPTION

[0027] The detailed description set forth below, in connection with the appended drawings, is intended as a description of various configurations and is not intended to represent the only configurations in which the subject matter described herein may bepracticed. The detailed description includes specific details to provide a thorough understanding of various embodiments of the present disclosure. However, it will be apparent to those skilled in the art that the various features, concepts and embodiments described herein may be implemented and practiced without these specific details. In some instances, well-known structures and components are shown in block diagram form to avoid obscuring such concepts. Various embodiments, implementations, advantages, configurations, and examples of the present disclosure can also be found in the

[0028] Embodiments of such methods, systems, software, and circuits are described below.

[0029] As used in this specification and the appended claims, the singular forms "a," "an," and "the" include plural referents unless the content clearly dictates otherwise. As used in this specification and the appended claims, the term "or" is generally employed in its sense including "and / or" unless the context clearly dictates otherwise.

[0030] As used herein, the capital letters 'C' and 'Q' indicate classical and quantum processing, respectively. The letters may be presented in pairs, where the first letter describes transmitter-side processing, and the second letter describes receiver-side processing. These concepts are illustrated with respect to FIG. 4. The CC strategy 401 uses separable encoding and decoding (only single qubit gates at the transmitter 402 and receiver 403, e.g., no interaction between qubits). The CQ strategy 404 uses separable encoding and general decoding (Single qubit gates at the transmitter 405, but allows for more general 2-qubit gates at the receiver 406). The QC strategy 407 uses general encoding and separable decoding at the receiver 409 (More general 2-qubit gates at transmitter 408, but single qubit gates at receiver 409). The QQ 410 strategy uses general encoding and decoding (2-qubit gates at both the transmitter 411 and receiver 412).

[0031] The optical implementation of 2-qubit gates is known to be a difficult problem as 2-qubit gates suffer significant loss and potential distortion in larger scale realization.Accordingly, approaches to quantum channel estimation using the Quantum Fourier Transform (QFT) at the receiver may suffer difficulties in practical realizations. Aspects of the disclosed technology may be implemented using 1-qubit quantum gates. For example, aspects may provide a semi-classical QFT (S-QFT) that employs 1-bit quantum gates, including 1-bit quantum gates that are controlled by measured states (e.g., classical states).

[0032] Implementations may provide quantum channel phase estimation, which is a critical functionality for quantum wireless communication and remote sensing systems. Implementations may be effective in estimation of a small phase offset and large phase offset. Accordingly, classic statistical analysis may be implemented via quantum circuitry that can effectively achieve the quantum-classic estimation of quantum channel phase offset. Additionally, the described technology may be implemented using NISQ (noisy intermediate-scale quantum era) devices without requiring the use of 2-qubit gates.

[0033] For ease of explanation, aspects of the described technology are provided in terms of a quantum-classical communications setup assuming a unitary quantum channel.Assumption of a unitary quantum channel is chosen to describe various methods of parameter estimation. The described technology may be adaptable for parameter estimation of non-unitary quantum channels. Additionally, the described technology may be implemented in a lossy quantum system, for example as a phase estimation stage in a multi-stage quantum receiver. In keeping with standard nomenclature, the terms "Alice" and "Bob" are used to describe operations or other aspects of a quantum transmitter system and a quantum receiver system, respectively. It should be understood that, unless indicated to the contrary, descriptions in terms of Alice and Bob do not refer to humans, human interactions, or manual human operations.

[0034] This description includes a discussion of standard quantum phase estimation to demonstrate various challenges within the context of quantum communication engineering. Three strategies for performing parameter estimation of a unitary quantum channel are described using the strategies of Figure 4: (i) LC-QCPE is described as a parallel CC method for performing parameter estimation of a quantum channel, (ii) A parallel CQ version of LC-QCPE is described using the QFT, and (iii) a parallel and sequential CQ strategy is described.

[0035] Further, the S-QFT is described which effectively converts CQ strategies to a CC strategy, and QQ strategies to a QC strategy, supporting practical optical implementation of a parameter estimation strategy at a quantum wireless (e.g., optical) communication receiver. Simulations of strategies (ii) and (iii) via the quantum SDK, Qiskit, are presented and the performance of strategies (i), (ii), and (iii) are compared.

[0036] Some aspects of the disclosed technology may provide a device, including an optical coupler coupled to an input quantum channel to receive a quantum state, the quantum state comprising a plurality of quantum encoded bit values. The device mayfurther comprise one or more optical channels, which may include a Hadamard gate to apply a first superposition operation to a first qubit of the plurality of qubits; and a first measurement gate to measure the first Hadamard gate output state to determine a first bit value corresponding to the first qubit. The device may further comprise a first controlled rotation gate controlled by the first bit value to apply a first controlled rotation operation to a second qubit of the plurality of qubits and a Hadamard gate to apply a second superposition operation to the first controlled rotation gate output state. The device may further comprise a measurement gate to measure the second Hadamard gate output state to determine a second bit value corresponding to the second qubit. Such a device may further include a phase estimator to estimate a phase error of the quantum channel based on the measured bit values.

[0037] Further aspects of the disclosed technology may provide a method for phase error estimation in a quantum communication system. For example, a method may include receiving a quantum state on a quantum channel, the quantum state comprising a plurality of quantum encoded bit values. The method may further include coupling the quantum state to a plurality of qubits. The method may further include applying a first superposition operation to a first qubit of the plurality of qubits and measuring the first superposition operation output state to determine a first bit value corresponding to the first qubit. The method may further include sequentially measuring respective bit values corresponding to respective qubits of the plurality of qubits. The sequential measurement operations may include applying a respective controlled rotation operation to a respective qubit based on preceding bit values corresponding to qubits measured prior to the respective qubit, applying a respective superposition operation to the respective controlled rotation operation output state, and measuring the respective superposition operation output state to determine the respective bit value corresponding to the respective qubit. The method may further include estimating a phase error of the quantum channel based on the measured bit values.

[0038] Still further aspects of the disclosed technology may provide a method for phase error estimation in an optical quantum communication system. For example, such a method may include receiving an optical quantum state via an input optical quantum channel. The method may further include directing the optical quantum state to a plurality of k optical quantum channels to couple the optical quantum state to a correspondingplurality of k polarization-encoded qubits. The method may further include applying a k-qubit semiclassical Quantum Fourier Transform to the k polarization-encoded qubits to obtain k polarization state measurements. Such a method may further include estimating a phase error of the input quantum optical channel based on the k polarization state measurements.

[0039] FIG. 1 illustrates an example quantum-classical communication system 101. Alice 102 encodes binary information at104 into quantum information | at) 105, and then transmits |cij)through a quantum channel 8106. Bob 103 receives and decodes the state | f) = £(a() 107 using measurements M =and generates binary information | dt108. A quantum-classical communications system 101 may include components such as a quantum transmitter 102 (Alice), a quantum channel 106, and the quantum receiver 103 (Bob). For example, a system 101 may comprise room-temperature quantum photonic circuits, such as fiber optics, quantum optical modulators, optical switches, wave plates (e.g., quarter wave plates or half wave plates), photodetectors, optical modulators (e.g., electro-optical modulators), amplifiers, processing circuitry, etc.

[0040] In this example, Alice 102 first encodes binary information atG {0,1} 104 into quantum information | at) G J-CA105 with probabilities p at= 0) = p and p at= 1) = 1 — P> 0), withprobability pa'1), withprobability 1 — pand transmits n quantum states | at) through 8106 to Bob 103. The quantum channel 8 106 maps | at) from Alice's Hilbert space J-CAto Bob's Hilbert space J-CB, 8\ J-CA— > J-CB106. As indicated above, a unitary quantum channel model is considered, that is, J-CA= J-CBand is described by the operatorIT > [cosA0 — sinA0]einA0 cosA0 JBob 103 receives the errored quantum state | dt) = Ug> | cij) and measures each one of Alice's qubits, \di), using quantum measurements, M =Bob decodes Alice's quantum information |dj), to binary information dtG {0,1} using M 107 such that Alice 102 and Bob 103 share symmetric bits= air) with probabilityTherefore, giving a probability of errorP(error) = 1 — P success) = sin2A0 (3) Once estimating A0 is achieved, Alice and Bob can implement mitigation procedures at either the receiver 102 or transmitter 103 with the aid of information feedback. For example, with information feedback, Alice 102 can transmit the rotated stateor Bob 103 can use rotated measurements without feedbackto compensate for the quantum channel U0106, achieving a probability of error, P(error) « 0. For Alice to implement rotated states |a£)g, or Bob to implement rotated measurements Mg, Alice or Bob estimates A0. Use of standard quantum phase estimation to estimate A0 along with difficulties with the approach within the context of quantum communications is described with respect to FIG. 2.

[0041] Given the quantum eigenvalue equationa goal of quantum channel phase estimation is to estimate the unknown parameter 6.Without limitation to communicating binary states <a,|, Alice generates a general quantum state, ip, which is an eigen vector of U0. In this example, it is assumed that Bob can perform quantum operations on the output of the quantum channel U0.

[0042] Alice generates this state ip, and transmits it to Bob. Bob receives a quantum state, couples it to a sequence of qubits (H|0))®” 201, applies a sequence of controlled operations of the quantum channel U0, 202 and then applies the inverse quantum Fourier transform (QFT-1) 203 to each respective qubit. After measurement, Bob extracts the magnitude of the phase 6 of Alice's quantum state with high probability.

[0043] As an explanation of the standard quantum phase estimation Bob may perform the following operations at the receiver:1. |<px): After Bob receives Alice's quantum state \ip), Bob couples | at) to a superposition of qubits (H |0))®mand Hadamard gates 201, generating the statewhich can be written as a binary decomposition of xmG {0,l}m!«>,) = (H|0))®” ® |«9m2. |<p2): Next, Bob applies the controlled-unitary gate UQ 202 such that U0(| 0) + |1)) = |0)to each qubit, generating the state3. |<p3^: Next, Bob applies the inverse quantum Fourier transform 203 to extract the phase information|«>3> = <2FT-1|«>2>4. |<p4): Bob measures the m ancilla qubits 204 to estimate the phase of Alice's state \i ) |<p4) = |2™0) ® | >>giving the estimated value6 = —2m ( '6)1where S G {0,l}m.

[0044] FIG. 3 illustrates the results of a Qiskit simulation of the operation of the quantum circuit illustrated in FIG. 2. Graphs depicted in FIG. 3A, FIG. 3B, FIG. 3C, and FIG. 3D illustrate Qiskit simulation results using a corresponding m qubits (m = 3, m = 5, m = 7, and m = 9) for a small phase rotation of A0 = For m = 3 and m = 5 qubits, the estimated value- 13TTof 0 is relatively poor. However, for m = 7 the estimated value is A07= —, which is 1.5%256off from the true value. For m = 9, A0q= -, which is off by 0.5% of the true value.

[0045] The approach to quantum channel phase estimation may provide a relatively accurate estimate of A0; however, the quantum phase estimation considers the controlled operation of U0as indicated. However, in a quantum communication or sensing system, the channel may have uncontrollable aspects (e.g., noise, including distortions or phase errors). Following is a description of general framework of parameter estimation to explainvarious implementations for performing quantum channel phase estimation within the context of quantum communications.

[0046] A goal of parameter estimation within the context of quantum communications is to estimate a parameter 6 of a quantum channel U0(e.g., a unitary quantum channel), so mitigation techniques can be employed at the transmitter or receiver. This is achieved by communicating classical information over a quantum channel using the quantum-classical communication system setup described with respect to FIGS. 1 and 2. To estimate 6, n quantum states | at) are transmitted over the quantum channel and is written asniii iw (?)Then Alice and Bob can employ a variety of parallel or sequential strategies in order to estimate 6. As indicated above and described with respect to FIG. 4, such strategies can be classified into four categories CC, CQ, QC, and QQ. Within the context of quantum-classical communications, this describes the encoding and decoding procedures used at the transmitter and receiver, respectively. In particular, a parallel CC strategy describes a parameter estimation strategy where Alice transmits separable quantum states and Bob employs separable or local measurements. With respect to quantum circuitry, in a CC strategy neither Alice or Bob necessarily employ 2-qubit gates at either the receiver or transmitter. A CQ strategy describes a parameter estimation strategy where Alice transmits separable quantum states, whereas Bob employs a non-local approach at the receiver. A QC strategy describes a strategy where Alice transmits non-separable quantum states (entangled states) to Bob, whereas Bob employs local measurements at the receiver. Finally, a QQ approach is an approach where Alice transmits non-separable quantum states, and Bob employs non-local measurements at the receiver.

[0047] Consider that Alice transmits to Bob a sequence of n bits,where Bob receives after the quantum channel U0. Then, according to equation 2 and simple counting suggests that Alice and Bob share ri symmetric bits, given pre-agreed sequence of bits between Alice and Bob (conceptually similar to the synchronization preamble in classic communication), ri = ncos2A0. Alice and Bob estimate the magnitude of the phase error by solving for A0A0 = arccoswhere n' is the total number of symmetric bits that Alice and Bob share and n is the total number of bits communicated. Then for each instance i that (a st (a, information about the magnitude of the phase error A0 is leaked to Alice and Bob. Moreover, Alice and Bob can use A0 to compensate for the quantum channel by implementing mitigation techniques at the transmitter or receiver. In particular, given A0, Alice can begin transmitting rotated statesor Bob can decode using rotated measurementsboth of which have the overall effect of mitigating the quantum channel Ue, i.e., P(error) = 0 and n' = n. A succinct description of the protocol follows.PARALLEL CC STRATEGY

[0048] 1. Alice generates n training bits {a, encodes them into orthogonal qubits | at) according to the encoding 0 -> |0), 1 — > 11) and transmits the qubits through a quantum channel described by the unitary operator U0.

[0049] 2. Bob receives |and then performs measurements using M, resulting in the sequence of outcomes {d.

[0050] 3. Bob compares his measurement results { with Alice's training bits {a and organizes the results into two categories:n denotes the number of bits which should satisfy the condition at= d((all n qubits); ri denotes the number of bits which actually satisfies the condition at=t(determined by magnitude of A0).

[0051] 4. Given the statistics n and ri, Bob estimates the phase A0 using equation 8

[0052] A description of the QFT and then a parallel CQ strategy for estimating 6 of a unitary quantum channel U0follows.

[0053] The QFT maps an arbitrary quantum state |X) = ^=0xj\j) to another quantumstate- 'nparticular, the QFT maps vectors 502 from one Hilbert space to another according toThis mapping has a matrix representation given byUsing the matrix representation of the QFT, quantum circuitry 501 for a QFT can be implemented as illustrated in FIG. 5 using a sequence of Hadamard gates 503 and controlled rotation gates 504, one or more swap gates 505, and measurement gates 506. As follows, an example parallel CQ strategy is described using the QFT 501 in the receiver to estimate 0. PARALLEL CQ STRATEGY

[0054] 1. Alice generates n training bits {a, encodes them into orthogonal qubits | at, according to the encoding 0 -> |0), 1 — > 11) and transmits the qubits through a quantum channel described by the unitary operator U0.

[0055] 2. Bob receives | at) and processes the qubits using the inverse QFT, QFT-1

[0056] 3. Bob then performs measurements using M, resulting in the sequence of outcomes {d.

[0057] 4. Bob compares his measurement results {d with Alice's training bits {a and organizes the results into two categories:n denotes the number of transmitted qubitsri denotes the number of correct bits.

[0058] 5. Given the statistics n and ri, Bob estimates the phase A0 using equation 8.

[0059] The following provides a general overview of a parallel and sequential CQ strategy. In a parallel and sequential CQ strategy, Bob directs outputs of the quantum channel at different processing paths at the receiver.PARALLEL & SEQUENTIAL CQ STRATEGY

[0060] 1. Alice generates n training bits Aencodes them into orthogonal qubits according to the encoding 0 -> |0), 1 -> |1), generating the state

[0061] 2. Alice transmits |,4) through the quantum channel

[0062] 3. Bob receives | A') routes the qubits accordingly, and processes the qubits using QF.

[0063] 4. Bob then performs measurements using M, resulting in the sequence of outcomes?!' = {di}i=1 n.

[0064] 5. If f? where fc 6 Z is an integer, then the output of the circuit will exactlygive 6 = 6

[0065] 6. However if 6 then a more strategic approach must be taken. In particular,accurately estimating 6 «requires a suitable QFTk. For example, to estimate # ~ “, requires k > 3 (QFT3). Given a value # ~ QFT^1, and n qubits, a weighted sum of theoverall statistics Atis used to estimate 6« = 2=,g)g) <14> where niis the number of qubits measured to be the integer St, and n is the total number of qubits transmitted.

[0066] Table 1 provides results of an evaluation of the performance of the above strategies to estimate a quantum channel U0with A6 =for n = 1000 qubits. The parallel and sequential CQ strategy outperforms the other strategies for the same number of qubits n.Table 1: Results of estimating A0 using different parameter estimation strategies

[0067] However, the CQ parameter estimations strategies at the receiver use 2-qubits gates. This may be difficult, particularly in an optical quantum communication system.

[0068] FIG. 6 illustrates an example implementation of QC circuitry 601 that implements a semi-classical QFT (SQFT). In some example, s the SQFT may effectively convert a CQprotocol to a CC protocol by removing the necessity for 2-qubit gates while retaining the advantage that CQ strategies provide.

[0069] FIG. 6 illustrates an example implementation of QC circuitry 601. For example, circuitry 601 may comprise a quantum communications receiver 601, such as an optical quantum receiver. As illustrated, circuitry 601 may be connected to a quantum transmitter 602 via a quantum channel 604. For example, circuitry 601 ("Bob") may aquire a phase offset by using quantum-classical processing at the receiver 601. In some examples, receiver 601 may include a switch 605 to direct the quantum channel 604 output toward distinct processing paths 606, 607, 608, 609 at the receiver 601. For example, switch 605 may support the receiver 601 performing a semiclassical QFT, which may provide an improvement in the efficiency of estimating 6 compared to prior approaches.

[0070] In summary, a transmitter 602 ("Alice") prepares a quantum state |i4) using a quantum encoder 603 to generate a quantum state including sequence of encoded bits for communication. This state is then transmitted through the quantum channel U0604 to Bob 601, who receives the resulting quantum state | A'), which may have suffered errors during transmission.

[0071] The circuitry 601 may be configured to perform a parallel and sequential technique for performing an SQFT. To process the received state, the receiver 601 may include a switch 605 to direct the outputs of the quantum channel 604 into parallel processing paths 606, 607, 608, 609. As illustrated, the state |a0) of the overall state |i4') is routed to path 0 606 of the S — QFTk, while | at) is routed to path 1 607, | a'2) is routed to path 2608, etc... and |n) is routed to path n 609. Subsequently, Hadamard 610, 613, 617, 622 and classically controlled rotation gates 612, 615, 616, 619,..., 620, 621, 622, 623 are applied sequentially to each of these paths 606, 607, 608,..., 609, followed by measurements 611, 614, 618,..., 623 in the computational basis. The outcomes 624, 625, 626,..., 627 of these measurements, denoted as {a0„ al..., anl}, are further processed to reveal the phase offset A0.

[0072] Referring to the QFT described above (e.g., as described with respect Figure 5), the QFT may be implemented a sequence of Hadamard gates (single qubit gates) and controlled rotation gates (2-qubit gates)where 0 = — r.2k

[0073] Because the controlled rotation operations commute with the measurements in the computational basis, a circuit 601 can be implemented where the controlled rotations 612, 615, 616, 619, 620, 621, 622 are not dependent on the value of the previous qubit, but rather the measurement 611, 614, 617 result of the previous qubit 606, 607, 608. This may support an implementation of controlled rotation gates 612, 615, 616, 619, 620, 621, 622, which use a control-bit rather than a control-qubit, implementing controlled rotation gates as single-qubit gates 612, 615, 616, 619, 620, 621, 622.

[0074] The classically controlled rotation gates 612, 615, 616, 619, 620, 621, 622 may emulate the behavior of 2-qubit logic gates, drawing upon the analogy between Quantum Fourier Transform (QFT) and Semi-classical Quantum Fourier Transform (SQFT). Such implementations may provide a more general quantum-classical processing paradigm while retaining the advantages of quantum mechanics. By substituting the two-qubit gates with classically controlled qubit gates, the disclosed technology may effectively transform a classical-"quantum" (CQ) estimation strategy into a classical-"classical" (CC) estimation implementation, preserving the benefits of quantum mechanics without the need of using 2-qubit gates. Accordingly, a receiver 601 may estimate 6 without relying on 2-qubit gates, which may provide a low complexity implementation in quantum optical processing.SIMULATION AND PERFORMANCE EVALUATION

[0075] The following section describes results of various Qiskit simulations of various above-discussed strategies and presents an overall performance evaluation of the simulation results. As a particular example, Alice transmits to Bob the state | (p) = 0^=1(H |0)) i through the quantum channel U0(A0 =A0 = ^). Then Bob receives the erroneous stateFor the QC strategy, Bob processes the qubits using SQFT and estimates 6 using Equation 8. This process is simulated in Qiskit as illustrated in Figure 7.

[0076] Now, rather than directly processing the qubit using the SQFT as before, Bob instead applies a sequence of switches and then applies SQFT as described in with respectto FIG. 6. Therefore, following this procedure, if Alice transmits n qubits, Bob should expect the output of the quantum circuit to be a particular integer value Stfor all n qubits (n = n) if 0 = then Bob performs estimation by using the statistics generated by themeasurement outcomes and performs a weighted sum as described in equation 14. This process is simulated in Qiskit as described in Figure 8.

[0077] FIGs. 9A-B present simulation performance results of estimating A0 using equation 8 and using equation 14. As illustrated, estimating the phase error A0 = and A0 =using both the CC strategy 901, the CQ strategy 902, and the semi-CQ strategy 903, and suggests that the semi-CQ (e.g., using the SQFT) that uses classical processing outperforms other methods in estimating the parameters of the quantum channel.EXAMPLE IMPLEMENTATIONS

[0078] The following section describes various example implementations comprising example quantum optical componentry.

[0079] FIG. 10 illustrates an example quantum transmitter 1000. In this example, transmitter 1000 comprises a photon source 1001 such as a laser. Transmitter 1000 further comprises one or more filters 1003, such as a neutral density filter. Transmitter 1000 further comprises a lens 1005. The example transmitter further comprises a polarizer 1007, such as a linear polarizer. In some examples, coherent light 1002 emitted by laser 1001 is passed through one or more filters to produce a weak coherent pulse (WCP) 1004. WCP 1004 may be conditioned by lens 1005 (e.g., collimated). The WCP 1006 may then be polarized using linear polarizer 1007 to generate a well-defined polarized single photon quantum state. Although illustrated as free-space components, in further implementations may be implemented via a semiconductor photonic platform. For example, laser 1001 may comprise a solid-state laser coupled to a waveguide to emit laser light 1002. In this example, filter 1003, lens 1005, and polarizer 1007 may similarly comprise solid-state semiconductor photonic device. In various examples, emitted photon states 1008 may be transmitted via any suitable quantum channel, such as free-space, an optical fiber, a semiconductor photonic waveguide, etc.

[0080] FIG. 11 illustrates example optical quantum gates 1101, 1102. For example, an optical Hadamard gate 1101 may comprise a half-wave plate (HWP) oriented at 22.5°. As another example, Hadamard gate 1101 may comprise a suitably coupled beam splitter andmirror, a universal quantum gate configured as a Hadamard gate, etc. Optical quantum gates may further include a classically controlled rotation gate 1102. For example, rotation gate 1102 may comprise an optical modulator 1104 coupled to a voltage input 1103. In this example, depending on applied voltage, optical modulator 1104 may modulate incoming photons or pass photons without modulation. For example, optical modulator 1104 may comprise an electro-optical modulator, a MEMS (micro-electromechanical system) voltage-controlled optical modulator, a thermo-optic modulator, etc.

[0081] FIG. 12 illustrates example measurement gate components 1201, 1202. For example, in a polarization-encoded quantum communication system, measurement gates may measure the polarization state of qubits (e.g., single photons). As an example, a polarization beam splitter 1201 may direct photons based on their polarization states, and photodetectors 1202 may be arranged to detect photons. For example, a photodetector 1202 may comprise a photon detector, such as a single-photon avalanche diode (SPAD), or other single-photon photon detector. Photodetector 1202 may further comprise a voltage output 1203 which carries a voltage signal induced by detection of a photon.

[0082] FIGS. 13A and 13B illustrate an example of a portion of quantum circuitry 1300 for to implement a 2-qubit SQFT. FIG. 13A illustrates quantum circuitry 1300 and FIG. 13B illustrates example components to implement the circuitry 1300.

[0083] Referring to FIG. 13A, in this example, circuitry 1300 comprises a pair of quantum channels 1304, 1305 to carry two corresponding polarization encoded qubits. For example, quantum channels 1304, 1305 may comprise free-space, optical fibers, semiconductor waveguides, etc. In this example, a Hadamard gate 1302 performs a superposition operation on a qubit carried on channel 1304. A polarization of the output state of Hadamard gate 1302 is measured at measurement gate 1301, outputting a measurement on a classical channel 1303, for example as a voltage carried on a wire. The measurement channel 1303 is coupled to the control input of a controlled rotation gate 1307.

[0084] In this example, controlled rotation gate 1307 performs a rotation on a qubit carried on quantum channel 1305 depending on the control input resulting from measurement gate 1301. The output state of rotation gate 1308 is coupled to a second Hadamard gate 1308. Second Hadamard gate 1308 may perform a superposition operation on the output state of controlled rotation gate 1308. The output of Hadamard gate 1308may be measured at measurement gate 1309, to provide a second classical measurement 1306 (e.g., a second voltage signal).

[0085] Referring to FIG. 13B, Hadamard gates 1302, 1308 may be implemented as halfwave plates as indicated above. For example, Hadamard gate 1302 may comprise a halfwave plate 1310 to implement a 22.5° (i.e., - radians). Hadamard gate 1308 may comprise a 8similar half wave plate (not illustrated).

[0086] In some examples, controlled rotation gate 1307 may be implemented via a decision circuit 1311 to generate an output voltage 1312 depending on a measurement result. For example, decision circuit 1311 may comprise a voltage source and a voltage control, such as a processing device and memory to implement a decision program, an application-specific integrated circuit (ASIC), suitable configured field-programmable-gate array (FPGA), etc. Controlled rotation gate 1307 may further comprise an optical modulator 1313 (e.g., an electro-optic modulator) to perform a rotation controlled by voltage input 1312.

[0087] In some example, measurement gates 1301, 1309 may be implemented via polarization beam splitters and photon detectors as described above. For example, measurement gate 1301 may comprise a polarization beam splitter 1314 pass a photon or reflect a photon based on its polarization. A first photodetector 1315 may be coupled to detect a photon in a first polarization state (e.g., | «— >) ). Similarly, a second photodetector 1316 may be coupled to detect a photon in a second polarization state (e.g., | I)).Measurement gate 1309 may be implemented via similar components (not pictured).

[0088] FIG. 14 illustrates an example 2-qubit SQFT circuit 1400. In this example, a laser driver 1412 is coupled to a pair of quantum encoders 1401, 1412. As an example, a horizontally polarized laser 1401, 1412 is heavily attenuated (e.g., via NDFs), generating individual photons with well-defined polarization states 1402, 1413 (both assumed to be | «— >) for purposes of explanation). A first quantum channel comprises a quarter wave plate 1407 arranged to couple the first state to the first channel, producing the coupled state | 1403. Similarly, a second quantum channel comprises a second quarter wave plate 1417 arranged to couple the second quantum state 1413 to the second quantum channel, producing coupled state |1414.

[0089] As illustrated, in the first channel, Hadamard gate 1404 performs a superposition operation on |1403, the output of which is measured via measurement gate 1405. For example, a polarization beam splitter 1406 may direct a horizontal state | «— >) to photon detector 1407 producing output voltage Doand a vertical polarization state | I) to a second detector 1409 producing output voltage Di. The detected output is provided to a decision circuit 1410 to generate ac control voltage 1411 corresponding to the detected state. For example, may determine a Boolean k = 0 if Di =1 (e.g., Do=O) and k= 1 if Di=0 (e.g., Do=l) (where '1' represents the detection of a photon and '0' represent no detection. The output voltage 1411 may depend on k. For example, the voltage V may equal 0 if k=0 (where '0' represents a voltage corresponding to no rotation, which may be 0V in some implementations or may be another voltage in other implementations) and the voltage V may equal a voltage Vnto implement a rotation of n.

[0090] As illustrated, in the second channel, an optical modulator 1420 receives control voltage 1411 and either performs a rotation of n if V = Vnor performs no rotation if =0. A second Hadamard gate 1416 is applied to the output state of optical modulator 1420 to perform a superposition operation. A measurement gate 1417 is coupled to Hadamard gate 1416 to measure a polarization state of the Hadamard output state. For example, a second beam splitter 1418 may direct a horizontal polarized state | «— >) to a third detector 1419 to produce a detection voltage D2 and a vertical polarization state | I) to a fourth detector 1419 producing output voltage D2. As described above, the measurements produced by measurement gates 1405, 1417 may be used to estimate a phase error of the channel acting on input states 1402, 1414.

[0091] FIG. 15 illustrates an example portion of a quantum circuit 1500 for a 3-qubit SQFT. In this example, an input quantum state is coupled to corresponding quantum channels 1505, 1507, 1508.

[0092] In a first channel 1505, a superposition operation is performed on the input state by a first Hadamard gate 1501. The output state is measured by a measurement gate 1502 and the resulting measurement is used to control 1503, 1504 corresponding rotation gates 1509, 1512 of the second 1507 and third channel 1508.

[0093] In a second channel 1507, a first controlled rotation operation is performed by controlled rotation gate 1509. A second Hadamard gate 1510 is coupled to controlled rotation gate 1509 to perform a superposition on the output state of controlled rotationgate 1509. A measurement gate 1511 is coupled to third Hadamard gate 1511 to measure a polarization of an output state of the second Hadamard gate 1510. The output of the measurement gate is coupled to control a controlled rotation gate 1513.

[0094] As illustrated a third channel 1508 comprises a second controlled rotation gate 1512 coupled to first measurement gate 1502 to perform a controlled rotation operation 1512 based on the output 1503. Third channel 1508 further comprises a third rotation gate 1513 coupled to measurement gate 1511 to perform a controlled rotation operation on the output state of controlled rotation gate 1512 based on the output 1506. Channel 1508 may further comprise a third Hadamard gate coupled to controlled rotation gate 1512 to perform a superposition operation on the output state of controlled rotation gate 1513. As illustrated, third channel 1508 further comprises a third measurement gate 1515 to measure an output polarization of third Hadamard gate 1514. As an example, measurements from measurement gates 1502, 1511, 1515 may be further processed as described above to estimate a phase error of a quantum input channel.

[0095] FIG. 16 illustrate an example implementation of a 3-qubit SQFT. In this example, a transmitter system 1601 may comprise a plurality of three single-photon polarization-encoded transmitters 1610, 1621, 1633. For example, transmitters 1610, 1621, 1633 may emit corresponding quantum states 1602, 1614, 1626 as polarized single photon quantum states. A corresponding plurality of quarter wave plates 1606, 1619, 1628 couple the incoming states 1602, 1614, 1626 to corresponding quantum channels, producing coupled states 1603, 1615, 1625.

[0096] As illustrated, a first quantum channel may be implemented as described with respect to the first channel described with respect to FIG. 14. For example, the first channel may comprise a Hadamard gate 1604 coupled to input coupling gate 1606 to perform a superposition operation on input coupled state 1603. A measurement gate may be coupled to Hadamard gate 1605 comprising a polarization beam splitter 1605 and a pair of photo detectors 1607, 1611 may measure the output polarization state of Hadamard gate 1604. In this example, the outputs of detectors 1607, 1611 are input to a decision circuit 1613 as inputs no and ni, respectively. Decision circuit 1613 may produce a control signal for a signal generator 1608 (e.g., a voltage source) based on inputs noand ni to generate a first voltage V to implement a controlled rotation operation. For example, circuit 1600 may comprise a signal conditioner 1609, such as a high-voltage amplifier to amplify the first voltage V orotherwise conform to the control input of a modulator 1620 to generate a control voltage Vi.

[0097] In the illustrated example, a second quantum channel may comprise a first controlled rotation gate 1620 (e.g., an optical modulator) coupled to input coupler 1619 to perform a controlled rotation operation 1620 on qubit 1615 according to the control voltage Vi. For example, if ki represents the detection output of detectors 1607, 1611:control voltage V; may comprise:> f 0 if k = 01 / 1if k, = 1'where Vi=0 represents a control voltage to perform no rotation on state 1615 and Vi=n represents a control voltage to perform a n rotation.

[0098] In this example, the second quantum channel may further comprise a second Hadamard gate 1616 coupled to controlled rotation gate 1620 to perform a superposition operation on the output state of gate 1620. The second channel may further comprise a measurement gate coupled to Hadamard gate 1616 (e.g., a polarization beam splitter 1617 coupled to a pair of photon detectors 1618, 1622) to measure a polarization state of the Hadamard gate output state. Photon detectors 1618, 1622 may be coupled to decision circuit 1613 to input the measured result as r?2, 03, respectively.

[0099] In the illustrated example, a third quantum channel may comprise a second controlled rotation gate 1630 coupled to input 1628. For example, controlled rotation gate 1630 may perform a controlled rotation based on the measured output of measurement gate 1605,1607, 1611. For example, decision circuit 1613 may generate a control signal based on the measured output, signal generator 1608 may generate a control voltage that is amplified by amplifier 1609 to produce a control voltage V21623.

[0100] As illustrated, the third quantum channel may comprise a third controlled rotation gate 1631 coupled to the second controlled rotation gate 1630. For example, the third controlled rotation gate 1631 may perform a controlled rotation based on the measured output of the second measurement gate 1617, 1618, 1622. For instance, decision circuit 1613 may generate a control signal for signal generator 1608 based on inputs r?2, 03. In this example, signal generator 1608 may generate a third control voltage that is amplified by amplifier 1609 to implement a corresponding controlled rotation.

[0101] As illustrated, the third quantum channel may further comprise a third Hadamard gate 1632 coupled to the third controlled rotation gate 1631 to perform a superposition operation on the output state of controlled rotation gate 1631. In this example, the third channel may further comprise a measurement gate 1627, 1629, 1634 coupled to Hadamard gate 1632 to measure the polarization of the Hadamard gate output state. As described above, the three measurements may then be used to estimate a phase offset of the input channel carrying input states 1602, 1614, 1626.

[0102] As indicated above, a 4-qubit, 5-qubit,... n-qubit SQFT circuit may be implemented as described with respect to Fig. 16 with additional channels. Each additional channel may include an additional rotation gate, so that each channel performs rotations based on the measurement results of the preceding channels.

[0103] FIGS. 17 illustrates an example of circuitry 1700 for a serial SQFT. For example, circuitry 1700 may comprise circuitry a serial 2-qubit SQFT. In this example, a quantum transmitter 1703 may transmit quantum states 1705 in a serial manner over a quantum input channel.

[0104] At a first iteration (t=l), a polarized single photon quantum state 1705 is coupled to a quantum channel via a coupler 1710 (e.g, a quarter wave plate) to produce a coupled quantum state 1706. SQFT circuit 1700 further comprises a controlled rotation gate 1711 controlled by a control input 1702. During the first iteration, the rotation gate 1711 may be controlled to not perform any rotation (e.g., Vi=0 for t=l). The quantum circuitry 1700 may further comprise a half wave plate 1709 (e.g., oriented at 0 = -) to perform a Hadamard 8superposition operation on the output state of rotation gate 1711. In the first iteration, the output state of the rotation gate 1711 may be equal or approximately equal to the coupled input state 1706. Accordingly, at the first iteration, the quantum circuitry 1700 may operate as described with respect to the first channel of quantum circuitry 1400 or 1600 as described with respect to FIGS. 14, 16. Quantum circuitry 1700 may further comprise a measurement gate 1707 coupled to Hadamard gate 1709 to produce a measurement result ki.

[0105] At a second iteration (t=2), controlled rotation gate 1711 may perform a controlled rotation on a second coupled input state 1706 based on the measurement results ki of the previous iteration. For example, a decision circuit 1704 may generate a voltage V that isamplified to produce a control voltage Vi on control line 1702. For example, Vi may be generated as:> (0 if Do= 11(1 if D1= rThe output state of the controlled rotation gate 1711 at the second iteration may be superimposed via Hadamard gate 1709 and measured via measurement gate 1707.Accordingly, the second iteration, quantum circuitry 1700 may perform the operations described with respect to the second channel of quantum circuitry 1400.

[0106] In some examples, these iterations may be repeated to measure a set of n qubits, where at odd iterations (e.g., iteration t=3, t=5, etc.) no rotation is applied at controlled rotation gate 1711 and at even rotations (e.g., iteration t=4, t=6, etc.), a rotation is applied based on the measurement results of the previous iteration (e.g., iteration 4 depends on iteration 3, iteration 6 depends on iteration 5, etc.). Accordingly, in this manner, a full set of n states may be measured by conducting a corresponding number of iterations. For example, in a 2-qubit SQFT implementation, n transmitted qubits may be coupled to n / 2 pairs of qubits 1705 (e.g., operated on via repetitions of two such iterations), where the first of the pair is not rotated and the second of the pair is rotated in a controlled manner.

[0107] FIG. 18 illustrates an example of quantum circuitry 1800 to implement a 3-qubit SQFT. In this example, quantum circuitry 1800 may be implemented as described with respect to quantum circuitry 1700. For instance, a quantum transmitter 1803 may comprise a single-photon polarized state source to produce a state 1805 as described with respect to transmitter 1703 and state 1705. The quantum state 1705 may be coupled to a quantum channel via a coupler 1810 to produce coupled quantum states 1806, as described with respect to coupler 1710. Similarly, quantum circuitry 1800 may comprise a classically controlled rotation gate 1811 as describe with respect to controlled rotation gate 1711, a Hadamard gate 1809 as described with respect to Hadamard gate 1809, and a measurement gate 1807.

[0108] In this example, quantum circuitry 1800 may further comprise a decision circuit 1804 coupled to a voltage source 1801 and an amplifier 1808. Compared to decision circuit 1704, decision circuit 1804 may generate control signals corresponding to a first, second, orthird iteration. For example, at a first iteration, decision circuitry 1804 may generate a null control signal such that controlled gate 1811 does not rotate the state 1806. In this example, at a second iteration, decision circuitry 1804 may generate a control signal for a first voltage Vi to implement a first rotation controlled by measurement results of the first iteration. Similarly, at a third iteration, decision circuitry 1805 may generate a control signal for a second voltage V2 to implement a second rotation controlled by measurement results of the second iteration.

[0109] As an example, at a first iteration (t=l), a measurement (without a rotation) may be performed to give a measurement result:> (0 if Do= 11(1 if D1= r

[0110] At a second iteration (t=2), a rotation gate may be applied via a control voltage Vi as a function of ki> f 0 if k = 01 / 1if k, = rFollowing the controlled rotation, a second measurement is performed, giving result k.(0 if Do= 12ll if D1= r

[0111] At a third iteration (t=3), a rotation gate may be applied via a control voltage V2 as a function of (ki, kf)Following the controlled rotation, a third measurement k3 may be performed. As described above with respect to FIG. 17, these iterations may be repeated to obtain n measurements. For example, when t=0 (mod 3), no controlled rotation is applied, when t=l (mod 3), a controlled rotation based on the previous measurement is applied, and when t=2 (mod 3) a controlled rotation based on the previous two measurements is applied. Subsequently, the n measurements may be used to estimate a phase offset of the input quantum channel as described above.

[0112] FIG. 19 illustrates example quantum circuitry 1900 for a general / V-qu bit SQFT. In this example, a transmitter 1903 may generate a serial quantum state 1905 that isiteratively coupled to quantum circuitry 1900 as states 1906. For example, circuitry 1900 may comprise a coupler 1910 as described above. Quantum circuitry 1900 may further comprise a controlled rotation gate 1911 (e.g., optical modulator), a Hadamard gate 1909 (e.g., half wave plate), and a measurement gate 1907 (e.g., a beam splitter and pair of detectors). Further, quantum circuitry 1900 may comprise a voltage source 1901 and an amplifier 1908 to generate control voltages Vi in response to control signals from decision circuit 1904.

[0113] In this example, circuitry 1900 may further comprise a decision circuit 1904 to generate control signals at an iteration that depends on the measurement results of all preceding iterations. For example, at a first iteration (t=l), the control voltage 1902 may be set to perform no rotation (e.g., V=0), and a measurement may be made to give ki:

[0114] Continuing the example, at a second iteration t=2, a voltage Vi may be applied to controlled rotation gate 1911:> f 0 if k = 01 / 1if k, = 1.Following the controlled rotation, a second measurement k2 may be made:_ (Q if D0= l2(1 if D =

[0115] In this example, iterations may be continued in this manner to obtain measurement results:> (0 if Do= 1n~ [lifD1= l.

[0116] Similarly, the control voltage at each iteration may depend on the previous measurement results, where at t=n:Accordingly, quantum circuitry 1900 may produce n measurements that may be used to estimate a phase offset as described above.

[0117] FIG. 20 shows a block diagram illustrating a quantum receiver 2000 comprising a SQFT quantum circuit 2002 and a control unit 2003. For example, quantum receiver 2000 may provide an example implementation of any quantum receiver described herein.

[0118] In this example, quantum receiver 2000 may comprise a quantum state input 2001. Quantum state input 2001 may couple quantum states received from a quantum state transmitter 2007 to one or more quantum channels. For example, in a parallel SQFT implementation, quantum state input 2001 may comprise a switch and a plurality of couplers to couple an input state to multiple parallel qubits. For example, in an optical implementation, quantum state input 2001 may comprise an optical switch and a coupling wave plate (e.g., a quarter wave plate). As another example, in a sequential SQFT, quantum state input 2001 may comprise a coupler to couple an input state to sequences of qubits on a quantum channel. In some cases, quantum state input 2001 may comprise various other components, such as an input buffer, etc.

[0119] Continuing with the example, quantum receiver 2000 may comprise an SQFT quantum circuit 2002. For example, SQFT circuit 2002 may comprise a parallel SQFT circuit, such as described above (e.g., with respect to FIGS. 6, 13A, 14, 15, 16). As another example, SQFT circuit 2002 may comprise a sequential SQFT circuit, such as described above (e.g., with respect to FIGS. 17, 18, 19). Of course, these are simply examples, and SQFT quantum circuit 2002 may be implemented in any suitable manner. For example, SQFT quantum circuit 2002 may comprise various universal quantum gates to implement the gate operations described above. As another example, SQFT quantum circuit 2002 may comprise a parallel and sequential circuit. For instance, a parallel 2-SQFT or 3-SQFT circuit as described with respect to FIGS. 14, 16 may comprise decision and control circuitry to operate in a sequential manner as described with respect to FIGS. 17, 18, 19. For example, a k-parallel n-sequential SQFT may comprise k parallel quantum channels to operate on batches of k qubits for at least n / k repetitions.

[0120] In some examples, circuitry 2000 may further comprise a receiver control unit 2003. Receiver control unit 2003 may include various componentry to control receiver 2000. For example, receiver control unit 2003 may comprise a processor 2004. In some embodiments, the processor 2004 can be any suitable hardware processor or combinationof processors, such as a central processing unit (CPU), a graphics processing unit (GPU), an application specific integrated circuit (ASIC), a field-programmable gate array (FPGA), a digital signal processor (DSP), a microcontroller (MCU), etc.

[0121] In further examples, receiver control unit 2003 may include a memory 2005.Memory 2005 can include any suitable storage device or devices that can be used to store suitable data and instructions that can be used, for example, by the processor 2004 to control operations of receiver 2000, receive commands, transmit measurements, etc. The memory 2005 can include any suitable volatile memory, non-volatile memory, storage, or any suitable combination thereof. For example, memory 2005 can include random access memory (RAM), read-only memory (ROM), flash memory, field programmable unit (FPU), electronically erasable programmable read-only memory (EEPROM), storage such as solid state or hard disk drives, etc.

[0122] In further examples, receiver control unit 2003 may include a communications system 2006. Communications system 2006 can include any suitable hardware, firmware, or software for communicating information over a communication network. For example, communications system 2006 can include one or more transceivers, one or more communication chips and / or chip sets, etc. In a more particular example, communications system 2006 can include hardware, firmware or software that can be used to establish a WiFi connection, a Bluetooth connection, a cellular connection, etc. In further examples, communications system 2006 may include hardware, firmware, or software to communicate with other circuit components. For example, communications system 2006 may comprise an I2C, I3C, or JTAG interface, or internal communication interface.

[0123] In further examples, receiver control unit 2003 may comprise a user interface. For example, receiver control unit 2003 may comprise a port for a wired connection to a user device (e.g., a USB port for a laptop or mobile device connection). As a further example, receiver control unit 2003 may comprise a graphical display screen, a button input interface, indicator lights, or other user interface components.

[0124] In some implementations, the processor 2004 can execute at least a portion of the methods described above. For example, processor 2004 may execute instructions stored in memory 2005 to calculate a phase offset estimate based on measurement results received from SQFT circuit 2002. As another example, processor 2004 may control operations of SQFT 2002. For instance, processor 2004 may implement a decision circuit as describedabove. In further implementations, processor 2004 may execute operations in conjunction with other connected systems. For example, processor 2004 may use communication system 2006 to transmit data to a remote location (e.g., a remote server or operator's laptop) for operations to be performed remotely. For instance, processor 2004 may use communication system 2006 to transmit data to a remote computer that performs phase estimation calculations.

[0125] The present technology may be embodied on various computing platforms that perform actions responsive to software-based instructions.

[0126] A computer readable medium or a memory may be a computer readable signal medium or a computer readable storage medium. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer readable storage medium would include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any non-transitory, tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device.

[0127] A computer readable signal medium may include a propagated data signal with computer readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof. A computer readable signal medium may be any computer readable medium that is not a computer readable storage medium and that can communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device.

[0128] Program code and software instructions stored on a computer readable medium or memory may be stored and retrieved using any appropriate medium, including but not limited to wireless, wire-line, optical fiber cable, radio frequency, etc., or any suitable combination of the foregoing. Computer program code for carrying out operations foraspects of the present disclosure may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, C#, C++, VHDL, or the like and conventional procedural programming languages, such as the "C" programming language or similar programming languages.

[0129] Aspects of the present technology are described above with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the technology. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0130] These computer program instructions may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium produce an article of manufacture including instructions which implement the function / act specified in the flowchart and / or block diagram block or blocks.

[0131] The computer program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices (such as through an application programming interface) to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0132] In the foregoing specification, implementations of the disclosure have been described with reference to specific example implementations thereof. It will be evident that various modifications may be made thereto without departing from the broader spirit and scope of implementations of the disclosure as set forth in the following claims. Thespecification and drawings are, accordingly, to be regarded in an illustrative sense rather than a restrictive sense.

Claims

CLAIMSWhat is claimed is:

1. A device, comprising:a coupler coupled to an input quantum channel to receive a quantum state, the quantum state comprising a plurality of quantum encoded bit values;a first channel comprising:a first Hadamard gate to apply a first superposition operation to a first qubit of the plurality of qubits; anda first measurement gate to measure the first Hadamard gate output state to determine a first bit value corresponding to the first qubit;a second channel comprising:a first controlled rotation gate controlled by the first bit value to apply a first controlled rotation operation to a second qubit of the plurality of qubits;a second Hadamard gate to apply a second superposition operation to the first controlled rotation gate output state; anda second measurement gate to measure the second Hadamard gate output state to determine a second bit value corresponding to the second qubit; and a phase estimator to estimate a phase error of the quantum channel based on the measured bit values.

2. The device of claim 1, further comprising:a third channel comprising:a second controlled rotation gate controlled by the first bit value to apply a second controlled rotation operation to a third qubit of the plurality of qubits; a third controlled rotation gate controlled by the second bit value to apply a third controlled rotation operation to the third qubit;a third Hadamard gate to apply a third superposition operation to the third qubit after the second and third controlled rotation operations; anda third measurement gate to measure the third Hadamard gate output state to determine a third bit value corresponding to the second third.

3. The device of claim 1, further comprising:a third channel comprising:a second controlled rotation gate controlled by the first bit value and thesecond bit value to apply a second controlled rotation operation to a third qubit of the plurality of qubits;a third Hadamard gate to apply a third superposition operation to the second controlled rotation gate output state; anda third measurement gate to measure the third Hadamard gate output state to determine a third bit value corresponding to the third qubit.

4. The device of claim 1, wherein:the first Hadamard gate comprises a first half-wave plate oriented to perform the first superposition operation; andthe second Hadamard gate comprises a second half-wave plate oriented to perform the second superposition operation.

5. The device of claim 1, wherein the first controlled rotation gate comprises a first modulator controlled by a first control voltage corresponding to the first bit value.

6. The device of claim 1, wherein the first measurement gate comprises:a halfwave plate;a first photodetector to measure a first polarity; anda second photodetector to measure a second polarity.

7. The device of claim 1, wherein:the first coupler comprises a first input quarter-wave plate; andthe second channel comprises a second input quarter-wave plate.

8. A method, comprising:receiving a quantum state on a quantum channel, the quantum state comprising a plurality of quantum encoded bit values;coupling the quantum state to a plurality of qubits;applying a first superposition operation to a first qubit of the plurality of qubits; measuring the first superposition operation output state to determine a first bit value corresponding to the first qubit;sequentially measuring respective bit values corresponding to respective qubits of the plurality of qubits, comprising:applying a respective controlled rotation operation to a respective qubit based on preceding bit values corresponding to qubits measured prior to the respective qubit;applying a respective superposition operation to the respective controlled rotation operation output state; andmeasuring the respective superposition operation output state to determine the respective bit value corresponding to the respective qubit; andestimating a phase error of the quantum channel based on the measured bit values.

9. The method 8, wherein the number k of the plurality of qubits is less than the number n of quantum encoded bit values.

10. The method of claim 9, further comprising:coupling the quantum state to sets of k qubits at least times;for each set of k qubits, repeating the steps of applying the first superposition operation, measuring the first superposition output state, and sequentially measuring respective bit values.

11. The method of claim 8, wherein the number k of the plurality of qubits is greater than or equal to the number n of quantum encoded bit values.

12. The method of claim 8, wherein estimating a phase error of the quantum channel based on the measured bit values comprising comparing the measured bit values to a set of expected bit values corresponding to the quantum encoded bit values.

13. The method of claim 12, wherein the quantum encoded bit values and the set of expected bit values comprise shared channel estimation bits.

14. The method of claim 8, wherein applying the respective controlled rotation operation comprises, for each preceding bit value, applying a controlled rotation gate controlled by the preceding bit value.

15. The method of claim 8, wherein applying the respective controlled rotation operation comprises applying a controlled rotation gate controlled by a combination of the preceding bit values.

16. The method of claim 8, wherein:applying the first superposition operation comprises applying a first Hadamard gate to the first qubit; andapplying the respective superposition operation to the respective controlled rotation operation output state comprises applying a respective Hadamard gate to the respectivecontrolled rotation operation output state.

17. A method, comprising:receiving an optical quantum state via an input optical quantum channel; directing the optical quantum state to a plurality of k optical quantum channels to couple the optical quantum state to a corresponding plurality of k polarization-encoded qubits;applying a k-qubit semiclassical Quantum Fourier Transform to the k polarization-encoded qubits to obtain k polarization state measurements; andestimating a phase error of the input optical quantum channel based on the k polarization state measurements.

18. The method of claim 17, wherein the received optical quantum state encodes at least n bits, where n>k, and further comprising:repeating the steps of directing the optical quantum state and applying the k-q u bit semiclassical Quantum Fourier Transform to obtain n polarization state measurements; and estimating the phase error of the quantum optical interconnect based on the n polarization state measurements.

19. The method of claim 18, wherein:the optical quantum state encodes n transmitted bits; andestimating the phase error comprises comparing n received bits corresponding to the n polarization state measurements to the n transmitted bits.

20. The method of claim 19, further comprising estimating the phase error aswhere 5 / are integers and niis the number of qubits measured tobe the integer St.

21. A device, comprising:a quantum state input;a controlled rotation gate coupled to the quantum state input;a Hadamard gate coupled to the controlled rotation gate;a measurement gate coupled to the Hadamard gate; anda decision circuit coupled to the measurement gate and a control input of the controlled rotation gate; wherein:the decision circuit is to implement a plurality of controlled rotation operations via thecontrolled rotation gate on a corresponding plurality of qubits obtained via the quantum state input;the decision circuit is to control the controlled rotation gate to not rotate a first qubit; andthe decision circuit is to control the controlled rotation gate to rotate a second qubit based on a measurement result of the first qubit.

22. The device of claim 21, wherein:the decision circuit is to control the controlled rotation gate to not rotate a third qubit; andthe decision circuit is to control the controlled rotation gate to rotate a fourth qubit based on measurement results of the third qubit.

23. The device of claim 21, wherein:the decision circuit is to control the controlled rotation gate to rotate a third qubit based on measurement results of the first qubit and the second qubit; andthe decision circuit is to control the controlled rotation gate to not rotate a fourth qubit following the third qubit.

24. The device of claim 21, wherein:the decision circuit is to control the controlled rotation gate to rotate a plurality of n qubits, where each rotation is based on measurement results of preceding qubits.

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