Patent Publication Number: US-11657312-B2

Title: Short-depth active learning quantum amplitude estimation without eigenstate collapse

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with Government support under Contract No.: FA8750-18-C-0098, awarded by U.S. Air Force, Office of Scientific Research (AFOSR). The Government has certain rights to this invention. 
    
    
     BACKGROUND 
     The subject disclosure relates to estimation of a quantum state&#39;s phase, and more specifically, to estimation of a quantum state&#39;s phase utilizing the estimation of the amplitude between two quantum states, and more specifically, to the estimation of the amplitude utilizing a hybrid of quantum and classical methods. 
     SUMMARY 
     The following presents a summary to provide a basic understanding of one or more embodiments of the invention. This summary is not intended to identify key or critical elements or delineate any scope of the particular embodiments or any scope of the claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that is presented later. In one or more embodiments described herein, devices, systems, computer-implemented methods, apparatus and/or computer program products facilitating estimation of a quantum state&#39;s phase, and more specifically, facilitating estimation of a quantum state phase utilizing a hybrid of quantum and classical methods. 
     According to an embodiment, a system is provided. The system can comprise a memory that stores computer executable components and a processor that executes the computer executable components stored in the memory. The computer executable components can include learning component that can utilize stochastic inference to determine an expectation value based on an uncollapsed eigenvalue pair. The computer executable components can further include an encoding component that can encode the expectation value associated with a quantum state. 
     According to an embodiment, a system is provided. The system can comprise a memory that stores computer executable components and a processor that executes the computer executable components stored in the memory. The computer executable components can include a learning component. The learning component can utilize stochastic inference to determine an expectation value based on an uncollapsed eigenvalue pair. The learning component can include a measuring component that can probabilistically measure the expectation value, with the measuring component being independent of an input state. 
     According to another embodiment, a computer-implemented method is provided. The computer-implemented method can comprise learning, by a system operatively coupled to a processor, an expectation value based on a quantum state based on an uncollapsed eigenvalue pair, by stochastic inference. In some embodiments, the learning can comprise measuring, by the system, the expectation value probabilistically, with the measuring being independent of an input state. The method can further comprise encoding, by the system, an expectation value associated with a quantum state, as a phase. 
     According to another embodiment, a computer-implemented method is provided. The computer-implemented method can comprise encoding, by a system operatively coupled to a processor an expectation value associated with a quantum state. The method can further comprise learning, by the system, by stochastic inference, the expectation value based on an uncollapsed eigenvalue pair. 
     According to yet another embodiment, a computer program product facilitating phase estimation of a quantum state can comprise a computer readable storage medium having program instructions embodied therewith. The program instructions can be executable by a processor and cause the processor to encode, by the processor, an expectation value associated with a quantum state. Further, the program instructions can cause the processor to learn, by utilizing a stochastic inference, the expectation value based on an uncollapsed eigenvalue pair. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
         FIG.  1    illustrates a block diagram of an example, non-limiting system that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. 
         FIG.  2 A  illustrates an example approach that can use a circuit to select a control parameter m that can be used by an encoding component to encode an expectation value in the phase of a quantum state, in accordance with one or more embodiments. 
         FIG.  2 B  illustrates a block diagram of an example, non-limiting system that shows the interaction of the encoding component and learning component in accordance with one or more embodiments described herein. 
         FIG.  3    illustrates an example, non-limiting circuit that shows an encoding component in accordance with one or more embodiments described herein. 
         FIG.  4    illustrates an example, non-limiting geometric representation of the example, non-limiting circuit of  FIG.  3    in accordance with one or more embodiments described herein. 
         FIG.  5    illustrates a block diagram of an example, non-limiting system that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. 
         FIG.  6    illustrates a flow diagram of an example, non-limiting computer-implemented method that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. 
         FIG.  7    illustrates a flow diagram of an example, non-limiting computer-implemented method that can encode a quantum state as a phase in accordance with one or more embodiments described herein. 
         FIG.  8    illustrates a flow diagram of a computer-implemented method that can learn an eigenvalue corresponding to an expectation value based on a quantum state in accordance with one or more embodiments described herein. 
         FIG.  9    illustrates a flow diagram of an example, non-limiting computer-implemented method that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. 
         FIG.  10    illustrates a block diagram of an example, non-limiting operating environment in which one or more embodiments described herein can be facilitated. 
         FIG.  11    illustrates an example, non-limiting graph showing an example probabilistic measurement of a phase in accordance with one or more embodiments described herein. 
         FIGS.  12 A and  12 B  illustrate example, non-limiting graphs showing the samples for a given error ϵ and circuit depth for a given number of qubits. 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description is merely illustrative and is not intended to limit embodiments and/or application or uses of embodiments. Furthermore, there is no intention to be bound by any expressed or implied information presented in the preceding Background or Summary sections, or in the Detailed Description section. 
     One or more embodiments are now described with reference to the drawings, wherein like referenced numerals are used to refer to like elements throughout. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of the one or more embodiments. It is evident, however, in various cases, that the one or more embodiments can be practiced without these specific details. 
     Quantum computing employs quantum physics principles such as the superposition principle and entanglement to encode information rather than binary digital techniques. For example, a quantum circuit can employ quantum bits (e.g., qubits) that operate according to the superposition principle of quantum physics and the entanglement principle of quantum physics. The superposition principle of quantum physics allows each qubit to represent both a value of “1” and a value of “0” at the same time. The entanglement principle of quantum physics states allows qubits in a superposition to be correlated with each other. For instance, a state of a first value (e.g., a value of “1” or a value of “0”) can depend on a state of a second value. As such, a quantum circuit can employ qubits to encode information rather than binary digital techniques based on transistors. However, design of a quantum circuit can be generally difficult and/or time consuming as compared to conventional binary digital devices. Furthermore, it is generally desirable to increase efficiency of a quantum circuit and/or a quantum computing process. As such, design of a quantum circuit and/or quantum computing processing can be improved. 
     To address these and/or other issues, embodiments described herein include systems, computer-implemented methods, and computer program products facilitating quantum computation. In one embodiment, provided is a system comprising a memory that stores computer executable components, and a processor that executes the computer executable components stored in the memory, wherein the computer executable components comprise an encoding component that encodes an expectation value in the phase of a quantum state; and a learning component that utilizes stochastic inference to determine an eigenvalue corresponding to the expectation value. The system can include a quantum processor. In an aspect, the encoding component can encode the expectation value as a phase. According to certain embodiments, the encoding component can encode the expectation value based on an amplitude of the expectation value. In some embodiments, the stochastic inference can employ Bayesian learning. The learning component can include a measuring component. The measuring component can utilize at least one ancilla qubit to determine the eigenvalue corresponding to the expectation value. The measuring component can produce an output comprising a probabilistic measurement. The probabilistic measurement can be based on the at least one ancilla qubit. 
     In another embodiment, provided is a system comprising a memory that stores computer executable components, and a processor that executes the computer executable components stored in the memory, wherein the computer executable components comprise a learning component that utilizes stochastic inference to determine an eigenvalue corresponding to an expectation value based on a the phase of a quantum state, wherein the learning component comprises a measuring component that probabilistically measures the expectation value, wherein the measuring component is independent of an input state. The system can include a quantum processor. The system can further comprise an encoding component that encodes the expectation value based on the phase of a quantum state. The encoding component can encode the expectation value as a phase. In some embodiments, the encoding component can encode the expectation value based on an amplitude of the expectation value. In some embodiments, the stochastic inference can comprise Bayesian learning, which can include Bayesian inference. According to certain embodiments, the measuring component can utilize at least one ancilla qubit to determine the eigenvalue corresponding to the expectation value. 
     In another embodiment, provided is a computer-implemented method comprising encoding, by a system operatively coupled to a processor, an expectation value associated with the phase of a quantum state; and learning, by the system, an eigenvalue corresponding to the expectation value by stochastic inference. The system can include a quantum processor. According to certain embodiments, the expectation value can be encoded as a phase. In some embodiments, the expectation value can be encoded based on an amplitude of the expectation value. In an aspect, the stochastic inference can include Bayesian learning, which can include Bayesian inference. Learning an eigenvalue can include measuring, by the system, the eigenvalue corresponding to the expectation value by utilizing at least one ancilla qubit. Measuring can produce an output comprising a probabilistic measurement, wherein the probabilistic measurement is based on the at least one ancilla qubit. 
     In another embodiment, provided is a computer-implemented method comprising: learning, by a system operatively coupled to a processor, an eigenvalue corresponding to an expectation value based on a quantum state by stochastic inference, wherein learning comprises measuring, by the system, the expectation value probabilistically, wherein the measuring is independent of an input state. The system can include a quantum processor. The method can further include encoding, by the system, the expectation value as a phase. The encoding can be based on an amplitude of the expectation value. The stochastic inference can include Bayesian learning, which can include Bayesian inference. 
       FIG.  1    illustrates a block diagram of an example, non-limiting system that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. According to multiple embodiments, the system  150  can include memory  165  can store one or more computer and/or machine readable, writable, and/or executable components and/or instructions that, when coupled with a processor  160  and/or quantum processor  162 , can facilitate performance of operations defined by the executable components and/or instructions. The quantum processor  162  can be a machine that performs a set of calculations based on principles of quantum physics. For instance, the quantum processor  162  can perform one or more quantum computations employing a set of quantum circuits. Furthermore, the quantum processor  162  can encode information using qubits which can store information in superposition. Memory  165  can store computer and/or machine readable, writable, and/or executable components and/or instructions that, when executed by a processor  160  or quantum processor  162 , can facilitate execution of the various functions described herein relating to the system  150 , including encoding component  164 , learning component  166 , and measuring component  168 . The encoding component  164  can be operatively coupled to both a processor  160  and quantum processor  162 . The quantum processor  162  can encode based on instructions that are formed and administrated to the quantum processor  162  by a processor  160 . The processor  160  can receive quantum measurements, perform Bayesian learning, determine experimental parameters, update probability/prior parameters, and translate them into instructions for the quantum processor  162 . 
     As described with  FIGS.  2 - 11    below, system  150  can generate an eigenvalue pair  145  from two quantum states  130  in accordance with one or more embodiments described herein. One having skill in the relevant art(s), given the description herein, would appreciate that, as discussed herein, without departing from the spirit of one or more embodiments described herein, eigenvalue can broadly refer to an eigenvector, an eigenvalue, as well as an eigenpair of these components. 
     In one or more embodiments, quantum state  130  can be any state of a quantized system that is represented by quantum numbers, wherein the quantum numbers can be numbers that describe the values of conserved quantities in a quantum system. The quantum state  130  can be employed with technologies such as, but not limited to, quantum processing technologies, quantum circuit technologies, quantum computing design technologies, artificial intelligence technologies, machine learning technologies, search engine technologies, image recognition technologies, speech recognition technologies, model reduction technologies, iterative linear solver technologies, data mining technologies, healthcare technologies, pharmaceutical technologies, biotechnology technologies, finance technologies, chemistry technologies, material discovery technologies, vibration analysis technologies, geological technologies, aviation technologies, and/or other technologies. As a quantum state  130  can be any state of a quantized system that is represented by quantum numbers, wherein the quantum numbers can be numbers that describe the values of conserved quantities in a quantum system, the quantum state  130  can represent properties of systems in these technologies. 
     In one or more embodiments, quantum state  130  can comprise an associated expectation value. The expectation value can be a statistical mean of measured values of an observable property, which can be the result of a linear operator operating on a Hilbert space. The encoding component  164  can encode an expectation value in the phase of a quantum state  130 . The encoding component  164  can receive an at least partial quantum state and encode the expectation value as a phase. The phase can be encoded by successive iterations of a circuit that effectively multiply a quantum state  130  by a complex exponential that includes an amplitude. The encoding component  164  can encode the expectation value based on an amplitude of the expectation value. The encoding component  164  can execute on a quantum processor  162 . The encoding component  164  can encode a spectrum or an eigenvalue as a phase which can be inferred utilizing the processor  160 . 
     The learning component  166  can utilize stochastic inference to determine an eigenvalue pair  145  corresponding to the expectation value. Stochastic inference can comprise Bayesian learning. The learning component can utilize a quantum processor  162  and a processor  160 . 
     Stochastic inference can include updating the posterior probability portion of Bayes&#39; law based on obtaining at least one updated sample in a Bayesian learning process. Exploiting Bayes&#39; law in this fashion, a measuring component  168  can measure the at least one ancilla qubit to provide a probabilistic output that reveals information about an eigenvalue pair  145 . The learning component can include a measuring component  168  that can utilize at least one ancilla qubit to determine the eigenvalue pair  145  corresponding to the expectation value. An ancilla qubit can be a qubit that can be used to store temporary information that can be neither an input qubit nor an output qubit. 
     The learning component  166  can receive at least one ancilla qubit  205  (e.g. by a learning component  166  comprising a quantum circuit  220 ). The quantum circuit  220  can include a Pauli rotation that depends on a control parameter θ. The control parameter θ can depend on the prior, including the approximation parameter, μ. The control parameter m of the circuit can determine an amount of controlled unitary action that can be utilized by an encoding component  164 . The control parameter m can be determined by the prior, including the variance, σ. 
       FIG.  2 A  illustrates an example approach that can use circuit  202  to select a control parameter m that can be used by encoding component  164  to encode an expectation value in the phase of a quantum state  130 , in accordance with one or more embodiments. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. 
     In one or more embodiments, the intuition for control parameter m can be the same as in the full phase estimation algorithm. One having skill in the relevant art(s) given the description herein, would appreciate that circuit  202  can be used to, peel away m copies of the eigenvalue. In another use of circuit  202 , using the full algorithm, control parameter m can run through powers of two, and select a value for control parameter m that can maximize the information gain at each step. 
     Considering the above in greater detail, circuit  202  can, in one or more embodiments, use formulas  203  for points  260  A-E, to determine control parameter m, in accordance with one or more embodiments. in an example of this process, point  260  D corresponds to a ‘phase kick-back’ that can leave the eigenvector register unchanged, but change the phase of the control qubit. Additional discussion of the encoding process of encoding component  164  is included with a discussion of different approaches to measuring, discussed below. 
       FIG.  2 B  illustrates a block diagram of an example, non-limiting system  200  that shows the interaction of the encoding component  164  and learning component  166  in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. 
     The quantum circuit  220  encodes at least one ancilla qubit  205  qubit into a quantum state for measuring  222  that can be received by the encoding component  164 . The quantum circuit  220  encodes at least one ancilla qubit  205  qubit into a quantum state for measuring  222  that can be received by the measuring component  168 . The quantum circuit  220  of the learning component can be executed on a quantum circuit operatively connected to a quantum processor  162 . 
     The measuring component  168  can produce an output comprising a probabilistic measurement. The probabilistic measurement can be based on at least one ancilla qubit. The learning component  166  can execute partially on a quantum processor  162  and partially on the processor  160 ; the measuring component  168  can interface between a quantum portion, utilizing the quantum processor  162 , and a classical portion, utilizing the processor  160 . 
     The measuring component  168  can learn/approximate the eigenvalue pair  145  by Bayesian learning, which can include Bayesian inference. Exploiting Bayes&#39; law, the probability of a particular phase  230  (φ), given outcome (E), a variance, σ, and a mean of the initial prior function. For example, Bayesian analysis can begin with a prior over ϕ, namely P 0 (ϕ) and can proceed by the formulas below: 
                 P   ⁡   (     0   |   ϕ   ;   θ   ;   m     )     =       1   +     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2       ⁢   
       P   ⁡   (     1   |   ϕ   ;   θ   ;   m     )     =       1   -     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2       ⁢   
         P     i   +   1       (     ϕ   |     E     i   +   1       ;     θ   i     ,     m   i       )     =         P   ⁡   (       E     i   +   1       |     ϕ   i     ;     θ   i     ,     m   i       )     ⁢       P   i     (   ϕ   )       N             
P(φ) can be updated each sample by the update component  504 . The measuring component  168  can send its output to an update component. The update component can receive probabilistic output from the measuring component  168  and utilize that probabilistic output to provide a new estimate of the eigenvalue pair  145  to the measuring component  168 . The measuring component  168  can include a classical statistical model based on a sampling filter, which can filter based on stochastic inference. The measuring component  168  can perform a quantum measurement on the output of the encoding component  164 . Based on a classical value output of the quantum measurement, successive iterations can produce a classical output  240 .
 
     The Quantum Phase Estimation method can employ a circuit depth that is technologically unfeasible with near term quantum computing. To circumvent this, a portion of the circuit depth is replaced with a classical statistical model based upon a sampling filter. In preparation, a quantum measurement is performed on the last stage of the encoding circuit  300 , as shown in  FIG.  3   , and the measurement is done in a certain basis, |k&gt;. After the quantum measurement is performed giving a classical value, iterations the realizations of the circuit to produce an ensemble. 
     One or more embodiments can perform amplitude estimation using different approaches. In one approach to amplitude estimation that saturates the Heisenberg limit is to use Quantum Phase Estimation. Quantum Phase Estimation can employ entanglement (via the inverse Quantum Fourier Transform) between repeated applications of U to extract the phase. An iterative version of Quantum Phase Estimation exists that can use feed-forward measurements instead of entanglement. 
     In one approach to using phase estimation to do amplitude estimation, first the amplitude can be encoded into the phase, second, an operator can be defined for which eigenvectors are known, e.g., because phase estimation relies on its preparation. In some circumstances, the U from amplitude estimation can be arbitrary, and neither eigenvalue pairs  145  nor eigenvectors are known a priori. 
     In one approach to defining the operator, a new rotation operator, S (related to U) can be built that, by design, can leave a certain subspace invariant. In this approach, because S can be a two-dimensional rotation the form of the two eigenvalue pairs  145  is: e ±iϕ . Furthermore, because in some circumstances there are only two eigenvalue pairs  145  which differ only with respect to the signs of their phases, as long as the initial state is prepared within the subspace, “both” phases can be determined in one run of full quantum phase estimation. One having skill in the relevant art(s), given the disclosure herein, would appreciate that one or more embodiments can use this dual eigenvalue property in a Bayesian iterative context, e.g., without using an iterative algorithm that can require projection onto one of the eigenvectors. 
     In an example below, the following operator can have the above described beneficial properties, with a rotation by ϕ=|&lt;ψ|U|ψ&gt;| that operates, in this example, only in a two-dimensional subspace (spanned by |ψ 0 &gt; and |ψ 1 &gt;):
 
 S=S   0   S   1  
 
 S   0 =Π−2|ψ 0 &gt;&lt;ψ 0 |
 
 S   1 =Π−2|ψ 1 &gt;&lt;ψ 1 |=Π−2 U|ψ   0 &gt;&lt;ψ 0   |U   † 
 
where S x |ψ&gt;, x∈{0,1} is a sign flip of the component of |ψ&gt; in the direction of |ψ x &gt;. Each S x  can operate in a two dimensional subspace spanned by the state being acted upon and |ψ x &gt;. Therefore, if the process is started in the sub-space spanned by |ψ 0 &gt; and |ψ 1 &gt;, the results can be determined by remaining in the sub-space.
 
     In one or more embodiments, a sign flip of the component in the direction of a vector can be a reflection around the line perpendicular to the vector, e.g., one or more embodiments can evaluate reflections about axes. In an example implementation, two reflections around the same axes can return a vector to where it began, and two reflections about two non-coinciding axes can rotate the vector by exactly twice the angle between the two axes. This can be illustrated by how a reflection about the second axis can be seen as rotation to the first axis, followed by reflection about the second axis, picking up a rotation of twice the angle between the axis followed by a rotation of the same angle as the first. Based at least on these reflections, the angle of separation can be transferred between two vectors (this being related to the inner-product between the two vectors) into the angle of rotation of any vector in the plane. In one or more embodiments, S can be a (real) rotation by an angle ϕ=2 cos −1 (| ψ 0 |ψ 1   |), therefore | ψ 0 |ψ 1   |=|cos(ϕ/2)|. 
     In one or more embodiments, sign flips along an arbitrary vector can be written as a sign flip Π along the zero vector |0&gt;, by rotating to the zero vector, performing the sign flip Π and rotating back. Therefore, if the construction of |ψ 0 &gt; is specified as the unitary V acting on |0&gt;, then:
 
 S   0   =VΠV   † 
 
 S   1   =UVΠV   †   U   † 
 
 S=VΠV   †   UVΠV   †   U   † 
 
     Thus, full phase estimation of S operating on |ψ 0 &gt; can reveal either ϕ or −ϕ and therefore can result in the desired amplitude. 
     In some circumstances a ‘phase kick-back’ can leave the eigenvector register unchanged but can be seen as changing the phase of the control qubit. 
     Therefore: 
     
       
         
           
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   0 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   + 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   1 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   - 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
           
         
       
     
     The measuring component  168  can produce an accurate measurement without many samples by measuring intelligently, e.g., in a strategic way. This probabilistic measurement by the measuring component  168  can utilize Bayesian learning and stochastic inference upon the quantum results. The measuring component  168  can perform these tasks by updating a prior distribution using Bayes&#39; law. 
     The eigenvalue pair  145  can be a value corresponding to the expectation value in the phase of the quantum state  130 . The eigenvalue pair  145  can be a value of an observable physical quality of a linear operator operating on a Hilbert space. A Hilbert space is a vector space that is topologically closed under finite vector addition and scalar multiplication with an inner product that allows properties like length and angle to be measured. For any observable physical quantity, there exists a linear operator, which can be a Hermitian operator, acting on the physical environment, which can be a Hilbert space. When we make a measurement of property in the environment, we obtain one of the eigenvalues of the operator. Therefore, any observable physical quantity can be modeled as the eigenvalue of a linear operator operating on a Hilbert space. 
     The encoding component  164  can receive a quantum state  130 . The encoding component  164  can encode a spectrum or an eigenvalue as a phase  230 , which can then be inferred classically, by a processor  160 . The classical inference can be performed by the measuring component  168  by exploiting Bayes&#39; law, as the probability of a particular phase  230  (φ), given outcome (E), a variance, σ, and a mean of the initial prior function. For example, Bayesian analysis can begin with a prior over ϕ, namely P 0 (ϕ) and can proceed by the formulas below: 
     
       
         
           
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   0 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   + 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   1 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   - 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 
                   P 
                   
                     i 
                     + 
                     1 
                   
                 
                 ( 
                 
                   ϕ 
                   | 
                   
                     E 
                     
                       i 
                       + 
                       1 
                     
                   
                   ; 
                   
                     θ 
                     i 
                   
                   , 
                   
                     m 
                     i 
                   
                 
                 ) 
               
               = 
               
                 
                   
                     P 
                     ⁡ 
                     ( 
                     
                       
                         E 
                         
                           i 
                           + 
                           1 
                         
                       
                       | 
                       
                         ϕ 
                         i 
                       
                       ; 
                       
                         θ 
                         i 
                       
                       , 
                       
                         m 
                         i 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       P 
                       i 
                     
                     ( 
                     ϕ 
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                 N 
               
             
           
         
       
     
     P(φ) can be updated each sample by the update component  504 . 
     The encoding component can utilize a quantum processor  162 . The encoding component can send the quantum state  130  to an encoding circuit  210  to encode the quantum state  130  as a phase  230  with an encoding circuit  300 , as shown in  FIG.  3   . The encoding circuit  210  can receive the quantum state  130  and input from the learning component  166 . Utilizing the quantum state  130  and input from the learning component  166 , the encoding circuit  210  can output a phase  230 . This encoding can be performed with an encoding circuit  300 . The encoding circuit  300  can be thought of as repeated iterations of S, where S=S 0 S 1 , wherein S 0 =VΠV † =I−2|ψ 0 &gt;&lt;ψ 0 | and S 1 =UVΠV † U † =I−2U|ψ 0 &gt;&lt;ψ 0 |U † . V represents a circuit preparation gate, V †  is the conjugate of V, U represents a unitary operator to infer a spectrum, U †  is the conjugate of U, and Π=I−2|0&gt;&lt;0|. Iterations of S lead to encoding the expectation value as a phase, φ(λ), wherein φ(λ)=2 arccos(&lt;ψ(λ)|H|ψ(λ)&gt;). This can encode an expectation value of the quantum state  130 , |0&gt;|ψ&gt; as e −i|&lt;ψ(λ)|H|ψ(λ)&gt;| |0&gt;|ψ&gt;, wherein |&lt;ψ(λ)|H|ψ(λ}&gt;| can be an amplitude of the expectation value. The learning component  166  receives at least one ancilla qubit  205  and sends the at least one ancilla qubit  205  to a quantum circuit  220 . In one or more embodiments, quantum circuit  220  can depend on control parameters m and θ which in turn can depend on the probability/prior parameters μ and σ. 
     The approximation parameter μ can be set by a Bayesian probability distribution with a tuning parameter, based on active learning (γ). In one or more parameters, active learning can be applied to select new control parameters, potentially revealing better quality information at every Bayesian update step, as discussed above. According to certain embodiments, γ∈[0,1]. The tuning parameter can enable a gradual steering between a quantum estimation and a classical estimation. In some embodiments, a quantum estimation can be used when γ=0 and a classical estimation can be used when γ=1. 
     Steering between a quantum estimation and a classical estimation by the encoding component  164  using a tuning parameter, can incorporate some of the lower costs of both quantum and classical algorithms, as shown in  FIGS.  12 A and  12 B . It should be noted that the eigenvector-collapse free method of amplitude estimation described in one or more embodiments herein, can be a subroutine of a determination of VQE, which can confer benefits in some circumstances, with these benefits including shorter depth circuits and fewer samples being required, as compared to naive sampling. In VQE the goal is to produce an approximation to the ground-state and calculate energy (the lowest possible) by varying variational parameters λ. 
     This is made possible by the variational principle, which states that energy of an arbitrary state is always greater than the ground state. Our amplitude estimation method can be used as a subroutine in VQE to calculate the energy level of the current quantum state, e.g., starting with the ansatz and refining. 
     The samples which can be employed to achieve a relatively low error ϵ are very high for VQE  1210 . However, even at relatively low errors, γ-QPE  1220  compares quite well with QPE  1230 , while not requiring the high fault tolerance of QPE. The circuit depth for a given number of processed qubits goes up very quickly for QPE  1240 . However, even at relatively high amounts of processing, γ-QPE  1260  compares favorably with VQE  1250 . Quantum algorithms can be used for the solution of linear systems of equations, factorization and eigenvalue decomposition, for example. However, a problem for realizing the potential of some quantum algorithms is that some quantum algorithms can scale proportionally to O (1/∈), wherein O denotes a function of the asymptotic upper bound approaching infinity, to obtain, a level of precision, ε. However, classical algorithms can have a much greater computational cost in terms of sample counts relative to quantum algorithms to form a proficient variational form for the representation of an eigenstate. The computational cost for a classical algorithm can grow proportional to O(1/ε 2 ), however, each realization can necessitate a repeat preparation of the state. By first encoding the expectation value in a quantum manner and preparing it for a classical inference, a substantial speed up can be achieved. 
     Quantum Phase Estimation (QPE) and Variational Quantum Eigen (VQE) solver are instrumental for the computation of eigenvalues. Quantum Phase Estimation can offer an exponential advantage over classical computation, in simulation of the evolution of the state of n qubits by matrix multiplication. The algorithm can be used in applications such as the solution of linear systems of equations, factorization, eigenvalue decomposition, etc. A key issue for realizing the potential of QPE is that the circuit depth scales like 
             𝒪   ⁡   (     1   ϵ     )         
in order to attain e precision. A popular alternative is the VQE algorithm that replaces the long coherence of QPE with multiple shorter coherence time expectation realizations. The drawbacks of this algorithm forming a proficient variational form for the representation of the eigenstate, and the cost of computation in terms of sample counts (which grow like
 
             𝒪   ⁡   (     1     ϵ   2       )         
for ϵ error, while repeated preparation of the state for each iteration can be employed for each realization).
 
     A variational quantum phase estimation approach is considered that combines ideas associated with rejection filtering and neural network inference. The present embodiments offer almost continuous steering between VQE and QPE, based upon the availability of computational resources. For this reason, the algorithm is termed herein γ-QPE. In addition to the benefits described above, merits of using a hybrid quantum classical approach include that the hybrid approach can utilize the strength of each domain to obtain the best tradeoff between the error ϵ, circuit depth and the computational complexity. 
     The quantum circuit  220  encodes the at least one ancilla qubit  205  into a quantum state for measuring  222  and provides input to the encoding circuit  210 . The quantum state for measuring  222  can be sent to the measuring component  168 , which can measure the ancilla qubit  205  to provide a classical output  240 , which can be a probabilistic output and can reveal information about an eigenvalue pair  145 . The quantum state for measuring  222  can be a complex valued probability function that represents a probability or probability density for a given value. 
       FIG.  3    illustrates an example, non-limiting encoding circuit  300  that shows an encoding component in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. 
     The encoding circuit  300  can receive a quantum state  130 . The encoding circuit  300  passes the eigenstate to  304 , which can be a conjugate of a unitary operator, P. P can be any unitary operator that can generate information that can be employed by the encoding circuit  300  to generate an inference of a spectrum. P can be a Hamiltonian. The output  305  of the conjugate of the unitary operator  304  goes to the conjugate of the circuit preparation circuit  306 . The output of the circuit preparation circuit  306  goes to the first order projection operator  308 . By way of example, the first order projection operator  308 , Π, can be represented as Π=I−2|0&gt;&lt;0|, wherein I is the identity matrix. The quantum state for measuring  222  based on the at least one ancilla qubit  205  can be passed to both the conjugate of the unitary operator  304  and to a Hadamard gate  302 . The Hadamard gate  302  can receive a single qubit and bring it to a state superposition where its output has an equal probability of being measured classically as either 0 or 1. The Hadamard gate can be represented as a unitary matrix that is a combination of two rotations, 180 degrees about the Z-axis and 90 degrees about the Y-axis. The output  303  of the Hadamard gate  302  can be also passed to the first order projection operator  308 , H, can be represented as Π=I−2|0&gt;&lt;0|, wherein I is the identity matrix. The first order projection operator  308  can be a vacuum projection operator. The output  309  of the first order projection operator  308  can be passed to both another Hadamard gate  302  and to the circuit preparation circuit  310 . As shown at the upper portion of  FIG.  3   , the output of the Hadamard gate  302  can be passed to both another Hadamard gate  302  and to a unitary operator  312 . The output  303  of the Hadamard gate  302  can be passed to another first order projection operator  308 , which can be represented as Π=I−2|0&gt;&lt;0|. As shown at the bottom portion of  FIG.  3   , the output  309  of the first order projection operator  308  goes to a circuit preparation circuit  310 . The output  311  of the circuit preparation circuit  310  goes to the unitary operator  312 . The output  313  of the unitary operator  312  can be passed to the conjugate of the circuit preparation circuit  306 . Like the output  303  of the Hadamard gate  302 , the output  307  of the conjugate of the circuit preparation circuit  306  can be passed to the first order projection operator  308 . 
     The output  303  of the first order projection operator  308  can be passed to a Hadamard gate  302  and to a circuit preparation circuit  310 . This encoding circuit  300  can be iterated as necessary to reduce error in encoding a phase. 
       FIG.  4    illustrates an example, non-limiting geometric representation of the example, non-limiting circuit of  FIG.  3   . The encoding circuit  300  can encode an absolute expectation value as a phase by a rotation S, where S=S 0 S 1 . The phase can be measured. The encoding circuit  300  can utilize partial projection operators. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. As shown on the Bloch sphere  400 , geometrically, the circuit of  FIG.  3    can be thought of as repeated iterations of S, where S=S 0 S 1 , wherein S 0 =VΠV † =I−2|ψ 0 &gt;&lt;ψ 0 | and S 1 =UΠV † U † =I−2U|ψ 0 &gt;&lt;ψ|ψ 0   † . S can be a rotation by an angle ϕ=2 arccos  ψ(Δ)|H|ψ(λ) . S 0  and S 1  can coincide with the geometrical axis shown in  FIG.  4    and can be a geometrical representation and rotation of subset projection operators. The first order projection operator  308  can be a vacuum operator which can be Π=I−2|0&gt;&lt;0|. Iterations of S lead to encoding the expectation value as a phase, φ(λ), wherein φ(λ)=2 arccos(&lt;ψ(λ)|H|ψ(λ)&gt;). 
       FIG.  5    illustrates a block diagram of an example, non-limiting system that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. 
     The quantum state  130  can be received by the encoding component  164 . The encoding component  164  can encode the quantum state  130  as a phase  230  based on an amplitude of the expectation value of the quantum state  130 . The phase  230  can be passed to the learning component  166 . The encoding component can be executed by a quantum processor  162 . 
     The learning component  166  can receive an at least one ancilla qubit  205 . The at least one ancilla qubit  205  can be received by the quantum circuit  220  of the learning component  166 . The quantum circuit  220  can be a Pauli rotation that includes a variance, σ, and a mean of the initial prior function, μ. The quantum circuit  220  encodes at least one ancilla qubit into a quantum state for measuring  222  that can be received by the encoding component  164 . The quantum circuit  220  of the learning component  166  can be executed on a quantum circuit operatively connected to a quantum processor  162 . 
     The learning component  166  can comprise a measuring component  168 . The measuring component  168  can receive a quantum state for measuring  222  from the quantum circuit  220 . The probability of a given outcome, which can be a value corresponding to the expectation value of the quantum state  130 , can be calculated explicitly on by the measuring component  168  according to the equation, below, 
                 P   ⁡   (     0   |   ϕ   ;   θ   ;   m     )     =       1   +     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2       ⁢   
       P   ⁡   (     1   |   ϕ   ;   θ   ;   m     )     =       1   -     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2             
thereby measuring the eigenvalue probabilistically based on the at least one ancilla qubit. The measuring component can receive the phase  230  from the encoding component  164 , represented as φ(λ)=2 arccos(&lt;ψ(λ)|H|ψ(λ)&gt;).
 
     The encoding component  164  can receive a quantum state  130 . The encoding component  164  can encode a spectrum or an eigenvalue as a phase  230 , which can then be inferred classically, by a processor  160 . The classical inference can be performed by the measuring component  168  by exploiting Bayes&#39; law, as the probability of a particular phase  230  (φ), given outcome (E), a variance, σ, and a mean of the initial prior function. For example, Bayesian analysis can begin with a prior over ϕ, namely P 0 (ϕ) and can proceed by the formulas below: 
     
       
         
           
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   0 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   + 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   1 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   - 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 
                   P 
                   
                     i 
                     + 
                     1 
                   
                 
                 ( 
                 
                   ϕ 
                   | 
                   
                     E 
                     
                       i 
                       + 
                       1 
                     
                   
                   ; 
                   
                     θ 
                     i 
                   
                   , 
                   
                     m 
                     i 
                   
                 
                 ) 
               
               = 
               
                 
                   
                     P 
                     ⁡ 
                     ( 
                     
                       
                         E 
                         
                           i 
                           + 
                           1 
                         
                       
                       | 
                       
                         ϕ 
                         i 
                       
                       ; 
                       
                         θ 
                         i 
                       
                       , 
                       
                         m 
                         i 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       P 
                       i 
                     
                     ( 
                     ϕ 
                     ) 
                   
                 
                 N 
               
             
           
         
       
     
     P(φ) can be updated each sample by the update component  504 . 
     The measuring component  168  can send its output to an update component  504 . The measuring component  168  can execute on both the quantum processor  162  and processor  160 . The update component  504  can receive the probabilistic output  508  of the measuring component  168  and utilize that probabilistic output  508  to provide a new upper bound  506  of the eigenvalue pair  145  to the measuring component  168 . |&lt;ψ(λ)|H|ψ(λ}&gt;| is the absolute value of the expectation value, which can be encoded as the phase of a quantum state, |0&gt;|ψ&gt;, quantum state  130  can be transformed by the encoding component  164  into e −i|&lt;ψ(λ)|H|ψ(λ)&gt;| |0&gt;|ψ&gt;. The expression &lt;ψ|H|ψ&gt; can also be expressed as: 
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       λ1 
                       , 
                       
                         λ2 
                         ∈ 
                         
                           Spec 
                           ⁡ 
                           ( 
                           H 
                           ) 
                         
                       
                     
                   
                   
                     &lt; 
                     
                       ψ 
                       | 
                       
                         ψ 
                         λ1 
                       
                       &gt; 
                       &lt; 
                       
                         ψ 
                         λ1 
                       
                       | 
                       H 
                       | 
                       
                         ψ 
                         λ2 
                       
                       &gt; 
                       &lt; 
                       
                         ψ 
                         λ2 
                       
                       | 
                       ψ 
                     
                     &gt; 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
     which is equivalent to 
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       λ 
                       ∈ 
                       
                         Spec 
                         ⁡ 
                         ( 
                         H 
                         ) 
                       
                     
                   
                   
                     λ 
                     | 
                     &lt; 
                     
                       ψ 
                       λ 
                     
                     | 
                     ψ 
                     &gt; 
                     
                       | 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                         
                     
                       ( 
                       5 
                       ) 
                     
                   
                 
               
             
           
         
       
     
     Equation (4) and Equation (5) can be always greater than or equal to 
                       ∑     λ   ∈     Spec   ⁡   (   H   )             E   0       ❘   &#34;\[LeftBracketingBar]&#34;     &lt;     ψ   λ     |   ψ   &gt;       ❘   &#34;\[RightBracketingBar]&#34;     2         =       E   0     .             Equation   ⁢           (   6   )                 
wherein E 0  is the minimum eigenvalue of H. VQE, employs this variational principle to search for the lowest eigenvalue by adjusting the variational parameters λ. At each stage of adjustment the expectation value needs to be calculated. The learning component  166  can utilize both stochastic inference and Bayesian inference to calculate the expectation value. After obtaining a sample, the posterior probability (P(φ|E, μ, σ)) of the Bayesian learning equation can be updated with a new variance (σ) and mean of the initial function (μ). Further, as σ and μ are updated by the update component  504 , the measuring component  168  can be independent of a given initial state.
 
       FIG.  6    illustrates a flow diagram of an example, non-limiting computer-implemented method that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. As shown by the diagram  600 , at  602 , a quantum state  130  can be encoded (e.g., via the encoding component  164 ). The quantum state  130  can be associated with technologies such as, but not limited to, quantum processing technologies, quantum circuit technologies, quantum computing design technologies, artificial intelligence technologies, machine learning technologies, search engine technologies, image recognition technologies, speech recognition technologies, model reduction technologies, iterative linear solver technologies, data mining technologies, healthcare technologies, pharmaceutical technologies, biotechnology technologies, finance technologies, chemistry technologies, material discovery technologies, vibration analysis technologies, geological technologies, aviation technologies, and/or other technologies. The quantum state  130  can be encoded as a phase  230 . The encoding component can receive a quantum state for measuring  222  from a learning component  166 . The encoding component can pass the phase  230  to the learning component  166 . The encoding component  164  can a quantum state  130 , which can be an at least partial eigenstate. The encoding component can send the quantum state  130  to an encoding circuit  210 . The encoding circuit  210  can receive the quantum state  130  and input from the learning component  166 . Utilizing the quantum state  130  and input from the learning component  166 , the encoding circuit  210  can output a phase  230 . 
     At  604 , in one or more embodiments, learning component  166  can learn an eigenvalue corresponding to the expectation value by stochastic inference. Learning component  166  can receive a phase  230  from the encoding component  164 . The learning component can pass a quantum state for measuring  222  based on the at least one ancilla qubit  205  to the encoding component  164 . In one or more embodiments, quantum circuit  220  can depend on control parameters m and θ which in turn can depend on the probability/prior parameters μ and σ. The quantum circuit  220  encodes the at least one ancilla qubit  205  into a quantum state for measuring  222  and provides input to the encoding circuit  210 . The quantum state for measuring  222  can be sent to the measuring component  168 , which can measure to provide a classical output  240 , which can be an eigenvalue pair  145 . In one or more embodiments, learning component  166  can learn an eigenvalue corresponding to the expectation value by stochastic inference by a process that comprises measuring the quantum state  606  to produce an output comprising a probabilistic measurement. 
       FIG.  7    illustrates a flow diagram of an example, non-limiting computer-implemented method that can encode a quantum state  130  as a phase  230  in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. As shown in the block diagram  700 , at  702 , a quantum state can be received (e.g., by an encoding component  164  comprising an encoding circuit  300 ). The encoding circuit  300  passes the eigenstate to  304 , which can be a conjugate of a unitary operator, P. P can be any unitary operator that can generate information that can be employed by the encoding circuit  300  to generate an inference of a spectrum. P can be a Hamiltonian. The output  305  of the conjugate of the unitary operator  304  goes to the conjugate of the circuit preparation circuit  306 . The output of the circuit preparation circuit  306  goes to the first order projection operator  308 . By way of example, the first order projection operator  308 , H, can be represented as Π=I−2|0&gt;&lt;0|, wherein I is the identity matrix. 
     At  704 , a quantum state for measuring  222  based on an at least one ancilla qubit  205  can be received (e.g., by an encoding component  164  comprising an encoding circuit  300 ). The quantum state for measuring  222  based on the at least one ancilla qubit  205  can be passed to both the conjugate of the unitary operator  304  and to a Hadamard gate  302 . The Hadamard gate  302  can receive a single qubit and bring it to a state superposition where its output has an equal probability of being measured classically as either 0 or 1. The Hadamard gate can be represented as a unitary matrix that is a combination of two rotations, 180 degrees about the Z-axis and 90 degrees about the Y-axis. The output  303  of the Hadamard gate  302  can be also passed to the first order projection operator  308 , H, can be represented as Π=I−2|0&gt;&lt;0|, wherein I is the identity matrix. 
     The output of the first order projection operator  308  can be passed to both another Hadamard gate  302  and to the circuit preparation circuit  310 . As shown at the upper portion of  FIG.  3   , the output of the Hadamard gate  302  can be passed to both another Hadamard gate  302  and to a unitary operator  312 . The output of the Hadamard gate  302  can be passed to another first order projection operator  308 , which can be represented as Π=I−2|0&gt;&lt;0|. As shown at the bottom portion of  FIG.  3   , the output  309  of the first order projection operator  308  goes to a circuit preparation circuit  310 . The output  311  of the circuit preparation circuit  310  goes to the unitary operator  312 . The output  313  of the unitary operator  312  can be passed to the conjugate of the circuit preparation circuit  306 . Like the output  303  of the Hadamard gate  302 , the output  307  of the conjugate of the circuit preparation circuit  306  can be passed to the first order projection operator  308 . 
     The output  303  of the first order projection operator  308  can be passed to a Hadamard gate  302  and to a circuit preparation circuit  310 . This encoding circuit  300  can be iterated as necessary to reduce error in encoding a phase. 
     At  706 , the quantum state can be encoded as a phase (e.g., by an encoding component  164  comprising an encoding circuit  300 ). The encoding component  164  can execute on a quantum processor  162 . Iterations of the encoding circuit  300 , can be represented mathematically as repeated iterations of S, where S=S 0 S 1 , wherein S 0 =VΠV † =I−2|ψ 0 &gt;&lt;ψ 0 | and S 1 =UVΠV † U † =I−2U|ψ 0 &gt;&lt;ψ 0 |U † . Iterations of S lead to encoding the expectation value as a phase, φ(λ), wherein φ(λ)=2 arccos(&lt;ψ(λ)|H|ψ(λ)&gt;). This can encode an expectation value of the quantum state  130 , |0&gt;|ψ&gt; as e −i|&lt;ψ(λ)|H|ψ(λ)&gt;| |0&gt;|ψ&gt;, wherein I&lt;ψ(λ)|H|ψ(λ)&gt;| can be an amplitude of the expectation value. 
       FIG.  8    illustrates a flow diagram  800  of a computer-implemented method that can learn an eigenvalue corresponding to an expectation value based on a quantum state in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. At  802 , the learning component  166  can receive an at least one ancilla qubit  205  (e.g. by a learning component  166  comprising a quantum circuit  220 ). In one or more embodiments, quantum circuit  220  can depend on control parameters m and θ which in turn can depend on the probability/prior parameters μ and σ. The quantum circuit  220  encodes at least one ancilla qubit  205  qubit into a quantum state for measuring  222  that can be received by the encoding component  164 . The quantum circuit  220  encodes at least one ancilla qubit  205  qubit into a quantum state for measuring  222  that can be received by the measuring component  168 . The quantum circuit  220  of the learning component can be executed on a quantum circuit operatively connected to a quantum processor  162 . 
     At  804 , the eigenvalue corresponding to the expectation value can be determined by measuring an ancilla qubit (e.g., via a measuring component  168 ). The learning component  166  can comprise a measuring component  168 . The measuring component  168  can receive a quantum state for measuring  222  from the quantum circuit  220 . The probability of a given eigenvalue pair  145 , which can be a value corresponding to the expectation value of the quantum state  130 , can be calculated explicitly on by the measuring component  168  according to the equations below, 
                 P   ⁡   (     0   |   ϕ   ;   θ   ;   m     )     =       1   +     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2       ⁢   
       P   ⁡   (     1   |   ϕ   ;   θ   ;   m     )     =       1   -     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2       ⁢   
         P     i   +   1       (     ϕ     ❘   &#34;\[LeftBracketingBar]&#34;       E     i   +   1       ;     θ   i     ;     m   i       )     =         P   ⁡   (       E     i   +   1       |   ϕ   ;     θ   i     ,     m   i       )     ⁢       P   i     (   ϕ   )       N             
thereby measuring the eigenvalue probabilistically based on the at least one ancilla qubit. The measuring component can receive the phase  230  from the encoding component  164 , represented as φ(λ)=2 arccos(&lt;ψ(λ)|H|ψ(λ)&gt;).
 
     The classical inference can be performed by the measuring component  168  by exploiting Bayes&#39; law, as the probability of a particular phase  230  (φ), given outcome (E), a variance, σ, and a mean of the initial prior function. For example, Bayesian analysis can begin with a prior over ϕ, namely P 0 (ϕ) and can proceed by the formulas below: 
     
       
         
           
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   0 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   + 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   1 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   - 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 
                   P 
                   
                     i 
                     + 
                     1 
                   
                 
                 ( 
                 
                   ϕ 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     E 
                     
                       i 
                       + 
                       1 
                     
                   
                   ; 
                   
                     θ 
                     i 
                   
                   ; 
                   
                     m 
                     i 
                   
                 
                 ) 
               
               = 
               
                 
                   
                     P 
                     ⁡ 
                     ( 
                     
                       
                         E 
                         
                           i 
                           + 
                           1 
                         
                       
                       | 
                       ϕ 
                       ; 
                       
                         θ 
                         i 
                       
                       , 
                       
                         m 
                         i 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       P 
                       i 
                     
                     ( 
                     ϕ 
                     ) 
                   
                 
                 N 
               
             
           
         
       
     
     P(φ) can be updated each sample by the update component  504 . 
     At  806 , the upper bound in the measuring component  168  can be updated (e.g., via an update component  504 ). The measuring component  168  can measure the eigenvalue pair  145  by Bayesian learning, which can include Bayesian inference. The classical inference can be performed by the measuring component  168  by exploiting Bayes&#39; law, as the probability of a particular phase  230  (φ), given outcome (E), a variance, σ, and a mean of the initial prior function. For example, Bayesian analysis can begin with a prior over ϕ, namely P 0 (ϕ) and can proceed by the formulas below: 
     
       
         
           
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   0 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   + 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 P 
                 ⁡ 
                 ( 
                 
                   1 
                   | 
                   ϕ 
                   ; 
                   θ 
                   ; 
                   m 
                 
                 ) 
               
               = 
               
                 
                   1 
                   - 
                   
                     cos 
                     ⁡ 
                     ( 
                     
                       m 
                       ⁡ 
                       ( 
                       
                         θ 
                         - 
                         ϕ 
                       
                       ) 
                     
                     ) 
                   
                 
                 2 
               
             
             ⁢ 
             
 
             
               
                 
                   P 
                   
                     i 
                     + 
                     1 
                   
                 
                 ( 
                 
                   ϕ 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     E 
                     
                       i 
                       + 
                       1 
                     
                   
                   ; 
                   
                     θ 
                     i 
                   
                   ; 
                   
                     m 
                     i 
                   
                 
                 ) 
               
               = 
               
                 
                   
                     P 
                     ⁡ 
                     ( 
                     
                       
                         E 
                         
                           i 
                           + 
                           1 
                         
                       
                       | 
                       ϕ 
                       ; 
                       
                         θ 
                         i 
                       
                       , 
                       
                         m 
                         i 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       P 
                       i 
                     
                     ( 
                     ϕ 
                     ) 
                   
                 
                 N 
               
             
           
         
       
     
     P(φ) can be updated each sample by the update component  504 . 
     The measuring component  168  can send its output to an update component  504 . The measuring component  168  can execute on both the quantum processor  162  and processor  160 . The update component  504  can receive the probabilistic output  508  of the measuring component  168  and utilize that probabilistic output  508  to provide a new upper bound  506  of the eigenvalue pair  145  to the measuring component  168 . |&lt;ψ(λ)|H|ψ(λ}&gt;| can be an amplitude on which the phase  230  can be encoded, as the quantum state  130 , |0&gt;|ψ&gt;, can be encoded by the encoding component  164  as e −i|&lt;ψ(λ)|H|ψ(λ)&gt;| |0&gt;|ψ&gt;. The expression &lt;ψ|H|ψ&gt; can also be expressed as: 
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       λ1 
                       , 
                       
                         λ2 
                         ∈ 
                         
                           Spec 
                           ⁡ 
                           ( 
                           H 
                           ) 
                         
                       
                     
                   
                   
                     &lt; 
                     
                       ψ 
                       ⁢ 
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             ψ 
                             λ1 
                           
                           &gt; 
                           &lt; 
                           
                             ψ 
                             λ1 
                           
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                       ⁢ 
                       H 
                       ⁢ 
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             ψ 
                             λ2 
                           
                           &gt; 
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     wherein E 0  can be representative of the quantum state  130 . In this way, the learning component  166  can utilize both stochastic inference and Bayesian inference. After obtaining a sample, the posterior probability (P(φ|E, μ, σ)) of the Bayesian learning equation can be updated with a new variance (σ) and mean of the initial function (μ). Further, as σ and μ are updated by the update component  504 , the measuring component  168  can be independent of an input state. 
       FIG.  9    illustrates a flow diagram of an example, non-limiting computer-implemented method that can generate an eigenvalue from a quantum state in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. At  902 , a quantum state  130  can be encoded as a phase. Encoding a quantum state as a phase  902  can include receiving a quantum state  904  and receiving a quantum state  906 . 
     The quantum state  130  can be received by the encoding component  164 . The encoding component  164  can encode the quantum state  130  as a phase  230  based on an amplitude of the expectation value of the quantum state  130 . The phase  230  can be passed to a learning component  166 . The encoding component can be executed by a quantum processor  162 . 
     At  908 , an eigenvalue corresponding to the expectation value can be learned, by the system, by stochastic inference (e.g., via a learning component  166 ). The learning component  166  can receive an at least one ancilla qubit  205 . The at least one ancilla qubit  205  can be received by the quantum circuit  220  of the learning component  166 . In one or more embodiments, quantum circuit  220  can depend on control parameters m and θ which in turn can depend on the probability/prior parameters μ and σ. The quantum circuit  220  encodes at least one ancilla qubit into a quantum state for measuring  222  that can be received by the encoding component  164 . The quantum circuit  220  of the learning component can be executed on a quantum circuit operatively connected to a quantum processor  162 . 
     The learning component  166  can comprise a measuring component  168 . The measuring component  168  can receive a quantum state for measuring  222  from the quantum circuit  220 . The probability of a given eigenvalue pair  145 , which can be a value corresponding to the expectation value of the quantum state  130 , can be calculated explicitly on by the measuring component  168  according to the equations below: 
                 P   ⁡   (     0   |   ϕ   ;   θ   ;   m     )     =       1   +     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2       ⁢   
       P   ⁡   (     1   |   ϕ   ;   θ   ;   m     )     =       1   -     cos   ⁡   (     m   ⁡   (     θ   -   ϕ     )     )       2       ⁢   
         P     i   +   1       (     ϕ     ❘   &#34;\[LeftBracketingBar]&#34;       E     i   +   1       ;     θ   i     ;     m   i       )     =         P   ⁡   (       E     i   +   1       |   ϕ   ;     θ   i     ,     m   i       )     ⁢       P   i     (   ϕ   )       N             
thereby measuring the eigenvalue probabilistically based on the at least one ancilla qubit. The measuring component can receive the phase  230  from the encoding component  164 , represented as φ(λ)=2 arccos(&lt;ψ(λ)|H|ψ(λ)&gt;).
 
     The measuring component  168  can measure the eigenvalue pair  145  by Bayesian learning, which can include Bayesian inference. Exploiting Bayes&#39; law, the probability of a particular phase  230  (φ) with a given eigenvalue (E), μ, and σ can be written as P(φ|E, μ, σ)=[P(E, μ, σ|φ) P(φ)]/[∫P(E, μ, σ|φ) P(φ)dφ]. P(φ) can be updated each sample by the update component  504 . 
     The measuring component  168  can measure the eigenvalue pair  145  by Bayesian learning, which can include Bayesian inference. Exploiting Bayes&#39; law, the probability of a particular phase  230  (φ) with a given eigenvalue (E), μ, and σ can be written as P(φ|E, μ, σ)=[P(E, μ, α|φ) P(φ)]/[∫P(E, μ, σ|φ) P(φ)dφ]. P(φ) can be updated each sample by the update component  504 . 
     The measuring component  168  can send its output to an update component  504 . The measuring component  168  can execute on both the quantum processor  162  and processor  160 . The update component  504  can receive the probabilistic output  508  of the measuring component  168  and utilize that probabilistic output  508  to provide a new upper bound  506  of the eigenvalue pair  145  to the measuring component  168 . |&lt;ψ(λ)|H|ψ(λ)&gt;| can be an amplitude on which the phase  230  can be encoded, as the quantum state  130 , |0&gt;|ψ&gt;, can be encoded by the encoding component  164  as e −i|&lt;ψ(λ)|H|ψ(λ)&gt;| |0&gt;|ψ&gt;. The expression &lt;ψ|H|ψ&gt; can also be expressed as: 
     
       
         
           
             
               
                 
                   
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     wherein E 0  can be representative of the quantum state  130 . In this way, the learning component  166  can utilize both stochastic inference and Bayesian inference. After obtaining a sample, the posterior probability (P(φ|E, μ, σ)) of the Bayesian learning equation can be updated with a new variance (a) and mean of the initial function (μ). Further, as σ and μ are updated by the update component  504 , the measuring component  168  can be independent of a given input state. 
     Moreover, because at least encoding an expectation value associated with a quantum state and learning, by the system, an eigenvalue corresponding to the expectation value by stochastic inference, etc. are established from a combination of electrical and mechanical components and circuitry, a human is unable to replicate or perform processing performed by the processor or quantum processor (e.g., the learning component, measuring component, encoding component, etc.) disclosed herein. For example, a human is unable to encode an expectation value, etc. 
     Moreover, because encoding, measuring, learning, updating, etc. and/or communication between processing components and/or an assignment component is established from a combination of electrical and mechanical components and circuitry, a human is unable to replicate or perform the subject measuring, learning, updating, etc. and/or the subject communication between processing components and/or a database component. For example, a human is unable to encode an expectation value associated with a quantum state as a phase on a system comprising a processor operatively coupled to memory, etc. Moreover, a human is unable to learn, by a system, an eigenvalue corresponding to the expectation value by stochastic inference, etc. 
     In order to provide a context for the various aspects of the disclosed subject matter,  FIG.  10    as well as the following discussion are intended to provide a general description of a suitable environment in which the various aspects of the disclosed subject matter can be implemented.  FIG.  10    illustrates a block diagram of an example, non-limiting operating environment in which one or more embodiments described herein can be facilitated. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. 
     With reference to  FIG.  10   , a suitable operating environment  1000  for implementing various aspects of this disclosure can also include a computer  1012 . The computer  1012  can also include a processing unit  1014 , a system memory  1016 , and a system bus  1018 . The system bus  1018  couples system components including, but not limited to, the system memory  1016  to the processing unit  1014 . The processing unit  1014  can be any of various available processors. Dual microprocessors and other multiprocessor architectures also can be employed as the processing unit  1014 . The system bus  1018  can be any of several types of bus structure(s) including the memory bus or memory controller, a peripheral bus or external bus, and/or a local bus using any variety of available bus architectures including, but not limited to, Industrial Standard Architecture (ISA), Micro-Channel Architecture (MSA), Extended ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus (USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and Small Computer Systems Interface (SCSI). 
     The system memory  1016  can also include volatile memory  1020  and nonvolatile memory  1022 . The basic input/output system (BIOS), containing the basic routines to transfer information between elements within the computer  1012 , such as during start-up, is stored in nonvolatile memory  1022 . Computer  1012  can also include removable/non-removable, volatile/non-volatile computer storage media.  FIG.  10    illustrates, for example, a disk storage  1024 . Disk storage  1024  can also include, but is not limited to, devices like a magnetic disk drive, floppy disk drive, tape drive, Jaz drive, Zip drive, LS-100 drive, flash memory card, or memory stick. The disk storage  1024  also can include storage media separately or in combination with other storage media. To facilitate connection of the disk storage  1024  to the system bus  1018 , a removable or non-removable interface is typically used, such as interface  1026 .  FIG.  10    also depicts software that acts as an intermediary between users and the basic computer resources described in the suitable operating environment  1000 . Such software can also include, for example, an operating system  1028 . Operating system  1028 , which can be stored on disk storage  1024 , acts to control and allocate resources of the computer  1012 . 
     System applications  1030  take advantage of the management of resources by operating system  1028  through program modules  1032  and program data  1034 , e.g., stored either in system memory  1016  or on disk storage  1024 . It is to be appreciated that this disclosure can be implemented with various operating systems or combinations of operating systems. A user enters commands or information into the computer  1012  through input device(s)  1036 . Input devices  1036  include, but are not limited to, a pointing device such as a mouse, trackball, stylus, touch pad, keyboard, microphone, joystick, game pad, satellite dish, scanner, TV tuner card, digital camera, digital video camera, web camera, and the like. These and other input devices connect to the processing unit  1014  through the system bus  1018  via interface port(s)  1038 . Interface port(s)  1038  include, for example, a serial port, a parallel port, a game port, and a universal serial bus (USB). Output device(s)  1040  use some of the same type of ports as input device(s)  1036 . Thus, for example, a USB port can be used to provide input to computer  1012 , and to output information from computer  1012  to an output device  1040 . Output adapter  1042  is provided to illustrate that there are some output devices  1040  like monitors, speakers, and printers, among other output devices  1040 , which require special adapters. The output adapters  1042  include, by way of illustration and not limitation, video and sound cards that provide a means of connection between the output device  1040  and the system bus  1018 . It should be noted that other devices and/or systems of devices provide both input and output capabilities such as remote computer(s)  1044 . 
     Computer  1012  can operate in a networked environment using logical connections to one or more remote computers, such as remote computer(s)  1044 . The remote computer(s)  1044  can be a computer, a server, a router, a network PC, a workstation, a microprocessor based appliance, a peer device or other common network node and the like, and typically can also include many or all of the elements described relative to computer  1012 . For purposes of brevity, only a memory storage device  1046  is illustrated with remote computer(s)  1044 . Remote computer(s)  1044  is logically connected to computer  1012  through a network interface  1048  and then physically connected via communication connection  1050 . Network interface  1048  encompasses wire and/or wireless communication networks such as local-area networks (LAN), wide-area networks (WAN), cellular networks, etc. LAN technologies include Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ring and the like. WAN technologies include, but are not limited to, point-to-point links, circuit switching networks like Integrated Services Digital Networks (ISDN) and variations thereon, packet switching networks, and Digital Subscriber Lines (DSL). Communication connection(s)  1050  refers to the hardware/software employed to connect the network interface  1048  to the system bus  1018 . While communication connection  1050  is shown for illustrative clarity inside computer  1012 , it can also be external to computer  1012 . The hardware/software for connection to the network interface  1048  can also include, for exemplary purposes only, internal and external technologies such as, modems including regular telephone grade modems, cable modems and DSL modems, ISDN adapters, and Ethernet cards. 
       FIG.  11    illustrates an example, non-limiting graph showing an example probabilistic measurement of a phase in accordance with one or more embodiments described herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. The graph  1100  shows plot points  1120  of the probabilistic measurement that can be done by the measuring component  168 . 
     Quantum Bayesian Amplitude Estimation: One or more embodiments described herein can combine quantum expectation estimation with quantum Bayesian phase estimation, while avoiding problems associated with eigenvector collapse. 
     In contrast to the use of a full phase estimation or iterative phase estimation, one or more embodiments can use a non-trivial variant of a quantum Bayesian phase estimation process. In some uses of quantum Bayesian estimation, a state is initialised into an eigenvector. In contrast, in one or more embodiments, an S rotation operator and the initial state |ψ 0 &gt; can be used by selecting certain θ i  values. One or more embodiments can avoid a problem where |ψ 0 &gt; has components in the direction of both eigenvectors, that is, picking up both phases ±ϕ, and potentially preventing the ability to construct a single pair of likelihoods that is independent of the unknown eigenvector decomposition of the initial state. To avoid this result, one or more embodiments can select θ such that the likelihood is invariant under a sign change of ϕ. In some implementations, this approach can leads to two possible choices
 
0∈{0,π/2 m} 
 
     These two choices allow that for any m (and higher m&#39;s can be used as precision improves) there can be significant statistical distinguishability between measuring a zero versus a one. 
     This can result, in some circumstances, because of differences between two outcome of either cos(mϕ) or sin(mϕ), with the larger of which being generally greater than 1/√{square root over (2)}. 
     One or more embodiments can be implemented where prior knowledge of the phase can be incorporated in order to speed up convergence. For example, in some uses of quantum homology, quantum expectation estimation can be used to calculate betti numbers, these being known to be integer values. 
     Maximizing Information Gain: One or more embodiments can analyze information gain as defined by the expected utility, where the utility is defined as the Kullback-Liebler divergence between the prior and posterior distributions. In one or more embodiments, this approach can yield useful formulations, as well as a novel understanding of the Kitaev protocol. 
     One or more embodiments can determine an expected utility by the difference between the entropy of the outcome of the experiment averaging out prior knowledge and the entropy taking into account prior knowledge. One approach to implementing this determination is to choose m and θ to maximise this expected utility. Because, in some circumstances, possible values of θ (given m) can be restricted to two, one or more embodiments can use the following approach to define the expected utility for these two choices of θ. For example, with θ=0: 
               EU   ⁡   (         m   i     ,     θ   i       =   0     )     =       H   ⁡   (       1   2     +       π   2     ⁢         P   ˆ     i   E     (   m   )         )     -       ∫     -   π     π         H   ⁡   (       1   2     +       cos   ⁡   (     m   ⁢   ϕ     )     2       )     ⁢       P   i   E     (   ϕ   )     ⁢   d   ⁢       ϕ               
where,
 
                     P   ˆ     i   E     (   m   )     :     =       1   π       ∫     -   π     π             
cos(mϕ)P i (ϕ)dϕ, is the even Fourier component (the cosine component) of the prior at frequency m and P i   E (ϕ): =(P i (ϕ)+P i (−ϕ))/2 is the even part of the prior.
 
     In another example, θ=π/2m: 
     
       
         
           
             
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     Thus, in this example approach, an example optimal choice of θ and m is the one that maximises the maximum of the above two expressions. 
     The measuring component  168  can produce an accurate measurement without many samples by measuring intelligently. This probabilistic measurement can utilize Bayesian learning and stochastic inference upon the quantum results. This can be performed by updating a prior distribution using Bayes&#39; law. 
     Based on this probabilistic distribution, the measuring component  168  can start from an initial Gaussian prior with mean μ and variance σ 2 . Bayesian inference can be seen as an update of the prior distribution to produce the posterior distribution. As a result of the updated posterior distribution, measuring component  168  can replace the prior distribution with the posterior distribution. This process can be iterated for each of the random measurements in a data set. 
     Bayesian inference can be based on a discrete distribution. The discrete distribution can be obtained by sampling from a discrete set of samples based on a prior distribution. This can be done on a classical processor  160 . 
     The measuring component  168  can measure for a given μ and σ, thereby observing the outcome E∈{0,1} for a given number of i samples. For each sample, the measuring component  168  can obtain a phase ϕ j  and assign ϕ j  to Φ accept  with probability P(E|ϕ j ,θ,M)/k E , where k E  ∈(0,1] is a constant constrained by P(E|ϕ j ,θ,M)/k E ≤1∀ϕ j , E. Thereby, the measuring component  168  can return 
             μ   =         𝔼   ⁡   (     Φ   accept     )     ⁢         and   ⁢         σ     =         𝕍   ⁡   (     Φ   accept     )       .             
As a first approximation and in order to reduce the Bayes risk, the measuring component  168  can choose μ=[1.25/σ].
 
       FIGS.  12 A and  12 B  illustrate example, non-limiting graphs showing the samples for a given error ϵ. and circuit depth for a given number of qubits. 
     As shown in  FIG.  12 A , the samples which can be employed to achieve a relatively low error ϵ are very high for VQE  1210 . However, even at relatively low errors, γ-QPE  1220  compares quite well with QPE  1230 , while not requiring the high fault tolerance of QPE. 
     As shown in  FIG.  12 B , the circuit depth for a given number of processed qubits goes up very quickly for QPE  1240 . However, even at relatively high amounts of processing, γ-QPE  1260  compares favorably with VQE  1250 . 
     While VQE and QPE scale well in different categories, the γ-QPE scales intermodally in both categories of precision and number of gates showing the tradeoff between precision (error) and circuit number (circuit depth). Allowing such continues steering lets one change the scaling depending on the tasks at hand giving an advantage in hard quantum inference tasks. 
     Eigenvalues can be a value of an observable physical quality of a linear operator operating on a Hilbert space. Eigenvalues can correspond to the expectation values of a quantum state in a Hilbert space. Classical and quantum algorithms facilitating the determination of these eigenvalues come with competing costs. 
     However, classical algorithms do provide some relative benefits to pure quantum algorithms. Where the ansatz of a quantum algorithm can be rigid and non-adaptive, the ansatz of a classical algorithm can be relatively flexible, bounded only by the ansatz span. Further, classical algorithms can be less dependent upon an input state than quantum algorithms, and do not require an extremely high fault tolerance. 
     While quantum algorithms have shown significant results in many areas, as the complexity of the problems increases, the rigidity and fault tolerance requirements of quantum algorithms becomes problematic. Therefore, a hybrid classical-quantum approach, such as γ-QPE, can incorporate some of the lower costs of both quantum and classical algorithms. 
     The system and/or the components of the system can be employed to use hardware and/or software to solve problems that are highly technical in nature (e.g., related to encoding, measuring, machine learning) that are not abstract and that cannot be performed as a set of mental acts by a human because they refer to specific machine processes. For example, a human cannot encode a quantum state as a phase on computer-readable medium, as a human cannot read a computer readable medium, or learn an eigenvalue associated with a quantum state on a computer-readable medium. Further, some of the processes, such as receiving a quantum state and measuring an ancilla probabilistically, can be performed by specialized computers facilitating carrying out defined tasks related to the quantum state estimation subject area. The system and/or components of the system can be employed to solve problems of greater complexity and that require greater computational power that arise through advancements in technology. The system can provide technical improvements to solving for eigenvalues by improving processing efficiency, reliability, and flexibility by providing a hybrid classical-quantum approach for quantum state estimation. In this way, the system can reduce errors in quantum state estimation by providing improved scaling and depth for a given processing cost, and thereby improve computer functionality in solving for eigenvalues which represent observables, etc. 
     The present invention can be a system, a method, an apparatus and/or a computer program product at any possible technical detail level of integration. The computer program product can include a computer readable storage medium (or media) having computer readable program instructions thereon for causing a processor to carry out aspects of the present invention. The computer readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. The computer readable storage medium can be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of the computer readable storage medium can also include the following: 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), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as punch-cards or raised structures in a groove having instructions recorded thereon, and any suitable combination of the foregoing. A computer readable storage medium, as used herein, is not to be construed as being transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission media (e.g., light pulses passing through a fiber-optic cable), or electrical signals transmitted through a wire. 
     Computer readable program instructions described herein can be downloaded to respective computing/processing devices from a computer readable storage medium or to an external computer or external storage device via a network, for example, the Internet, a local area network, a wide area network and/or a wireless network. The network can comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers and/or edge servers. A network adapter card or network interface in each computing/processing device receives computer readable program instructions from the network and forwards the computer readable program instructions for storage in a computer readable storage medium within the respective computing/processing device. Computer readable program instructions for carrying out operations of the present invention can be assembler instructions, instruction-set-architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state-setting data, configuration data for integrated circuitry, or either source code or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, C++, or the like, and procedural programming languages, such as the “C” programming language or similar programming languages. The computer readable program instructions can execute entirely on the user&#39;s computer, partly on the user&#39;s computer, as a stand-alone software package, partly on the user&#39;s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer can be connected to the user&#39;s computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection can be made to an external computer (for example, through the Internet using an Internet Service Provider). In some embodiments, electronic circuitry including, for example, programmable logic circuitry, field-programmable gate arrays (FPGA), or programmable logic arrays (PLA) can execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry, in order to perform aspects of the present invention. 
     Aspects of the present invention are described herein with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. 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 readable program instructions. These computer readable program instructions can 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. These computer readable program instructions can also be stored in a computer readable storage medium that can direct a computer, a programmable data processing apparatus, and/or other devices to function in a particular manner, such that the computer readable storage medium having instructions stored therein comprises an article of manufacture including instructions which implement aspects of the function/act specified in the flowchart and/or block diagram block or blocks. The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational acts to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process, such that the instructions which execute on the computer, other programmable apparatus, or other device implement the functions/acts specified in the flowchart and/or block diagram block or blocks. 
     The flowchart and block diagrams in the Figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in the flowchart or block diagrams can represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logical function(s). In some alternative implementations, the functions noted in the blocks can occur out of the order noted in the Figures. For example, two blocks shown in succession can, in fact, be executed substantially concurrently, or the blocks can sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts or carry out combinations of special purpose hardware and computer instructions. 
     While the subject matter has been described above in the general context of computer-executable instructions of a computer program product that runs on a computer and/or computers, those skilled in the art will recognize that this disclosure also can or can be implemented in combination with other program modules. Generally, program modules include routines, programs, components, data structures, etc. that perform particular tasks and/or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the inventive computer-implemented methods can be practiced with other computer system configurations, including single-processor or multiprocessor computer systems, mini-computing devices, mainframe computers, as well as computers, hand-held computing devices (e.g., PDA, phone), microprocessor-based or programmable consumer or industrial electronics, and the like. The illustrated aspects can also be practiced in distributed computing environments in which tasks are performed by remote processing devices that are linked through a communications network. However, some, if not all aspects of this disclosure can be practiced on stand-alone computers. In a distributed computing environment, program modules can be located in both local and remote memory storage devices. 
     As used in this application, the terms “component,” “system,” “platform,” “interface,” and the like, can refer to and/or can include a computer-related entity or an entity related to an operational machine with one or more specific functionalities. The entities disclosed herein can be either hardware, a combination of hardware and software, software, or software in execution. For example, a component can be, but is not limited to being, a process running on a processor, a processor, an object, an executable, a thread of execution, a program, and/or a computer. By way of illustration, both an application running on a server and the server can be a component. One or more components can reside within a process and/or thread of execution and a component can be localized on one computer and/or distributed between two or more computers. In another example, respective components can execute from various computer readable media having various data structures stored thereon. The components can communicate via local and/or remote processes such as in accordance with a signal having one or more data packets (e.g., data from one component interacting with another component in a local system, distributed system, and/or across a network such as the Internet with other systems via the signal). As another example, a component can be an apparatus with specific functionality provided by mechanical parts operated by electric or electronic circuitry, which is operated by a software or firmware application executed by a processor. In such a case, the processor can be internal or external to the apparatus and can execute at least a part of the software or firmware application. As yet another example, a component can be an apparatus that provides specific functionality through electronic components without mechanical parts, wherein the electronic components can include a processor or other means to execute software or firmware that confers at least in part the functionality of the electronic components. In an aspect, a component can emulate an electronic component via a virtual machine, e.g., within a cloud computing system. 
     In addition, the term “or” is intended to mean an inclusive “or” rather than an exclusive “or.” That is, unless specified otherwise, or clear from context, “X employs A or B” is intended to mean any of the natural inclusive permutations. That is, if X employs A; X employs B; or X employs both A and B, then “X employs A or B” is satisfied under any of the foregoing instances. Moreover, articles “a” and “an” as used in the subject specification and annexed drawings should generally be construed to mean “one or more” unless specified otherwise or clear from context to be directed to a singular form. As used herein, the terms “example” and/or “exemplary” are utilized to mean serving as an example, instance, or illustration. For the avoidance of doubt, the subject matter disclosed herein is not limited by such examples. In addition, any aspect or design described herein as an “example” and/or “exemplary” is not necessarily to be construed as preferred or advantageous over other aspects or designs, nor is it meant to preclude equivalent exemplary structures and techniques known to those of ordinary skill in the art. 
     As it is employed in the subject specification, the term “processor” can refer to substantially any computing processing unit or device comprising, but not limited to, single-core processors; single-processors with software multithread execution capability; multi-core processors; multi-core processors with software multithread execution capability; multi-core processors with hardware multithread technology; parallel platforms; and parallel platforms with distributed shared memory. Additionally, a processor can refer to an integrated circuit, an application specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), a discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. Further, processors can exploit nano-scale architectures such as, but not limited to, molecular and quantum-dot based transistors, switches and gates, in order to optimize space usage or enhance performance of user equipment. A processor can also be implemented as a combination of computing processing units. In this disclosure, terms such as “store,” “storage,” “data store,” data storage,” “database,” and substantially any other information storage component relevant to operation and functionality of a component are utilized to refer to “memory components,” entities embodied in a “memory,” or components comprising a memory. It is to be appreciated that memory and/or memory components described herein can be either volatile memory or nonvolatile memory, or can include both volatile and nonvolatile memory. By way of illustration, and not limitation, nonvolatile memory can include read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM). Volatile memory can include RAM, which can act as external cache memory, for example. By way of illustration and not limitation, RAM is available in many forms such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM). Additionally, the disclosed memory components of systems or computer-implemented methods herein are intended to include, without being limited to including, these and any other suitable types of memory. 
     What has been described above include mere examples of systems and computer-implemented methods. It is, of course, not possible to describe every conceivable combination of components or computer-implemented methods for purposes of describing this disclosure, but one of ordinary skill in the art can recognize that many further combinations and permutations of this disclosure are possible. Furthermore, to the extent that the terms “includes,” “has,” “possesses,” and the like are used in the detailed description, claims, appendices and drawings such terms are intended to be inclusive in a manner similar to the term “comprising” as “comprising” is interpreted when employed as a transitional word in a claim. 
     The descriptions of the various embodiments have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.