Amplitude estimation method

Abstract

To provide an amplitude estimation method for estimating a parameter efficiently in quantum calculation.SOLUTION: In an amplitude estimation method, parameters θ,p are estimated by steps of: forming a variation quantum circuit 503 controlled by a parameter θ` and adjusting a measurement base for a quantum state after amplitude amplification; setting an initial value of the parameter θ` in the variation quantum circuit; and carrying out prior measurement with addition of the variation quantum circuit 513 with the initial value set. In the method, the parameters θ,p are estimated by steps of: employing an estimated value of the parameter θ obtained from a result of the prior measurement as the parameter θ` to reconstruct the variation quantum circuit for adjusting the measurement base; and carrying out succeeding measurement using the measurement base adjusted with addition of the reconstructed variation quantum circuit 523 to use the result of the succeeding measurement.SELECTED DRAWING: Figure 4

Description

[Technical field]

[0001] The present invention relates to an amplitude estimation method for estimating the amplitude of a quantum state realized on a quantum computer using a noisy quantum computer. [Background technology]

[0002] Quantum computers are expected to be used to solve problems that could not be solved in a realistic time or on a realistic scale by conventional computers. Therefore, various application technologies using quantum computers are being considered (for example, Patent Document 1).

[0003] The amplitude estimation problem, which estimates parameters embedded in the amplitude of a quantum state, is one of the representative problems for quantum computers. For example, the amplitude estimation problem appears as a subproblem when derivative price calculations in the financial field or energy calculations in the chemical field are performed on a quantum computer. The amplitude estimation problem is solved by estimating the state |ψ(θ)> of the following equation, which is defined using an operator A that performs the operation defined by the following [Equation 1]: n+1 The problem is to estimate the parameter θ embedded in the amplitude of

[0004]

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[0005] As shown in Fig. 15, it is known that in a quantum computer, by performing an operation using the Grover operator G, it is possible to estimate the parameter θ with the theoretically highest accuracy for the number of times the operator A is called (Heisenberg limit). The operation using the Grover operator G shown in the fourth equation of [Mathematical formula 2] is called amplitude amplification.

[0006]

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[0007] Here, m is the number of times amplitude amplification is performed (the number of times the Grover operator G is called). Also, Ng=2m+1 is the number of times the operator A is called when the operation shown in the circuit of FIG. 15 is performed once.

[0008] As an amplitude estimation method using amplitude amplification, a method using maximum likelihood estimation when estimating amplitude is known (see Patent Document 2). In the technology described in this document, the amplitude is estimated by performing maximum likelihood estimation based on a combination of measurement data generated from quantum circuits with different numbers of Grover operators G (amplitude amplification operations).

[0009] As shown in FIG. 16, in amplitude estimation using maximum likelihood estimation, the Grover operator G is applied to each quantum circuit a different number of times m1, m2, ..., mM to amplify the amplitude, and the number of repetitions (number of shots) N is measured for each quantum circuit. Then, the estimated value θ that maximizes the likelihood is calculated by using maximum likelihood estimation on the measurement results. In a noise-free quantum computer, the best estimation accuracy is obtained when the number of calls to the operator A is constant by using amplitude estimation using this maximum likelihood estimation. When the number of shots N is sufficiently large, the theoretical limit of estimation accuracy is almost achieved (see Patent Document 2).

[0010] Quantum states are usually disturbed by noise resulting from the interaction of the system with the environment. The effect of such noise is modeled, for example, by a model in which depolarization noise is generated every time the Grover operator is applied, as in [Equation 3], where ρ noise is a density operator that represents the quantum state after depolarization noise occurs. In this model, it is assumed that the correct operation occurs with probability p, and that the completely mixed state occurs with probability [1-p], depending on the noise parameter p, which gives the strength of the noise. The value of the noise parameter p is unknown because it changes depending on the details of the operation, but since the measurement results change depending on the noise parameter p, the effect of the noise parameter p must be taken into account in order to accurately estimate the parameter θ. Here, d=2 (n+1) denotes the dimension of the Hilbert space.

[0011]

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[0012] In estimating the parameter θ, the measurement basis that can theoretically estimate θ most accurately is the SLD operator L S It is known that the measurement basis can be obtained by spectral decomposition of the above (see Non-Patent Document 1). Here, the measurement basis refers to a state that serves as a reference in measuring a quantum state.

[0013] In this problem setting, the SLD operator L S The eigenvectors of are expressed as follows: d is given by:

[0014]

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[0015] Eigenvectors λ0, λ1, λ d In theory, it is possible to estimate the parameter θ with the highest accuracy if measurements can be made in a measurement basis composed of the following. On the other hand, in order to make measurements in an arbitrary measurement basis using a quantum computer, it is generally necessary to perform a large number of gate operations immediately before the measurement.

[0016] One possible method for adjusting the measurement direction with relatively few operations is to use a variational quantum circuit. A variational quantum circuit is a method that is expected to provide an operation that approximates a desired operation with relatively few gate operations by using a circuit in which gate operations are controlled by parameters. For example, Hardware Efficient Ansatz and Alternating Layered Ansatz have been proposed as variational quantum circuits that are easy to implement on hardware in response to the physical constraints of quantum computers (see Non-Patent Documents 2 and 3). [Prior art documents] [Patent documents]

[0017] [Patent Document 1] JP-2017-59074 [Patent Document 2] Patent No. 7005806

Non-licensed documents

[0018]

Non-patent document 1

Non-patent document 2

Non-patent document 3

[0019] As mentioned above, the optimal measurement basis in the presence of noise is the SLD operator L S However, in quantum computers, measurements are performed in a specific measurement basis called the computational basis, so it has been difficult to achieve the theoretical limit of the estimation accuracy in estimating the parameter θ. [Means for solving the problem]

[0020] The amplitude estimation method of the present disclosure includes an operator A that embeds a parameter θ into the amplitude of a quantum state, a Grover operator that amplifies the amplitude of the quantum state embedded by the operator A, and a result obtained by measuring the quantum state after the amplitude is amplified by the Grover operator, and estimates the parameter θ and a noise parameter p indicating the intensity of noise generated by the operation. Here, a variational quantum circuit controlled by a parameter θ' that adjusts a measurement basis for the quantum state after the amplitude is amplified is configured, an initial value of the parameter θ' in the variational quantum circuit is set, and a prior measurement is performed by adding the variational quantum circuit to which the initial value is set, thereby estimating the parameter θ and the noise parameter p, and the variational quantum circuit that adjusts the measurement basis is reconfigured by adopting an estimated value obtained from the result of the prior measurement as the parameter θ', and the reconfigured variational quantum circuit is added, and a subsequent measurement is performed using the adjusted measurement basis, and the parameter θ and the noise parameter p are estimated using the result of the subsequent measurement. That is, this is realized by performing three calculations: a pre-calculation to adjust the parameter α of the variational quantum circuit, a pre-calculation to estimate the parameter θ, and a post-calculation. First, the pre-calculation adjusts the parameter α of the variational quantum circuit so as to approximately provide the desired operation. Next, the pre-calculation obtains a rough estimate of the parameter embedded in the amplitude using a relatively small number of shots. Finally, in the post-calculation, the parameter α of the variational quantum circuit and the estimated value of the amplitude parameter are used to adjust the measurement basis, thereby obtaining a final estimate of the parameter θ. Effect of the Invention

[0021] According to the present invention, in quantum computing, a parameter embedded in the amplitude of a quantum state can be estimated with high accuracy. [Brief description of the drawings]

[0022] [Figure 1] FIG. 1 is a schematic diagram of a system according to an embodiment. [Diagram 2] FIG. 2 is an explanatory diagram of a hardware configuration of the embodiment. [Diagram 3]FIG. 2 is an explanatory diagram illustrating an outline of a processing procedure according to an embodiment. [Figure 4] 1A is an explanatory diagram of a quantum circuit according to an embodiment, in which (a) is an explanatory diagram of a circuit using an operator B503, (b) is an explanatory diagram of a circuit using an operator B513, and (c) is an explanatory diagram of a circuit using an operator B523. FIG. [Diagram 5] FIG. 1 is an explanatory diagram of a quantum circuit according to an embodiment. [Figure 6] 1A and 1B are explanatory diagrams of a quantum circuit according to an embodiment, in which FIG. 1A is an explanatory diagram of a first gate and FIG. [Figure 7] FIG. 2 is an explanatory diagram of a processing procedure according to an embodiment. [Figure 8] FIG. 1 is an explanatory diagram of a quantum circuit according to an embodiment. [Figure 9] FIG. 1 is an explanatory diagram of a quantum circuit according to an embodiment. [Figure 10] FIG. 1 is an explanatory diagram of a quantum circuit according to an embodiment. [Figure 11] FIG. 1 is an explanatory diagram of a quantum circuit according to an embodiment. [Figure 12] FIG. 11 is an explanatory diagram of the effect of the embodiment. [Figure 13] FIG. 2 is an explanatory diagram of a quantum circuit for verification according to the embodiment. [Figure 14] FIG. 1 is an explanatory diagram of a quantum circuit according to an embodiment. [Figure 15] FIG. 1 is an explanatory diagram of a conventional quantum circuit. [Figure 16] FIG. 1 is an explanatory diagram of a conventional processing procedure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0023] An embodiment of the amplitude estimation method will be described below with reference to Figures 1 to 14. In this embodiment, in a situation where a parameter p indicating the intensity of noise is unknown, a parameter θ embedded in the amplitude of a quantum state by an operator A is estimated.

[0024] In this embodiment, an “adaptive” measurement basis optimization strategy is adopted that can achieve the theoretical limit of estimation accuracy in estimating the parameter θ. Here, the parameter estimation is divided into two stages. Specifically, the parameter θ,p is roughly estimated from the measurement results obtained in the preceding measurement. Then, the measurement basis is optimized using this result, and the subsequent measurement is performed to obtain the final estimate of the parameter θ,p. For this purpose, as shown in FIG. 1, a user terminal 10 and a computing device 20 connected via a network are used.

[0025] (Hardware configuration description) 2, the hardware configuration of the information processing device H10 constituting the user terminal 10 and the arithmetic device 20 (management unit 211, classical calculation unit 212) will be described. The information processing device H10 includes a communication device H11, an input device H12, a display device H13, a storage device H14, and a processor H15. Note that this hardware configuration is an example, and it is also possible to realize it using other hardware.

[0026] The communication device H11 is an interface that establishes a communication path with another device and transmits and receives data, and is, for example, a network interface or a wireless interface.

[0027] The input device H12 is a device that accepts input from a user, etc., and is, for example, a mouse, a keyboard, etc. The display device H13 is, for example, a display that displays various information. The storage device H14 is a device that stores data and various programs for executing various functions of the user terminal 10 and the arithmetic device 20. Examples of the storage device H14 include a ROM, a RAM, and a hard disk.

[0028] The processor H15 uses programs and data stored in the storage device H14 to control each process in the user terminal 10 and the computing device 20. Examples of the processor H15 include a CPU and an MPU. The processor H15 loads programs stored in a ROM or the like into a RAM and executes various processes for each service.

[0029] The processor H15 is not limited to a processor that performs software processing for all the processes it executes. For example, the processor H15 may include a dedicated hardware circuit (e.g., an application specific integrated circuit) that performs hardware processing for at least a part of the processes it executes. That is, the processor H15 may be configured as a circuit including the following:

[0030] (1) One or more processors that operate according to a computer program (2) One or more dedicated hardware circuits that perform at least some of the various processes (3) Combinations of these The processor includes a CPU and memory, such as RAM and ROM, that stores program codes or instructions configured to cause the CPU to execute processes. Memory, i.e., computer-readable media, includes any available media that can be accessed by a general-purpose or special-purpose computer.

[0031] (System configuration of the arithmetic device 20) Next, the system configuration of the user terminal 10 and the arithmetic device 20 will be described with reference to FIG. The user terminal 10 is a computer terminal used by a user.

[0032] The arithmetic unit 20 is a computer for performing amplitude estimation. The arithmetic device 20 performs processes (such as management stage, classical computation stage, quantum computation stage, etc.) described below. By executing a program for that purpose, the arithmetic device 20 functions as a management unit 211, a classical computation unit 212, and a quantum computation unit 22.

[0033] The management unit 211 manages the quantum computation and the classical computation. The classical calculation unit 212 executes statistical processing in maximum likelihood estimation. Furthermore, the classical calculation unit 212 executes an update process for updating parameters used in optimizing the measurement basis.

[0034] The quantum computing unit 22 performs quantum computing. The quantum computing unit 22 includes an operation unit 221, a state holding unit 222, and a measurement unit 223. The operation unit 221 performs quantum operations according to a program on the quantum bits of the state holding unit 222. In this case, the operation unit 221 operates (creates) the state held by the state holding unit 222 by using a quantum circuit configured of quantum gates and the like.

[0035] The state holding unit 222 has a plurality of quantum bits and holds any quantum state. Each quantum bit holds a superposition state of multiple values ​​in any physical state, such as an electron level, an electron spin, an ion level, each spin, a photon, etc. The quantum bits are not limited to the above as long as they can hold a superposition state.

[0036] The measurement unit 223 observes the eigenstates of the superposition states of the quantum bits in the state holding unit 222. The measurement unit 223 records the number of hits according to the state of the quantum bits in the state holding unit 222.

[0037] (Overview of parameter estimation process) Next, an overview of the parameter estimation process will be described with reference to FIGS. As shown in FIG. 4(a), in the following parameter estimation process, amplitude estimation using maximum likelihood estimation is performed on a quantum circuit 500 consisting of an operator A (501), a Grover operator G (502), and an operator B(θ') (503) to estimate parameters. The operator B(θ') is an operator for adjusting the measurement basis, and is controlled by the parameter θ'. As shown in [Equation 5], the operator B(θ') ideally calculates eigenvectors λ0 and λ1 from the calculation basis |0> n |0>,|0> nConvert to |1>.

[0038]

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[0039] As shown in Fig. 5, a quantum circuit 600 in which the operator B(θ') is divided into an operator C0 (first gate), an operator C1 (second gate), and an Ry gate can be used. n quantum bits are input to the operators C0 and C1. The Ry gate is a gate that provides a rotation operation around the y axis on the Bloch sphere when focusing on the control bit.

[0040] However, the operators C0 and C1 satisfy the following conditions.

[0041]

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[0042] Here, we consider constructing the operators C0 and C1 that satisfy the condition of [Mathematical formula 6] approximately using a variational quantum circuit controlled by a parameter α. Note that the configuration of the variational quantum circuit is arbitrary, and specifically, it is possible to use the Hardware Efficient Ansatz described in Non-Patent Document 2 or the Alternating Layered Ansatz described in Non-Patent Document 3.

[0043] For example, the operator C0 and the operator C1 in the case of n=2 will be described. 6(a) and 6(b), it is possible to use quantum circuits 610 and 611 as operators C0 and C1, respectively. The operator C0 can be configured by an Ry gate including parameters α1 to α4, and the operator C1 can be configured by an Ry gate including parameters α5 to α8.

[0044] The parameter θ is estimated using the above quantum circuit according to the outline of the procedure shown in Fig. 3. First, the parameter α of the variational quantum circuit is adjusted by pre-calculation so as to approximately provide a desired operation to the operators C0 and C1 configured using the variational quantum circuit as described above (step S11).

[0045] Next, the arithmetic device 20 executes the advance calculation process (step S12). Specifically, the management unit 211 uses the classical calculation unit 212 and the quantum calculation unit 22 to perform amplitude estimation by maximum likelihood estimation.

[0046] (Pre-calculation process) As shown in FIG. 4(b), in the advance calculation process, first, operator A (operator A501), operator G m A quantum circuit 510 is configured using an operator B(θ0) (Grover operator 502) and an operator B(θ0) (operator B 513). The parameter θ0 included in the operator B 513 is the initial value of the parameter θ' included in the operator B 503, and an appropriate value is set as described below. Then, the management unit 211 executes calculations N0 times while changing the number of calls m of the Grover operator by the quantum computing unit 22, performs measurements, and records the prior measurement results. After that, from the acquired prior measurement results, amplitude estimation is performed by maximum likelihood estimation as described in Patent Document 2. Then, the management unit 211 acquires θ1 as an estimate of the parameter θ.

[0047] Next, the arithmetic device 20 executes a process of optimizing the measurement basis (step S13). Here, the parameter θ1 acquired in the advance calculation process is used to configure an operator B523 that adjusts the measurement basis.

[0048] Next, the arithmetic device 20 executes the subsequent calculation process (step S14). Here, too, the management unit 211 uses the classical calculation unit 212 and the quantum calculation unit 22 to perform amplitude estimation by maximum likelihood estimation.

[0049] (Subsequent calculation processing) As shown in FIG. 4(c), in the subsequent calculation process, first, operator A (operator A501), operator G mA quantum circuit 520 is configured using operator B(θ1) (Grover operator 502) and operator B(θ1) (operator B 523). As in the advance calculation process, the management unit 211 executes calculations N1 times while changing the number of times m of invocations of the Grover operator using the quantum calculation unit 22, performs measurements, and records the subsequent measurement results. After that, amplitude estimation is performed from the acquired subsequent measurement results using maximum likelihood estimation as described in Patent Document 2. Then, the management unit 211 acquires θ2 as an estimate of the parameter θ.

[0050] (Specific processing of parameter estimation processing) Next, the process of FIG. 3 will be described in more detail with reference to FIG. First, the arithmetic device 20 optimizes the parameter α that controls the operators C0 and C1 (step S21). Here, the operators C0 and C1 are configured as shown in FIG. 6 so that the variational quantum circuit can provide a desired operation, and the parameters α1 to α8 are optimized. In this case, optimization is performed to satisfy the following conditions.

[0051]

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[0052] As shown in FIG. 8, operators C0 and C1 are combined to configure operator B' (quantum circuit 550). As a cost function, the number of times each bit indicated as n-qubit does not become "0" is counted in the measurement results of the quantum circuit shown in FIG. 8. Here, the lowest bit is the control bit, and a white circle means that a gate operation (C0) associated with the control bit being 0 is executed, and a black circle means that a gate operation (C1) associated with the control bit being 1 is executed. The measurement results used to optimize parameter α can be used to estimate parameter θ.

[0053] Next, the arithmetic unit 20 judges whether the sign is inverted (step S22). Here, B'|ψ(θ)> can give a transformation to a state with a different sign as follows:

[0054]

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[0055] For this reason, if the conversion is as follows, a Z-gate is required to invert the bits:

[0056]

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[0057] To determine this sign reversal, an H gate is added and a measurement is performed.

[0058]

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[0059] Here, since 0≦θ≦π / 2, if (cosθ+sinθ)|0>n|0>+(cosθ-sinθ)|0>n|1>, then |0> n |0> and |0> n The probability of obtaining |1> and |2> is (p0, p1), and p0>p1 holds. Therefore, if the measurement result shows that p0>p1, a Z gate is added.

[0060] In this case, as shown in FIG. 9, a quantum circuit 560 to which a Z gate is added is used as an operator B' depending on the result of code discrimination. Next, the arithmetic device 20 constructs an operator B(θ') (step S23). Here, Ry(-2Ngθ'-π / 2) is added to the operator B' to construct the operator B(θ') shown in FIG. 5 (quantum circuit 600).

[0061] When bit inversion is required, an operator B(θ′) with a Z gate added is used (quantum circuit 620) as shown in FIG. Next, the arithmetic unit 20 constructs an operator B(θ0) (step S24). Here, an initial value θ0 of the parameter θ′ is set. Specifically, here, [−2N gSet an initial value θ0 that satisfies θ0−π / 2=0.

[0062] Next, the arithmetic device 20 estimates the parameter θ,p (step S25). Here, the management unit 211 uses the quantum calculation unit 22 to perform N0 measurements on the quantum circuit 510 (FIG. 4(b)) to which the operator B(θ0) has been added, while changing the number of times that amplitude amplification is performed (the number of times that the Grover operator G is called). Then, the classical calculation unit 212 estimates the parameter θ,p by amplitude estimation using maximum likelihood estimation, and obtains an estimated value θ1 of θ.

[0063] Next, the arithmetic device 20 constructs an operator B(θ1) (step S26). Here, the operator B(θ1) is constructed by adopting the estimated value θ1 of θ obtained from the measurement result as a new parameter θ1.

[0064] Next, the parameter θ,p is estimated using the optimized measurement basis (step S27). Here, the management unit 211 uses the quantum calculation unit 22 to perform N1 measurements on the quantum circuit 520 (FIG. 4(c)) to which the operator B(θ1) has been added, while changing the number of times amplitude amplification is performed (the number of times the Grover operator G is called). Then, the classical calculation unit 212 estimates the parameter θ,p by amplitude estimation using maximum likelihood estimation, and obtains an estimated value θ2 of θ.

[0065] According to this embodiment, the following effects can be obtained. (1) In this embodiment, the arithmetic device 20 executes a pre-calculation process (step S11), a prior calculation process (step S12), a measurement basis optimization process based on the prior calculation result (step S13), and a subsequent calculation process (step S14). This allows the amplitude of the quantum state to be estimated with high accuracy by adaptively optimizing the measurement basis.

[0066] For example, it is possible to estimate a parameter θ that gives S as given by the following equation:

[0067]

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[0068] For this estimation, a quantum circuit 700 shown in FIG. 11 is used. In this case, as shown in FIG. 12, it is indeed possible to estimate the amplitude of the quantum state with an accuracy approaching the theoretical limit in the presence of noise.

[0069] (2) In this embodiment, the arithmetic device 20 determines whether the sign is inverted (step S22). This allows the quantum circuit to be configured in consideration of the inversion of the sign of the quantum state after the operation of the operators C0 and C1 configured in the variational quantum circuit.

[0070] (3) In this embodiment, in estimating the parameter θ, the amplitude of the following state is estimated.

[0071]

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[0072] The above state is "|ψ0> n |0>" and "|ψ1>" n Since the circuit is constructed from a basis of |1>", it is possible to estimate the amplitude by knowing the output of the n+1th quantum bit shown in Fig. 13. As shown in Fig. 14, since the operators C0 and C1 do not affect the state of the n+1th quantum bit, the output of this circuit can be used to estimate the parameter θ.

[0073] This embodiment can be modified as follows: This embodiment and the following modifications can be combined with each other to the extent that there is no technical contradiction. In this embodiment, the calculation device 20 executes a pre-calculation process (step S11), a prior calculation process (step S12), a measurement basis optimization process based on the prior calculation result (step S13), and a subsequent calculation process (step S14). As long as the measurement basis of the subsequent calculation can be optimized by the prior calculation, the number of steps is not limited to two.

[0074] In the above embodiment, the variational quantum circuit is configured using the operators C0, C1, and Ry gates, but is not limited to these. In the above embodiment, the operators C0 and C1 are configured using Ry gates, but the present invention is not limited to this. [Explanation of symbols]

[0075] 10...user terminal, 20...arithmetic unit, 211...management unit, 212...classical calculation unit, 22...quantum calculation unit, 221...operation unit, 222...state holding unit, 223...measurement unit.

Claims

1. An operator A that embeds a parameter θ into the amplitude of a quantum state; a Grover operator that amplifies the amplitude of the quantum state embedded by the operator A; In the amplitude estimation method, a parameter θ and a noise parameter p indicating the intensity of noise generated in a calculation are estimated using a result obtained by measuring a quantum state after the amplitude is amplified by the Grover operator, A variational quantum circuit is configured to adjust a measurement basis for the quantum state after the amplitude is amplified, the measurement basis being controlled by a parameter θ′; Setting an initial value of the parameter θ′ in the variational quantum circuit; By performing advance measurement by adding a variational quantum circuit to which the initial value is set, the parameter θ and the noise parameter p are estimated; reconfiguring the variational quantum circuit to adjust the measurement basis by adopting the estimated value obtained from the result of the previous measurement as the parameter θ′; adding the reconstructed variational quantum circuit to perform a subsequent measurement using the adjusted measurement basis; and estimating the parameter θ and the noise parameter p using results of the subsequent measurements.

2. 2. The method of claim 1, wherein the adjusted measurement basis is determined from eigenvectors calculated by spectral decomposition of an SLD operator in the definition of an SLD quantum Fisher information matrix.

3. The variational quantum circuit has an operation controlled by the parameter θ′, 2. The method of claim 1, wherein the measurement basis is adjusted by adjusting the parameter θ′.

4. The variational quantum circuit inputs n+1 quantum bits, A parameter α that acts on n quantum bits when the control bit is “1” １ The first gate C controlled by １ and, A parameter α that acts on n quantum bits when the control bit is “0” ０ The second gate C controlled by ０ and, an Ry gate acting on said control bit and controlled by said parameter θ′; [0010] 2. The method for estimating amplitude according to claim 1, further comprising optimizing the parameter α so as to satisfy the following:

5. determining a sign at an output of the variational quantum circuit; 5. The method according to claim 4, further comprising the step of adding a Z-gate to said control bit in response to said sign discrimination result.

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