Scalable discrete fourier transform using logical phi-bits as nonlinear acoustic qubit analogues

Abstract

A method for implementing multiple logical phi-bit large scale unitary operations in an acoustic waveguide array analogous to quantum circuits includes: inducing a π phase jump rotation operation by changing driving conditions, and interpreting detected signals in terms of a multiple phi-bit representation which maps the state of multiple phi-bits in their supporting nonlinear acoustic metastructure and their representations as complex state vectors in exponentially scaling Hilbert spaces. This allows evaluation of a target operation (such as a Fourier Transform) by exploiting how the array simultaneously rotates phases of detected signals in response to changes in driving conditions, and by enforcing an equivalence between a result of a transformation from the target operation and a result of the transformation from the π phase jump rotation operation.

Description

085067-799365 (UA23-226) SCALABLE DISCRETE FOURIER TRANSFORM USING LOGICAL PHI-BITS AS NONLINEAR ACOUSTIC QUBIT ANALOGUES CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This is a PCT application that claims benefit to U.S. provisional application serial number 63 / 507582 filed on June 12, 2023 which is incorporated by reference in its entirety. GOVERNMENT SUPPORT

[0002] This invention was made with government support under Grant No.2204400 awarded by the National Science Foundation. The government has certain rights in the invention. FIELD

[0003] The present disclosure generally relates to quantum-analogue computing, and in particular relates to providing a scalable, large, multiple logical phi-bit quantum analogue system for practical implementation of a scalable discrete Fourier transform. BACKGROUND

[0004] Quantum computing utilizes the properties of quantum superposition and quantum correlation. Quantum superposition can provide the support for encoding massive amounts of information as or using coherent superpositions of states. For example, the coherent superpositions of states can correspond to a composite quantum system constituted of subsystems (e.g., qubits). Quantum correlation can provide the correlation between the qubits and the capability of processing information in a parallel manner. An ultimate goal of quantum computing is to realize large scale unitary operations or gates on multiple qubits.

[0005] It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed. 95391587.2085067-799365 (UA23-226) BRIEF DESCRIPTION OF THE DRAWINGS

[0006] FIG.1 is a schematic illustration of an example experimental nonlinear acoustic waveguide platform for implementation of a multi phi-bit Fourier Transform operation;

[0007] FIGS.2A and 2B are a pair of graphical representations depicting corrected phase information for example phi-bits as functions of the driving frequency of the externally driven array of nonlinearly coupled waveguides (e.g., of the platform of FIG.1);

[0008] FIG.3A is a simplified diagram showing inducing a ^^ phase jump rotation operation using the platform of FIG.1;

[0009] FIG.3B is a simplified diagram showing representation of an input multi phi-bit value that can be used with the platform of FIG.1 and correlates the ^^ phase jump rotation operation of FIG.3A with a Fourier Transform operation;

[0010] FIG.3C is a simplified diagram showing representation of an output multi phi-bit value that correlates the output multi phi-bit value of FIG.3B, the ^^ phase jump rotation operation of FIG.3A and a Fourier Transform operation;

[0011] FIGS.4A and 4B are a pair of process flow diagrams showing a method for implementation of a multi phi-bit Fourier Transform operation using the platform of FIG.1 and the systems shown in FIGS.3A-3C; and

[0012] FIG.5 is a simplified diagram showing an example computing system that may perform some aspects of the method of FIGS.4A and 4B for supporting implementation of the multi phi-bit Fourier Transform operation using the platform of FIG.1.

[0013] Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims.95391587.22085067-799365 (UA23-226) DETAILED DESCRIPTION

[0014] The present disclosure relates to systems and techniques for providing multiple logical phi-bit large scale unitary operations. In particular, the systems and techniques can be used to realize multiple logical phi-bit large scale unitary operations analogous to quantum circuits. Logical phi-bits are nonlinear acoustic analogues of qubits. As used herein, the terms “phi-bits” and “logical phi- bits” may be used interchangeably.

[0015] An N-qubit gate operates on states supported by an exponentially scaling basis of dimension 2N. It is this exponential scaling that can potentially yield the advantage of quantum computing over classical computing. However, the fragility of quantum superpositions of large numbers of qubits is a challenge associated with achieving quantum computing. For instance, quantum superpositions of large numbers of qubits is achieved by using decomposition of large-scale unitary matrices into quantum circuits involving sequences of elementary single- and two-qubit gates, which may form a universal set of gates. An additional challenge is to then determine a minimal quantum circuit capable of realizing the desired large-scale operation.

[0016] The quantum Fourier transform (QFT) is, for example, the discrete Fourier transform implemented on qubits. The QFT can be implemented with a circuit utilizing a single qubit Hadamard gate and a phase gate. The QFT is also an important component of quantum algorithms such as Shor’s algorithm for finding prime factors.

[0017] Described herein are approaches that can be used to map the state of multiple phi-bits (e.g., in a supporting nonlinear acoustic metastructure corresponding to the multiple phi-bits) and their representations as complex state vectors in exponentially scaling Hilbert spaces. For instance, as will be described in greater detail below, based on physically actuating ^^ changes in phi-bit phases, and by engineering appropriate multiple phi-bits representations, the systems and techniques can be used to realize a scalable phi-bit-based quantum Fourier transform (QFT) or another target operation, such as a permutation operation (which can encompass, for example, CNOT or SWAP).

[0018] Aspects of the disclosure provide analogies between superpositions of states of acoustic waves and quantum waves, potentially offering a decoherence-free, acoustic, phase-based computing alternative to quantum systems95391587.23085067-799365 (UA23-226) that may be used for at least some quantum information processing applications. A phi-bit (e.g., logical phi-bit) is a two-state degree of freedom of an acoustic wave, which can be in a coherent superposition of states with complex amplitude coefficients. A phi-bit is a classical acoustic analogue of a qubit. Single phi-bit quantum-like phase and Hadamard gates have been experimentally demonstrated, as has the two phi-bit C-NOT (controlled NOT) gate. These gates are elementary components of a universal set of gates that can be used to generate multiple phi-bit quantum-like circuits.

[0019] There is a need to develop large scale phi-bit gates, which do not need to be decomposed into circuits of elementary gates. In some approaches, a non-trivial three phi-bit unitary transformation can be realized to swap components of a 23vector about a pivot. This operation is scalable with the number of phi-bits. Existing approaches to phi-bit based gates have largely been limited to existence demonstrations.

[0020] An aspect of the present disclosure is to demonstrate that multiple phi-bit large scale unitary operations can be realized by design. As will be described in greater detail below, the systems and techniques disclosed herein can engineer the correspondence between the state of the logical phi-bits in their supporting physical space and a representation in an exponentially scaling Hilbert space that transforms according to the desired gate upon changes in the experimental driving conditions. This approach is applied to the design of scalable multiple phi-bit QFT operations.

[0021] While the examples herein pertain to implementing a Fourier Transform operation by exploiting a physical response to actuating ^^ changes in phi- bit phases, note that a different target operation may be implemented in a similar manner, e.g., by modifying or replacing a unitary matrix ^^ to correlate with the target operation and by identifying a translation matrix ^^ that enforces an equivalence between a result ^^^′^of the transformation from the target operation and a result ^^^^′of the transformation from a ^^ phase jump rotation operation. A target operation may encompass an operation for which a translation matrix ^^ can be found to relate the unitary matrix ^^ to a physical manipulation of the system (e.g., such as the ^^ phase jump rotation operation). Alternative target operations can include, for example, permutation operations such as a CNOT operation or a SWAP operation.95391587.24085067-799365 (UA23-226) Logical Phi-bits

[0022] A logical phi-bit is a two-state degree of freedom in the spectral domain of nonlinear acoustic modes supported by an externally driven array of nonlinearly coupled waveguides. FIG.1 shows an example of a physical platform 100 which includes an array of acoustically coupled waveguides 110 (also referred to herein as “array 110”). In some embodiments, the array of acoustically coupled waveguides 110 is a metastructure comprising three acoustic waveguides (e.g., which can be provided as a quantity of aluminum rods) coupled with epoxy resin. External driving of the array of nonlinearly coupled waveguides is performed by one or more piezoelectric transducers (driving transducers 122a and 122b) attached to the ends of the rods. Each driving transducer 122a or 122b can be actuated by a separate signal generator and / or amplifier, and each respective driving transducer 122a and 122b applies a driving signal having a driving frequency. For instance, a first signal generator / amplifier 126a can be used to excite the first driving transducer 122a and a second signal generator / amplifier 126b can be used to excite the second driving transducer 122b. As illustrated, the first driving transducer 122a can be located or otherwise coupled to a waveguide at the edge of the array 110 and the second driving transducer 122b can be located or otherwise coupled to a waveguide in the middle of the array 110.

[0023] Detecting transducers 132a, 132b, and 132c are attached to the ends of the rods opposite to the driving transducers 122a and 122b. Transmission from the (driving and detecting) transducers to the rods can be enhanced by a thin layer of ultrasonic coupling agent. For instance, in some embodiments a thin layer of honey can serve as the ultrasonic coupling agent. The transducer / couplant / rod assemblies are maintained under constant pressure. For instance, in one approach, constant pressure can be provided by three independent rubber bands.

[0024] The first and second driving waveguides can be excited (e.g., driven by the driving transducers 122a, 122b) with sinusoidal signals with frequencies f1 and f2, respectively. In some embodiments, a detecting transducer 132a, 132b, 132c can be located or otherwise coupled to each rod (e.g., waveguide) at the opposite distal end from the driving transducers. The detecting transducers 132a-c can be connected to an oscilloscope or other data acquisition system 140 which collects the data output by the detecting transducers 132a-c.95391587.25085067-799365 (UA23-226)

[0025] In some embodiments, the waveguide(s), transducer(s), signal generator(s), amplifier(s), and acquisition system assembly described herein can be seen to behave nonlinearly. The origin of the nonlinearity may be mechanical or electrical, although it is noted that the following description can be applied equally or otherwise independent of the origin of the nonlinear behavior. The nonlinearity can enable multiple approaches for mixing the drivers’ frequencies. The measured displacement field at the detection end of the waveguides can be determined as the Fourier sum of a large number of linear and nonlinear modes, each with its own characteristic frequency. For example, the characteristic frequencies of the nonlinear modes can result from the mixing of the drivers’ frequencies.

[0026] The entire array of nonlinearly coupled waveguides is suspended by thin threads for isolation. By exciting the waveguides using different frequencies (e.g., ^^1and ^^2), the resulting displacement field measured at the waveguide’s detection ends is the Fourier sum of modes with the primary frequencies ^^1and ^^2, as well as secondary nonlinear modes. The nonlinear modes have frequencies which are linear combinations of the driving frequencies: ^^ ^^1+ ^^ ^^2, where ^^ and ^^ are integers. Each nonlinear mode represents a logical phi-bit. The displacement field of a logical phi-bit at the end of three waveguides can be referred and renormalized to that of a waveguide at the edge of the array (e.g., referred to as waveguide 1), to obtain a 2 × 1 vector: ^^( ^^, ^^) = (^^̂2 ^^ ^^ ^^ ( ^^ ^^12)^^ ^^ ^^ ^^) ^^ ^^ ^^ ( ^^( ^^ ^^1+ ^^ ^^2) ^^) Eq. (1)

[0027] Here, ^^^^= 2 ^^ ^^^^; ^^ = 1,2; and the phase difference between the displacement fields at waveguides 2 and 3, relative to waveguide 1, are ^^12= ^^2− ^^1and ^^13= ^^3− ^^1(respectively). The phases ^^1, ^^2and ^^3arise from the nature of the resonant amplitudes of the driven system with dissipation as well as any nonlinear effect combining these resonant amplitudes. ^^̂2and ^^̂3are the magnitudes of the acoustic field at the ends of the waveguides 2 and 3, respectively, normalized to the magnitude at the end of waveguide 1. The state of the logical phi- bit { ^^, ^^} can be redefined in terms of phase differences of the displacement field only, for instance by constructing the nontemporal part of the field as the normalized complex amplitude state vector:95391587.26085067-799365 (UA23-226) (^ 1^^ ^^ ^^ ( ^^ ^^12) ^^^, ^^)=√2(^^ ^^ ^^ ( ^^ ^^13)) Eq. (2)

[0028] A singlein this form, spans the Bloch sphere and is analogous to a quantum bit (e.g., qubit). By manipulating the frequency, phase and / or magnitude of the driving signals, the logical phi-bit state vector, ^^( ^^, ^^), can be rotated. Notably, the state of each logical phi-bit (e.g., nonlinear mode) is strongly correlated to all other logical phi-bits through the nonlinearity of the physical system. As such, a single change in the driving conditions of the physical system can result in simultaneous changes in the state vectors of all individual logical phi-bits. The single physical action can subsequently rotate a state of ^^ logical phi-bits, represented in the tensor product space of individual phi-bits. This space scales exponentially as 2^^. A change in the conditions of the physical systems changes all 2^^components of a ^^ phi-bit representation. Logical phi-bit correlations and manipulation of the supporting physical system can therefore be used to operate in a massively parallel manner on the components of exponentially scaling multiple phi-bit state representations.

[0029] Referring to FIGS.2A, 2B, and 3A, in some aspects, experimental observations indicate that changes in the driving conditions of the array of nonlinearly coupled waveguides can lead to ^^ jumps in the logical phi-bit phases ^^12and ^^13. These phase jumps may originate based on nonlinear interactions of driven acoustic waves in the metastructure. The phases as functions of driving conditions can be decomposed into a driver following phase and sharp ^^ jumps, and may be referred to as non-resonant and resonant contributions to the phi-bit phase (respectively).

[0030] Considering a phi-bit { ^^, ^^}, the non-resonant phases can be expressed as linear combinations ^^ ^^12( ^^1) + ^^ ^^12( ^^2) and ^^ ^^13( ^^1) + ^^ ^^13( ^^2) of the phase differences associated with the linear displacement of thewaveguides at the frequencies ^^1and ^^2. That is, these phases can be subtracted for each phi-bit to obtain a corresponding resonant contribution to ^^12and ^^13.

[0031] FIGS.2A and 2B are a pair of graphical representations depicting corrected phase information for example phi-bits as functions of the driving frequency of an externally driven array of nonlinearly coupled waveguides. For example, FIGS.2A and 2B illustrate experimentally measured behavior of the95391587.27085067-799365 (UA23-226) resonant phases for two logical phi-bits. In particular, a first graph 200A of FIG.2A, depicts corrected phases ^^12and ^^13for phi-bit { ^^ = 4, ^^ = −2}, and a second graph 200B of FIG.2B depicts corrected phases ^^12and ^^13for phi-bit { ^^ = 3, ^^ = −1}. The corrected phases ^^12and ^^13are shown as functions of the driving frequency ^^1. Initially, rod 1 is driven at the frequency ^^1= 62 kHz, and rod 2 is driven at ^^2= 66 kHz. ^^1is tuned by increasing its value by increments of 50 Hz.

[0032] In some rods 1 and 2 are driven at two different initial frequencies. The frequency ^^1is then tuned and the phases ^^12and ^^13for each phi-bit are obtained from the complex components of the Fourier spectrum at the ends of the rods. As seen in FIGS.2A and 2B, both phi-bits exhibit simultaneous ^^ jumps in their corresponding phases ^^12and ^^13.

[0033] As contemplated herein, it is this type of simultaneous phase jump that can be used to design single and multiple phi-bit representations that will enable QFT operations. Recall that the QFT for ^^ bits is associated with the unitary matrix: 11 11 ^^ ^^2 1^^3 …… ^^ ^ 1^−1(3)

[0034] Here, ^^ = 2^^and ^^ = ^^^^ ^^. Single Phi-bit Quantum Fourier Transform (QFT)

[0035] With additional reference to FIGS.3B and 3C, consider a variation on the single logical phi-bit analytical representation of Eq. (2) that could be achieved by some unitary transformation in the single phi-bit Hilbert space. A new single logical phi-bit representation can be defined as: ^^ = ^^^^ ^^2 sin ^^1(4)

[0036] Here, ^^1= ^^12⁄2 and ^^2= ^^13⁄2.95391587.28085067-799365 (UA23-226)

[0037] A physical action of the supporting metastructure leading to a ^^ jump in both ^^12and ^^13will produce a rotation of ^^0to a new state ^^0′such that: ^^0′= ^^2×2^^0Eq. (5)

[0038] Here, the^^ 0In obtaining ^^2×2, the following trigonometric relations be sin( ^^ + ^^) =sin ^^ cos ^^ + cos ^^ sin ^^ and cos( ^^ + ^^) = − sin ^^ sin ^^ + cos ^^ .

[0039] A new representation, ^^, can be designed that would transform upon ^^ jumps according to the single phi-bit QFT (or Hadamard gate, shown in Eq. (6)): ^^2=11 1 √2( ) Eq. (6) 1 −1

[0040] This new^^ = ^^2×2^^0= (^^^^11 ^^1221 ^^22) ^^^^ ^^2(sin ^^cos ^^11) Eq. (7)

[0041] When vector ^^ (e.g., given above in Eq. (7)) transforms as: ^^^^′= ^^2×2^^0′= ^^2×2^^2×2^^0Eq. (8)

[0042] Upon the application of the QFT in Eq. (6), the vector ^^ (e.g., of Eq. (7)) transforms according to: ^^^′^= ^^2^^ = ^^2^^2×2^^0Eq. (9)

[0043] A translation matrix ^^2×2can then be found, such that ^^^′^ = ^^^^′(e.g., such that ^^2^^2×2^^0= ^^2×2^^2×2^^0) in order to identify a representationwould transform upon ^^ jumps according to the single phi-bit QFT operation. Using Eqs. (8) and (9), this condition is satisfied if: ^^2^^2×2− ^^2×2^^2×2= 0 Eq. (10)95391587.29085067-799365 (UA23-226)

[0044] Eq. (10) is a special form of Sylvester equation. The solution to Eq. (10) can be obtained as follows. The translation matrix ^^2×2can be rewritten as a vector composed of the columns of the matrix: ^^ ^^ ^^ ^^2×2= ( ^^11, ^^21, ^^12, ^^22)^^Eq. (11)

[0045] Here, the superscript ^^ denotes the transpose (e.g., such that ^^ ^^ ^^ ^^2×2is a column vector).

[0046] To isolate the translation matrix ^^2×2, Eq. (10) can then be rewritten as: ^^ ^^ ^^ ^^ ^^2×2= 0 Eq. (12)

[0047] In Eq. (12), the matrix ^^ is given by: ^^ = ^^2×2^ ^^2− ^^2^^×2^ ^^2×2Eq. (13)

[0048] In Eq. (13), ^^2×2is the identity matrix. The condition for the existence of non-trivial solutions for ^^ ^^ ^^ ^^2×2in Eqs. (11) and (12) (or translation matrix ^^2×2) is that the determinant of ^^ be equal to zero.

[0049] Accordingly, inserting single phi-bit QFT (Hadamard gate) representation of ^^2from Eq. (6) and the ^^^^Pauli matrix (which in this case represents the unitary matrix ^^2×2) into Eq. (13) yields: 1 1 − ^^√2 0 ö ÷ ø

[0050] It can be shown that det( ^^) = 0. There are two vectors ^^ ^^ ^^ ^^2×2forming an orthonormal basis for the null space of ^^, namely: ( ^^ ^^ ^^ ^^2×2)(1)= (−0.205, −0.6613, 0.6447 ^^, −0.2905 ^^)^^and(^^ ^^ ^^ ^^2×2)(2)= (0.6613, −0.2505, −0.2905 ^^, −0.6447 ^^)^^95391587.210085067-799365 (UA23-226)

[0051] Any linear combination of the two vectors above will also be a solution to Eq. (12). Therefore, these vectors (or a linear combination of these vectors) can be recast in matrix form and inserted into Eq. (7) as ^^2×2to obtain a representation of the phi-bit state ( ^^ = ^^2×2^^0= ^^^^′= ^^^′^, ) which upon physically actuating ^^ phase jumps will transform an phi-bit state ^^0of the array into a ′′transformed phi-bit state ^^ = ^^^^= ^^^^of the according to the one-bit QFT operation. Note that the transformation will be independent of the initial value of the phases ^^12and ^^13from the original phi-bit state ^^0in Eq. (4), as well as of any coefficients in a linear combination of the null space basis of ^^. As such, these coefficients and initial phase values can be used to define the input values to the QFT operation that are being operated on. Two Phi-bit QFT

[0052] The representation of a single phi-bit “( ^^)” can be redefined as: (^^)^^( ^^)0 = ^^^^ ^^( ^^)2(sin ^^1)Eq. (14) (^^

[0053] Here, ^^)1 = ^^12⁄4 and ^^2= ^^13⁄4. The superscript ( ^^) = 1,2 refers to the phi-bits.divided by 4 instead of 2, as was the case for a single phi-bit QFT as discussed in the immediately preceding section (see Eqs. (4)-(5) and (7)-(9)). The two phi-bit state can be defined as the tensor product of single phi-bit states, namely: ^^0= ^^(1) (2)0 ⊗ ^^0Eq. (15)

[0054] Expanding Eq. (15) yields: sin ^^(1) (2)1 sin ^^195391587.211085067-799365 (UA23-226)

[0055] Using trigonometric relations and building upon the single phi-bit analysis in Eq. (5), upon a ^^ jump for the four phi-bit phases ^^(1) (1) (2) (2)12, ^^13and ^^12, ^^13, the vector ^^0transforms to:^^0′= ^^4×4^^0Eq. (17)

[0056] In anticipation of scaling the approach to designing a 11 1 1representation that results in the QFT, it is noted that ^^4×4 =^^2(−1 1 1 1−1 −1 1 1) = 1 −1 −1 1 1 1 1 1 √2( ) ⊗ ( 1 1) where ^^ =11 1 is now the transformation for t −1 1√2−2×2 √2( ) he 1 1 −1 1 .new ^^ can be introduced in terms of a translation matrix ^^, similar to Eq. (7) above: ^^11 ^^12 ^^13 ^^14^^ ^^ ^^^^21 ^^22 ^^23 ^^24^^0Eq. (18)

[0058] pre- ^^0is separable (e.g., it is a tensor product), the (post-transformation) representation ^^ may not be factorizable into a tensor product (e.g., it may be non-separable or classically entangled). In contrast to quantum systems, where entanglement is necessary to correlate qubit states, classical entanglement of phi-bits may only be needed to expand the accessible regions of multiple phi-bit Hilbert spaces and subsequently increase the number of possible states and the range of information that can be encoded and processed in those states. Recall that logical phi-bits are correlated via the nonlinearity of the elasticity of the physical system (e.g., the array of acoustically coupled waveguides).

[0059] When acting physically on the system to produce ^^ jumps, similar to Eq. (8) above, the vector ^^ transforms in terms of the translation matrix ^^ as: ^^^^′= ^^4×4^^0′= ^^4×4^^4×4^^0Eq. (19) 95391587.212085067-799365 (UA23-226)

[0060] Using Eq. (3), the QFT operation, ^^4, for two bits ( ^^ = 2 and ^^ =22) is written as:11 1 1^^4 =1(1 ^^ −1 − ^^) Eq. (20)

[0061] bit QFT, application of the QFT operation to the vector ^^ yields: ^^^′^ = ^^4^^ = ^^4^^4×4^^0Eq. (21)

[0062] The translation matrix ^^4×4can be sought (e.g., such that ^^4^^4×4^^0= ^^4×4^^4×4^^0) in order to identify a representation that would transform upon ^^ jumps according to the two phi-bit QFT operation, such that ^^^′^= ^^^^′. This condition is satisfied when: ^^4^^4×4− ^^4×4^^4×4= 0 Eq. (22)

[0063] Similar to Eq. (11) of the previous section (e.g., describing a single phi-bit QFT), the translation matrix ^^4×4can be rewritten as the vector composed of the columns of the matrix: ^^ ^^ ^^ ^^4×4= ( ^^11, ^^21, ^^31, ^^41, … , ^^14, ^^24, ^^34, ^^44)^^Eq. (23)

[0064] Equation (22) is then rewritten as: ^^ ^^ ^^ ^^ ^^4×4= 0 Eq. (24) where the matrix C is given by: ^^ = ^^4×4^ ^^4− ^^4^^×4^ ^^4×4Eq. (25)

[0065] Inserting the expression for ^^4×4and Eq. (20) pertaining to ^^4into Eq. (24), it can be verified (e.g., using Matlab©) that det( ^^) = 0. There exist 5 solutions that form an orthogonal basis for the null space of ^^ (as will be described in95391587.213085067-799365 (UA23-226) greater detail below). Similar to the single phi-bit case, a linear combination of these solutions upon physically actuating ^^ phase jumps will transform an original phi-bit state ^^0of the array into a transformed phi-bit state ^^ = ^^^^′= ^^^′^ of the array according to the two-bit QFT operation. Again, this transformation will bendependent of the initial value of the phases ^^( ^i^)12and^^)13from the original phi-bit state ^^0in Eqs. (14) and (15) as well as of the coefficients in a linear combination of the null space basis of ^^. These coefficients and initial phase values can be used to define the input values to the two phi-bit QFT operation that are being operated on. Multiple Phi-bit QFT

[0066] With continued reference to FIGS.3A-3C, described below is an example multiple phi-bit QFT operation based on a system composed of ^^ logical phi-bits, building on the previous discussions with respect to the single phi-bit QFT operation in Eqs. (4)-(13) and the two phi-bit QFT operation in Eqs. (14)-(25). The representation of a single phi-bit “( ^^)” among the ^^’s can be defined as: ^^( ^^)0 = ^^2(sin ( ^^)( ^^) ^^ ^^ ^^1) Eq. (26)

[0067] Here, ^^(1= ^^ ^^ ^^12⁄ 2 and ^^2 = ^^13⁄ 2 . The superscript ( ^^) =1, … , ^^ refers to the phi-

[0068] Similar to Eq. (15), the ^^ phi-bit state is represented as the tensor product of single phi-bit states, namely: ^^0= ^^(1) ( ) (0 ⊗ ^^02⊗ … ⊗ ^^0^^)Eq. (27)

[0069] Similar to Eq. (17) for the two phi-bit ^^ phase jump rotation operation, upon a ^^ jump for all the phi-bit phases ^^( ^^) ( ^^)12, ^^13, the vector ^^0transforms to: ^^0′= ^^2^^×2^^ ^^0Eq. (28)

[0070] ^^^^ ^^= ^^^^ ^^ ^^2 ^^^^ ^^ ^^ =^^ ^^). The quantities, ^^ = sin^^^^^^95391587.214085067-799365 (UA23-226)

[0071] Similar to Eq. (18), the target representation can be defined in terms of the translation matrix ^^ as: ^^ = ^^2^^×2^^ ^^0Eq. (29)

[0072] The components of the translation matrix ^^2^^×2^^are ^^^^ ^^with the indices ^^ = 1, … , 2^^and ^^ = 1, … , 2^^.

[0073] The physical action to produce ^^ jumps (similar to Eq. (19) above) applies the transformation: ^^^^′= ^^2^^×2^^ ^^0′= ^^2^^×2^^ ^^2^^×2^^ ^^0Eq. (30)

[0074] Similar to Eq. (21) for the two phi-bit QFT, the ^^ phi-bit QFT operation, ^^2^^, yields: ^^^′^= ^^2^^ ^^ = ^^2^^ ^^2^^×2^^ ^^0Eq. (31)

[0075] ^^^^′will be equal to ^^^′^when:^^2^^ ^^2^^×2^^ − ^^2^^×2^^ ^^2^^×2^^ = 0 Eq. (32)

[0076] Following the preceding sections, the translation matrix ^^2^^×2^^ is reformulated as the vector ^^ ^^ ^^ ^^2^^×2^^. The solutions of: ^^ ^^ ^^ ^^ ^^2^^×2^^ = 0 Eq. (33)

[0077] where the matrix ^^ is now given by: ^^ = ^^2^^×2^^^ ^^2^^− ^^2^^^^×2^^ ^ ^^2^^×2^^Eq. (34)

[0078] The solutions of Eq. (34) that form an orthonormal basis can be used to construct the representation ^^ of Eq. (29) that will transform according to the ^^ phi-bit QFT operation when operations on the physical system lead to ^^ jumps in the logical phi-bit phases.95391587.215085067-799365 (UA23-226)

[0079] It is contemplated herein that Eq. (33) possesses solutions for all number of phi-bits. The existence of 8 solutions forming the basis of the null space of ^^ has been verified for a three phi-bit system. The basis of the null space of ^^ for four phi-bits includes 9 vectors. This conjecture is supported by the fact that the representations ^^ and ^^ ^^2 ^^ 0 involve, upon ^^ jumps, powers of terms of the form ^^2^^which are also the components of the QFT matrix (Eq. (3)).

[0080] The general approach described above and described herein can enable the design of representations for inputting data into a multiple phi-bit QFT operation that can be physically realized by actuating an acoustic metastructure of a quantum analogue (e.g., an array such as the array 110 of FIG.1). The presently disclosed systems and techniques establish a link between multiple phi-bit gates in exponentially complex Hilbert spaces and operations on the physical system scaling linearly with the number of phi-bits. In the particular example above, the QFT operation of the input data is achieved through the ^^ jumps in the phi-bit phases. It is noted that the phi-bit-based QFT operation described herein can be defined and executed for any input, without departing from the scope of the present disclosure. Example: Two Phi-Bits

[0081] The concept of a phi-bit-based QFT is illustrated below with a simple two phi-bit case. In some embodiments, the phi-bits of FIGS.2A and 2B can be utilized as the physical platform. Here, instead of determining an orthonormal basis for the null space of the matrix ^^, a rational basis can be calculated for that null space. This basis is not usually orthonormal. However, linear combinations of these basis vectors will also provide a mean of defining inputs for the QFT. The five translation matrices below form the rational basis for the matrix ^^ of the two phi-bit case: 00 0 095391587.216085067-799365 (UA23-226) 0− ^^ / 2 − ^^ / 2 0^^[3]4×4 = (0 0 0 1

[0082] Inthe Fourier transform of 0the input vector ^^ ^^ = (10 ) can be calculated, for example choosing the −1 representation ^^ = ^^[1]4×4^^0.

[0083] input vector takes the form:0(1) (2)^^(1)^^(2)acoustic metastructure such that all phi-bit phases change by ^^, (e.g., by changing the driving frequency ^^1of FIGS.2A and 2B from 62kHz to 64.2kHz that is being applied to the array by the driving transducers), the vector ^^ transforms into: 0 )95391587.217085067-799365 (UA23-226) 0

[0085] To within a general phase, the output vector ^^ ^^ = (2 ^^0 ) is the^^ Fourier transform of ^^^^. Example: Evaluating period of a function using QFT

[0086] In some cases, it may be desirable to determine if a function possesses a specific period. In the example below, the function is represented as the −^^input vector: ^^ ^^ = (1 [3]^^). To realize this input vector, the representation ^^ = ^^4×4^^0−1can the input vector takes the form: ^^ −sin ^^(1) cos ^ (2) ^^ (1) (2)21 ^1 − 2 cos ^^1 sin ^^1(1) (2)

[0087] By initializing the two phi-bit system such that ^^(1) (2)12= ^^12= ^^, the sines and cosines are equal to each other (e.g., sin^^ ^^ 14 = cos 4 =√2) and ^^ reduces to: (1) − ^^^^ ^^ ^^(2) 1)

[0088] This initial state can be achieved by driving the two phi-bits of FIG.1 with a frequency ^^1~64.2kHz by the driving transducers. A change of driving conditions by the driving transducers achieved by raising ^^1to 65kHz adds ^^ to the phi-bit phases, and leads to the following transformations: sin ^^1(1)sin ^^(2)1 → sin ^^1(1)sin ^^(2) (1 + sin ^^11)cos ^^1(2)+ cos ^^(1) (2) (1) (1 sin ^^1+ cos ^^1cos ^^12)95391587.218085067-799365 (UA23-226) sin ^^( ) ( )11cos ^^12→ −sin ^^1(1) (2) ( ) ( ) (1) (2(1)sin ^cos (^1 + sin ^^11 cos ^^21)− cos ^^ )1 sin ^^1+cos ^^ 2(2)112representation given in Eq. zero. The fourth component becomes4 / 2 = −2. Upon ^^ jumps, the prefactor term, ^^^^(1)+ ^^(2) −( ^^22), transforms into (1 ^^ ^^ ^^( ^^ ) (2) 2 + ^^2), yielding: (1) ^(2 0^^ ^^ + ^ ))

[0090] To within a transform output can be 0obtained.

[0091] After the steps of initialization and physical operation by the driving transducers of the array that leads to ^^ jumps in the phi-bit phases, by calculating the fourth component of the chosen two phi-bit representation, the systems and techniques disclosed herein can determine if the input function possesses a specific period. For instance, a non-zero value of the first component of the output vector indicates a constant function over the domain of four discrete points of the function. A non-zero value of the third component of the output vector corresponds to a period extending over half of the domain of the function. A non-95391587.219085067-799365 (UA23-226) ^ zero value of the fourth component corresponds to a function of the form ^^^^^ 2^^with the domain of the function defined through the integer ^^ = 1,2,3,4. The QFT approach implemented on multiple phi-bit arrays can therefore be used to extract the periods of sequences of real but also complex numbers.

[0092] It is noted that after performing or otherwise implementing a QFT on a qubit-based computing platform, there may be no effective way to recover the Fourier coefficients which are stored in quantum superpositions of probability amplitudes (e.g., quantum wave functions). Because of quantum wave function collapse, a measurement of the state of qubits will only return the computation basis and not the superposition. In contrast, a logical phi-bit-based QFT operates on complex amplitudes of nonlinear acoustic modes which are not subjected to wave function collapse, and the result of the logical phi-bit-based QFT can be measured at the detecting transducers of the array. Retrieving all Fourier coefficients on ^^ phi-bits would require 2^^calculations, thus creating significant computational overhead. However, phi-bit-based QFT may realize its full potential when one needs the calculation of only one or a few Fourier components enabling the measurement of the period(s) of a function. Phi-bit-based QFT will also demonstrate its full power when combined with other multiple phi-bit gates and / or algorithms. General Approach Summary

[0093] To summarize, FIGS.3A-3C outline a general approach for performing a Fourier Transform (FT) operation (or another target operation) on a multi phi-bit value input value using the physical platform 100 shown in FIG.1. As discussed above, the FT operation takes advantage of a unique property found during experimental observation: when a certain change in driving conditions is applied, measured frequencies simultaneously jump (or, rotate) by a value of ^^. This is referred to herein as a “ ^^ phase jump rotation operation”, illustrated in diagram 300A of FIG.3A.

[0094] The goal of the systems and methods outlined herein is to be able to apply the FT operation (or another target operation) to an input phi-bit value to obtain an output phi-bit value, and to be able to scale the FT operation to any number of phi-bits. As discussed, the ^^ phase jump rotation operation can be exploited by relating the FT operation (or another target operation) to a result of the ^^ phase jump rotation operation which can be applied to an input multi phi-bit value.95391587.220085067-799365 (UA23-226)

[0095] Diagram 300B of FIG.3B illustrates how the multi phi-bit input value can be defined in terms of the phi-bit Fourier Transform representation (e.g., as ^^0in Eqs. (26)-(27)) which incorporates a translation matrix ( ^^). The translation matrix ^^ relates a unitary matrix ^^ for a ^^ phase jump rotation operation on the input multi phi-bit value to a unitary matrix ^^ for a quantum Fourier transform operation on the input multi phi-bit value.

[0096] Referring to diagram 300C of FIG.3C, upon application of a change in driving conditions to induce a ^^ phase jump rotation, the output multi phi- bit value is obtained in terms of the phi-bit Fourier Transform representation as ^^ = ^^^^′ = ^^′^^ = ^^2 ^^×2 ^^ ^^2 ^^×2 ^^ ^^0 = ^^2 ^^ ^^2 ^^×2 ^^ ^^0. Notably, the output multi phi-bit value{ ^^, ^^} correlates with a result ^^ ^^′ of the ^^ phase jump rotation operation on the inputmulti phi-bit value and a result ^^^′^of the quantum Fourier transform operation on the input multi phi-bit value. The phi-bit Fourier Transform representation represents the transformed phi-bit state of the array ^^ in terms of the original phi-bit state of the array ^^0and the translation matrix ^^, where the translation matrix ^^ ensures that a result ^^^^′of the transformation from the ^^ phase jump rotation operation is equal to what would be a result ^^^′^of the transformation from the Fourier transform operation.

[0097] This naturally leads to the condition where ^^^^′will be equal to ^^^′^when ^^2^^^^2^^×2^^− ^^2^^×2^^^^2^^×2^^= 0. To solve for the translation matrix ^^2^^×2^^, the known unitary matrices ^^ and ^^ can be collected into a matrix ^^ = ^^2^^×2^^ ^ ^^2^^ − ^^2^^^^×2^^ ^ ^^2^^×2^^, which can be used to solve for ^^2^^×2^^ as ^^ ^^ ^^ ^^ ^^2^^×2^^ = 0 (where ^^ ^^ ^^ ^^2^^×2^^ is simply a reformulated version of ^^2^^×2^^). In other words, the translation matrix ^^ of the phi-bit Fourier Transform representation must be selected such that the result ^^^^′of the ^^ phase jump rotation operation on the input multi phi-bit value is equal to the result ^^^′^of the quantum Fourier transform operation on the input multi phi-bit value when the input multi phi-bit value is represented in terms of the phi-bit Fourier Transform representation.

[0098] As such, for a given value of ^^ (corresponding to quantity of phi- bits) this formulation can be used to represent the multi phi-bit values in terms of the phi-bit Fourier Transform representation, which maps the physical response of the array (resulting from the change in driving conditions that induce a ^^ phase jump rotation operation) to the output multi phi-bit value represented by the physical response.95391587.221085067-799365 (UA23-226)

[0099] This allows evaluation of the Fourier Transform operation by exploiting how the acoustic waveguide array simultaneously rotates phases of detected signals in response to a certain change in driving conditions, and by enforcing an equivalence between the result ^^^′^of the transformation from the Fourier transform operation and the result ^^^^′of the transformation from the ^^ phase jump rotation operation.

[0100] In some examples, the target operation may be a different type of operation than the Fourier Transform operation. In that case, the process of designing a phi-bit representation which takes advantage of the ^^ phase jump rotation operation would involve a different unitary matrix to modify or replace the unitary matrix ^^ and adapting the translation matrix ^^ to enforce an equivalence to the ^^ phase jump rotation operation in a similar manner. Methods

[0101] FIGS.4A and 4B outline a method 400 for determining a Fourier Transform (or other target operation) of a multi phi-bit value according to the systems outlined herein. Some steps of the method 400 may be computer-implemented, e.g., by a computing device 500 shown in FIG.5 and / or using phi-bit or q-bit computing systems.

[0102] Referring to FIG.4A, step 402 of method 400 includes applying first driving signals having a first set of driving frequencies to an array of acoustically coupled waveguides by a plurality of driving transducers coupled to the array. Step 404 of method 400 includes measuring first detected signals that correlate with an input multi phi-bit value using a plurality of detecting transducers coupled to the array, the first detected signals being resultant of the first driving signals and having first phase difference values that correspond to the input multi phi-bit value.

[0103] Step 406 of method 400 includes determining the input multi phi- bit value based on the first detected signals measurable at the plurality of detecting transducers, the input multi phi-bit value being defined in terms of a phi-bit Fourier Transform representation (or, more generally, a phi-bit representation for a target operation) incorporating a translation matrix ^^ that relates a unitary matrix ^^ for a ^^ phase jump rotation operation on the input multi phi-bit value to a unitary matrix ^^ for a quantum Fourier transform operation (or another target operation) on the input multi phi-bit value. The translation matrix ^^ of the phi-bit Fourier Transform95391587.222085067-799365 (UA23-226) representation is selected such that a result ^^^^′of the ^^ phase jump rotation operation on the input multi phi-bit value is equal to a result ^^^′^ of the quantum Fourier transform operation on the input multi phi-bit value when the input multi phi- bit value is represented in terms of the phi-bit Fourier Transform representation.

[0104] Step 408 of method 400 includes applying, by the plurality of driving transducers coupled to the array, a change in driving conditions that induces the ^^ phase jump rotation operation and results in second detected signals measurable at the plurality of detecting transducers that correlate with an output multi phi-bit value. With additional reference to FIG.4B, step 408 can encompass steps 410 and 412. Step 410 includes applying second driving signals having a second set of driving frequencies to the array of acoustically coupled waveguides by the plurality of driving transducers, the second set of driving frequencies being different from a first set of driving frequencies of first driving signals associated with the input multi phi-bit value. Step 412 includes measuring the second detected signals that correlate with the output multi phi-bit value using the plurality of detecting transducers, the second detected signals being resultant of the second driving signals and having second phase difference values that correspond to the output multi phi-bit value, the second set of driving frequencies being selected to simultaneously induce a ^^ phase jump rotation in the second phase difference values relative to first phase difference values of the first detected signals.

[0105] Step 414 of method 400 includes determining the output multi phi-bit value based on the second detected signals and the phi-bit Fourier Transform representation with respect to the input multi phi-bit value, the output multi phi-bit value correlating with the result ^^^^′of the ^^ phase jump rotation operation on the input multi phi-bit value and the result ^^^′^ of the quantum Fourier transform operation on the input multi phi-bit value, e.g., such that ^^ = ^^2^^×2^^ ^^0= ^^^′^ = ^^^^′.

[0106] The phi-bit Fourier Transform representation maps a physical response of the array to the output multi phi-bit value. The physical response of the array results from the change in driving conditions and is measurable as the second detected signals by the plurality of detecting transducers coupled to the array.

[0107] Further, the ^^ phase jump rotation operation on the input multi phi-bit value is related to the output multi phi-bit value through the translation matrix ^^ and the unitary matrix ^^ associated with the ^^ phase jump rotation operation. The quantum Fourier transform operation on the input multi phi-bit value is related to the95391587.223085067-799365 (UA23-226) output multi phi-bit value through the translation matrix ^^ and a unitary matrix ^^ associated with the quantum Fourier transform operation. Unitary matrices ^^ and ^^ are known, and the translation matrix ^^ can be solved for by collecting terms associated with the unitary matrices ^^ and ^^ into a matrix ^^ = ^^2^^×2^^ ^ ^^2^^ − ^^2^^^^×2^^ ^ ^^2^^×2^^, which can be used to solve for ^^2^^×2^^as ^^ ^^ ^^ ^^ ^^2^^×2^^= 0 (where ^^ ^^ ^^ ^^2^^×2^^ is simply a reformulated version of ^^2^^×2^^). For examples where the target operation is a different operation than the Fourier Transform, unitary matrix ^^ may be modified or replaced accordingly.

[0108] Note that in some examples, the target operation may be different from the Fourier Transform operation. In such a case, the input multi phi-bit value can be defined in terms of a phi-bit representation incorporating a translation matrix ^^ that relates a unitary matrix ^^ for a ^^ phase jump rotation operation on the input multi phi-bit value to some other unitary matrix (which may modify or replace unitary matrix ^^) for the target operation on the input multi phi-bit value. The translation matrix ^^ of the phi-bit representation is selected such that a result ^^^^′of the ^^ phase jump rotation operation on the input multi phi-bit value is equal to a result ^^^′^of the target operation on the input multi phi-bit value. Conclusions

[0109] The systems and techniques described herein correspond to an approach that establishes a correspondence between the state of multiple logical phi-bits, acoustic qubit analogues, in their supporting physical space (e.g., held or otherwise represented by the array of acoustic waveguides) and their representations as complex state vectors in exponentially scaling Hilbert spaces. The physical response of the correlated logical phi-bits to changes in the experimental conditions is mapped onto a multiple phi-bit discrete Fourier transformation unitary matrix analogous to the quantum Fourier transform (QFT). This scalable QFT operates in parallel on the components of multiple phi-bit complex state vectors, based on only a single physical action on the nonlinear acoustic metastructure that supports the phi-bits. This approach forms the foundation for the design and development of other multiple phi-bit unitary operations and / or algorithms that, in contrast to qubits, are unaffected by measurements and decoherence.95391587.224085067-799365 (UA23-226)

[0110] For instance, aspects of the present disclosure can be applied to or otherwise utilized in the design of useful multiple phi-bit unitary matrices and gates based on: (a) a knowledge of phi-bit phase response to changes in driving conditions of the supporting acoustic metastructure; (b) analytical or numerical representations of single phi-bit and multiple phi-bits in exponentially scaling Hilbert space initializable into all (or as many possible) separable or non-separable states; and (c) an algebraic or numerical mapping between the effect of the desired unitary matrix and the change in driving conditions on the representations (e.g., by changing a frequency applied to the waveguides by the driving transducers). Advantageously, the strong nonlinear correlation between phi-bits enables the systems and techniques to bypass the need of qubit-based computing platforms for decomposition of large-scale unitary matrices into quantum circuits of elementary gates. Finally, a significant limitation in the design of a quantum algorithm is related to its mapping to a specific quantum processor. Indeed, current quantum computers are constituted of qubits occupying different parts of the processor imposing physical constraints on the possible coupling between qubits. As such, the design of an efficient quantum algorithm needs to be adapted to a specific quantum processor map of qubit coupling (e.g., the coupling map). As nonlinear acoustic modes, logical phi-bits reside within the same physical space of the acoustic metastructure utilized by aspects of the present disclosure. Accordingly, strongly correlated logical phi-bits occupy the same physical space and advantageously, the design of phi-bit-based gates and algorithms will not suffer from spatial physical constraints. Computing Device

[0111] Referring to FIG.5, some aspects of the systems outlined herein, including some aspects of the method 400 shown in FIGS.4A and 4B, may be implemented using a computing system such as computing device 500, or a phi- bit or q-bit analog of computing device 500. Computing device 500 can aid in selection and / or application of the first or second driving signals to the array 110 of FIG.1, and / or can aid in interpretation of the first or second detected signals measured by the detecting transducers 132a-132c to obtain information about the phase values that can be used to interpret the input multi phi-bit value or the output multi phi-bit value. In some examples, computing device 500 can also aid in selection or construction of the unitary matrices ^^ and ^^ as well as selection or95391587.225085067-799365 (UA23-226) construction of the translation matrix ^^ which may be used to represent the input multi phi-bit value or the output multi phi-bit value in terms of the phi-bit representation.

[0112] Importantly, note that while computing device 500 is recited here as aiding in practical implementation and evaluation of the Fourier Transform operation (or other target operation) as applied to multi phi-bit value, the actual calculation of the Fourier Transform operation (or another target operation) as applied to a multi phi-bit value is performed using the physical platform 100 of FIG.1, which leverages nonlinearity to perform phi-bit large scale unitary operations.

[0113] Computing device 500 comprises one or more network interfaces 510 (e.g., wired, wireless, PLC, etc.), at least one processor 520, and a memory 540 interconnected by a system bus 550, as well as a power supply 560 (e.g., battery, plug-in, etc.). Computing device 500 can also include or otherwise communicate with a display device 530 for user interaction.

[0114] Network interface(s) 510 include the mechanical, electrical, and signaling circuitry for communicating data over the communication links coupled to a communication network. Network interfaces 510 are configured to transmit and / or receive data using a variety of different communication protocols. As illustrated, the box representing network interfaces 510 is shown for simplicity, and it is appreciated that such interfaces may represent different types of network connections such as wireless and wired (physical) connections. Network interfaces 510 are shown separately from power supply 560, however it is appreciated that the interfaces that support PLC protocols may communicate through power supply 560 and / or may be an integral component coupled to power supply 560.

[0115] Memory 540 includes a plurality of storage locations that are addressable by processor 520 and network interfaces 510 for storing software programs and data structures associated with the embodiments described herein. In some embodiments, device 500 may have limited memory or no memory (e.g., no memory for storage other than for programs / processes operating on the device and associated caches). Memory 540 can include instructions executable by the processor 520 that, when executed by the processor 520, cause the processor 520 to implement some aspects of the method 400 outlined herein.

[0116] Processor 520 comprises hardware elements or logic adapted to execute the software programs (e.g., instructions) and manipulate data structures95391587.226085067-799365 (UA23-226) 545. An operating system 542, portions of which are typically resident in memory 540 and executed by the processor, functionally organizes device 500 by, inter alia, invoking operations in support of software processes and / or services executing on the device. These software processes and / or services may include phi-bit operation support processes / services 590, which can include some aspects of method 400 and / or implementations of various modules described herein. Note that while phi-bit operation support processes / services 590 is illustrated in centralized memory 540, alternative embodiments provide for the process to be operated within the network interfaces 510, such as a component of a MAC layer, and / or as part of a distributed computing network environment.

[0117] It will be apparent to those skilled in the art that other processor and memory types, including various computer-readable media, may be used to store and execute program instructions pertaining to the techniques described herein. Also, while the description illustrates various processes, it is expressly contemplated that various processes may be embodied as modules or engines configured to operate in accordance with the techniques herein (e.g., according to the functionality of a similar process). In this context, the term module and engine may be interchangeable. In general, the term module or engine refers to model or an organization of interrelated software components / functions. Further, while the phi-bit operation support processes / services 590 is shown as a standalone process, those skilled in the art will appreciate that this process may be executed as a routine or module within other processes.

[0118] It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto.95391587.227

Claims

085067-799365 (UA23-226) CLAIMS What is claimed is:

1. A system, comprising: a processor in communication with a memory, a plurality of driving transducers coupled to an array of acoustically coupled waveguides, and a plurality of detecting transducers coupled to the array of acoustically coupled waveguides, the memory including instructions executable by the processor to: determine an input multi phi-bit value based on first detected signals measurable at the plurality of detecting transducers, the input multi phi-bit value being defined in terms of a phi-bit Fourier Transform representation incorporating a translation matrix ^^ that relates a unitary matrix ^^ for a ^^ phase jump rotation operation on the input multi phi-bit value to a unitary matrix ^^ for a quantum Fourier transform operation on the input multi phi-bit value; applying, by the plurality of driving transducers coupled to the array, a change in driving conditions that induces the ^^ phase jump rotation operation and results in second detected signals measurable at the plurality of detecting transducers that correlate with an output multi phi-bit value; and determine the output multi phi-bit value based on the second detected signals and the phi-bit Fourier Transform representation with respect to the input multi phi-bit value, the output multi phi-bit value correlating with a result of the ^^ phase jump rotation operation on the input multi phi-bit value and a result of the quantum Fourier transform operation on the input multi phi-bit value.

2. The system of claim 1, the phi-bit Fourier Transform representation mapping a physical response of the array to the output multi phi-bit value, the physical response of the array resulting from the change in driving conditions and95391587.228085067-799365 (UA23-226) being measurable as the second detected signals by the plurality of detecting transducers coupled to the array.

3. The system of claim 1, the translation matrix ^^ of the phi-bit Fourier Transform representation being selected such that the result of the ^^ phase jump rotation operation on the input multi phi-bit value is equal to the result of the quantum Fourier transform operation on the input multi phi-bit value when the input multi phi-bit value is represented in terms of the phi-bit Fourier Transform representation.

4. The system of claim 1, the ^^ phase jump rotation operation on the input multi phi-bit value being related to the output multi phi-bit value through the translation matrix ^^ and the unitary matrix ^^ associated with the ^^ phase jump rotation operation.

5. The system of claim 1, the quantum Fourier transform operation on the input multi phi-bit value being related to the output multi phi-bit value through the translation matrix ^^ and the unitary matrix ^^ associated with the quantum Fourier transform operation.

6. The system of claim 1, the memory further including instructions executable by the processor to: apply first driving signals having a first set of driving frequencies to the array of acoustically coupled waveguides by the plurality of driving transducers; and measure the first detected signals that correlate with the input multi phi-bit value using the plurality of detecting transducers, the first detected signals being resultant of the first driving signals and having first phase difference values that correspond to the input multi phi-bit value.

7. The system of claim 1, the memory further including instructions executable by the processor to: apply second driving signals having a second set of driving frequencies to the array of acoustically coupled waveguides by the plurality of driving95391587.229085067-799365 (UA23-226) transducers, the second set of driving frequencies being different from a first set of driving frequencies of first driving signals associated with the input multi phi-bit value; and measure the second detected signals that correlate with the output multi phi-bit value using the plurality of detecting transducers, the second detected signals being resultant of the second driving signals and having second phase difference values that correspond to the output multi phi-bit value; the second set of driving frequencies being selected to simultaneously induce a ^^ phase jump rotation in the second phase difference values relative to first phase difference values of the first detected signals.

8. A method, comprising: determining an input multi phi-bit value based on first detected signals measurable at a plurality of detecting transducers coupled to an array of acoustically coupled waveguides, the input multi phi-bit value being defined in terms of a phi-bit Fourier Transform representation incorporating a translation matrix ^^ that relates a unitary matrix ^^ for a ^^ phase jump rotation operation on the input multi phi-bit value to a unitary matrix ^^ for a quantum Fourier transform operation on the input multi phi-bit value; applying, by a plurality of driving transducers coupled to the array, a change in driving conditions that induces the ^^ phase jump rotation operation and results in second detected signals measurable at the plurality of detecting transducers that correlate with an output multi phi- bit value; and determining the output multi phi-bit value based on the second detected signals and the phi-bit Fourier Transform representation with respect to the input multi phi-bit value, the output multi phi-bit value correlating with a result of the ^^ phase jump rotation operation on the input multi phi-bit value and a result of the quantum Fourier transform operation on the input multi phi-bit value.95391587.230085067-799365 (UA23-226) 9. The method of claim 8, the phi-bit Fourier Transform representation mapping a physical response of the array to the output multi phi-bit value, the physical response of the array resulting from the change in driving conditions and being measurable as the second detected signals by the plurality of detecting transducers coupled to the array.

10. The method of claim 8, the translation matrix ^^ of the phi-bit Fourier Transform representation being selected such that the result of the ^^ phase jump rotation operation on the input multi phi-bit value is equal to the result of the quantum Fourier transform operation on the input multi phi-bit value when the input multi phi-bit value is represented in terms of the phi-bit Fourier Transform representation.

11. The method of claim 8, the ^^ phase jump rotation operation on the input multi phi-bit value being related to the output multi phi-bit value through the translation matrix ^^ and the unitary matrix ^^ associated with the ^^ phase jump rotation operation.

12. The method of claim 8, the quantum Fourier transform operation on the input multi phi-bit value being related to the output multi phi-bit value through the translation matrix ^^ and the unitary matrix ^^ associated with the quantum Fourier transform operation.

13. The method of claim 8, further comprising: applying first driving signals having a first set of driving frequencies to the array of acoustically coupled waveguides by the plurality of driving transducers; and measuring the first detected signals that correlate with the input multi phi- bit value using the plurality of detecting transducers, the first detected signals being resultant of the first driving signals and having first phase difference values that correspond to the input multi phi-bit value.95391587.231085067-799365 (UA23-226) 14. The method of claim 8, further comprising: applying second driving signals having a second set of driving frequencies to the array of acoustically coupled waveguides by the plurality of driving transducers, the second set of driving frequencies being different from a first set of driving frequencies of first driving signals associated with the input multi phi-bit value; and measuring the second detected signals that correlate with the output multi phi-bit value using the plurality of detecting transducers, the second detected signals being resultant of the second driving signals and having second phase difference values that correspond to the output multi phi-bit value; the second set of driving frequencies being selected to simultaneously induce a ^^ phase jump rotation in the second phase difference values relative to first phase difference values of the first detected signals.

15. A method, comprising: determining an input multi phi-bit value based on first detected signals measurable at a plurality of detecting transducers coupled to an array of acoustically coupled waveguides, the input multi phi-bit value being defined in terms of a phi-bit representation incorporating a translation matrix ^^ that relates a unitary matrix ^^ for a ^^ phase jump rotation operation on the input multi phi-bit value to a unitary matrix ^^ for a target operation on the input multi phi-bit value; applying, by a plurality of driving transducers coupled to the array, a change in driving conditions that induces the ^^ phase jump rotation operation and results in second detected signals measurable at the plurality of detecting transducers that correlate with an output multi phi- bit value; and determining the output multi phi-bit value based on the second detected signals and the phi-bit representation with respect to the input multi phi- bit value, the output multi phi-bit value correlating with a result of the ^^ phase jump rotation operation on the input multi phi-bit value and a result of the target operation on the input multi phi-bit value.95391587.232085067-799365 (UA23-226) 16. The method of claim 15, the phi-bit representation mapping a physical response of the array to the output multi phi-bit value, the physical response of the array resulting from the change in driving conditions and being measurable as the second detected signals by the plurality of detecting transducers coupled to the array.

17. The method of claim 15, the translation matrix ^^ of the phi-bit representation being selected such that the result of the ^^ phase jump rotation operation on the input multi phi-bit value is equal to the result of the target operation on the input multi phi-bit value when the input multi phi-bit value is represented in terms of the phi-bit representation.

18. The method of claim 15, further comprising: applying first driving signals having a first set of driving frequencies to the array of acoustically coupled waveguides by the plurality of driving transducers; and measuring the first detected signals that correlate with the input multi phi- bit value using the plurality of detecting transducers, the first detected signals being resultant of the first driving signals and having first phase difference values that correspond to the input multi phi-bit value.

19. The method of claim 15, further comprising: applying second driving signals having a second set of driving frequencies to the array of acoustically coupled waveguides by the plurality of driving transducers, the second set of driving frequencies being different from a first set of driving frequencies of first driving signals associated with the input multi phi-bit value; and measuring the second detected signals that correlate with the output multi phi-bit value using the plurality of detecting transducers, the second detected signals being resultant of the second driving signals and having second phase difference values that correspond to the output multi phi-bit value;95391587.233085067-799365 (UA23-226) the second set of driving frequencies being selected to simultaneously induce a ^^ phase jump rotation in the second phase difference values relative to first phase difference values of the first detected signals.

20. The method of claim 15, the target operation being a Fourier Transform operation and the unitary matrix ^^ correlating to a unitary matrix for a quantum Fourier Transform operation.95391587.234

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