KR102591035B1 - Method of performing quantum state tomography in maximally noisy quantum depolarizing channel and apparatus thereof - Google Patents

Abstract

A tomography device according to an embodiment of the present invention includes a quantum switch unit that is controlled by a control qubit and receives a quantum state to be estimated, a tomography unit that performs quantum state tomography on the state output from the quantum switch unit, and It includes an estimation unit that estimates the initial state based on the output of the tomography unit and algebra, and quantum state imaging can be performed even in the maximum noise channel.

Description

Method and apparatus for performing quantum state tomography in a maximally noisy QUANTUM DEPOLARIZING CHANNEL AND APPARATUS THEREOF}

The present invention relates to a quantum tomography device and method based on a quantum switch, and more specifically, to a quantum state tomography device and method that accurately estimate the quantum state even in the maximum noise depolarization channel by using a quantum switch. .

The ability to characterize the state of quantum systems is becoming essential with advances in research fields such as quantum computing, communications, and metrology. Quantum State Tomography (QST) uses a series of identically prepared measurements of unknown quantum states to statistically reconstruct a density matrix representing the state of a quantum system.

Additionally, recent advances in QST have facilitated the characterization of more complex quantum systems, such as those involving entanglement and higher dimensions, and QST is expected to serve as a cornerstone of future quantum technologies.

QST techniques utilize a set of measurement observations, statistical models and quantum resources, one of the earliest methods being to uniquely determine quantum states using simple linear transformations of the measured data.

Korean Patent Publication No. 2020-0050841 “Method for UD-based qubit state tomography” Korean Patent No. 2098285, “Quantum process analysis device and method”

The purpose of the present invention is to enable quantum state tomography (QST) in a channel with maximum noise.

The purpose of the present invention is to enable accurate quantum state estimation.

A tomography apparatus according to an embodiment of the present invention includes a quantum switch unit that is controlled by a control qubit and receives a quantum state to be estimated, and a tomography unit that performs quantum state tomography on the state output from the quantum switch unit. and an estimation unit that estimates the initial state based on the output of the tomography unit and algebra, and quantum state imaging can be performed even in the maximum noise channel.

According to one embodiment of the present invention, the quantum switch unit can combine quantum channels.

According to one embodiment of the present invention, the maximum noise channel may be set to infinite causal order (ICO).

According to one embodiment of the present invention, the maximum noise channel may be a quantum depolarization channel.

A tomography method according to an embodiment of the present invention includes transmitting a quantum state to be estimated to a quantum switch, performing quantum state tomography in a state output from the quantum switch, and outputting the quantum state tomography. and estimating the initial state based on algebra, and quantum state imaging can be performed even in the maximum noise channel. According to one embodiment of the present invention, the quantum switch combines quantum channels and can be controlled based on control qubits.

According to one embodiment of the present invention, the maximum noise channel may be set to infinite causal order (ICO).

According to one embodiment of the present invention, the maximum noise channel may be a quantum depolarization channel.

According to one embodiment, by using a quantum switch, it is possible to perform QST in a channel with maximum noise.

According to one embodiment, more accurate quantum state estimation is possible by performing QST based on a quantum switch.

Figure 1 is a diagram illustrating a mechanism for performing QST in a quantum depolarization channel with maximum noise by adopting a quantum switch in qubit transmission.
Figure 2 shows the Log infidelity of the reconstructed state and the actual state for pure and mixed states.
Figure 3 is a block diagram of a tomographic imaging device according to an embodiment of the present invention.
Figure 4 is a flowchart of a tomography method according to an embodiment of the present invention.

Specific structural or functional descriptions of the embodiments according to the concept of the present invention disclosed in this specification are merely illustrative for the purpose of explaining the embodiments according to the concept of the present invention. They may be implemented in various forms and are not limited to the embodiments described herein.

Since the embodiments according to the concept of the present invention can make various changes and have various forms, the embodiments will be illustrated in the drawings and described in detail in this specification. However, this is not intended to limit the embodiments according to the concept of the present invention to specific disclosed forms, and includes changes, equivalents, or substitutes included in the spirit and technical scope of the present invention.

Terms such as first or second may be used to describe various components, but the components should not be limited by the terms. The above terms are used only for the purpose of distinguishing one component from another component, for example, a first component may be named a second component, without departing from the scope of rights according to the concept of the present invention, Similarly, the second component may also be referred to as the first component.

When a component is said to be "connected" or "connected" to another component, it is understood that it may be directly connected to or connected to the other component, but that other components may exist in between. It should be. On the other hand, when it is mentioned that a component is “directly connected” or “directly connected” to another component, it should be understood that there are no other components in between. Expressions that describe the relationship between components, such as “between”, “immediately between” or “directly adjacent to”, should be interpreted similarly.

The terminology used herein is only used to describe specific embodiments and is not intended to limit the invention. Singular expressions include plural expressions unless the context clearly dictates otherwise. In this specification, terms such as "include" or "have" are intended to designate the presence of a described feature, number, step, operation, component, part, or combination thereof, and one or more other features or numbers, It should be understood that this does not exclude in advance the possibility of the presence or addition of steps, operations, components, parts, or combinations thereof.

Unless otherwise defined, all terms used herein, including technical or scientific terms, have the same meaning as commonly understood by a person of ordinary skill in the technical field to which the present invention pertains. Terms as defined in commonly used dictionaries should be interpreted as having meanings consistent with the meanings they have in the context of the related technology, and unless clearly defined in this specification, should not be interpreted in an idealized or overly formal sense. No.

Hereinafter, embodiments will be described in detail with reference to the attached drawings. However, the scope of the patent application is not limited or limited by these examples. The same reference numerals in each drawing indicate the same members.

Quantum State Tomography (QST) is a quantum information processing task to determine unknown quantum states. This work is performed by measuring an ensemble of identically prepared quantum systems and inferring their states using various estimation methods. When the quantum state passes through the maximum noise depolarization channel with a clear causal sequence (classical trajectory) input before measurement, the quantum state switches to the maximum mixed state, making QST impossible.

Instead of propagating quantum states through classical trajectories, the proposed method here sets the channels in an infinite causal order (ICO). In this specification, we show that by setting a depolarizing channel in ICO, QST can be performed even if the channel is maximally noisy. And/or, this specification examines the impact of ICO on QST through linear regression estimation when the channel has no maximum noise. And/or, the present disclosure shows that configuring channels in an ICO arrangement is advantageous when the channels have maximum noise.

Quantum work that determines the state of an unknown system can be called quantum state tomography (QST). QST can consist of two steps.

First, measurements are performed on identically prepared copies of the quantum state.

Next, the quantum state is reconstructed using classical statistical estimation methods. Methods to construct a quantum state are linear inversion (MLE), maximum likelihood estimation, Bayesian estimation method, and linear regression. There may be.

Linear inversion can be performed assuming an ideal environment. Therefore, if an error occurs during qubit measurement, the recovered state may not be the physical state. MLE finds the density matrix that provides the highest probability of generating a set of measurement data determined by numerical optimization. The Bayesian method uses the Bayesian theorem to determine the posterior probability distribution p(ρ|D) with knowledge of the prior probability distribution p(ρ). Here D means measurement data.

In traditional quantum communication, quantum messages are a series of communication channels. and When transmitted through a channel, the order of messages transmitted through the channel is After passing or first After passing through It can be defined as passing through.

Quantum theory could allow new ways to combine quantum channels by creating superpositions of these channels. Through this, the information transfer order does not correspond to the previous two states. That is, the causal order between quantum channels is infinite. The process of infinite causal order can be achieved by a quantum switch where the order of operations is controlled by quantum bits. And/or, information can be transmitted through a combination of fully depolarizing channels, superimposing two alternative sequences. Quantum switches can be used to improve quantum metrology, enhance channel capacity, identify quantum depolarization channels, and enhance the quantum internet.

Herein, we model QST through linear regression under the assumption that the quantum state is transmitted through a quantum depolarization channel with maximum noise prior to measurement. Additionally, in this specification, we examine the impact of setting quantum channels in infinite causal order on QST. And, in this specification, the infidelity of the estimated state and the actual state is compared when transmitted through a channel assembled with a definite causal sequence and ICO. The latter method can support QST for quantum depolarization channels with maximum noise.

Hereinafter, in this specification, we first look at quantum state tomography using a linear regression method and a quantum switch, and then look at QST performance when transmitting quantum states through a channel established according to a clear causal order and ICO.

Quantum State Tomography

First, we look at a linear regression approach to estimate unknown quantum states.

Kronecker's delta function, equal to 1 if can be used. means tracking the matrix (1) and (2) Hermitian operator that satisfies It can be specified as . For example, in the single-qubit case, may be the same as Equation 1 below.

The qubit density operator can be parameterized as shown in Equation 2 below.

here, and the measurement set These basis It can be assumed that it can be parameterized as in Equation 3 below.

here, ego, am. The measurement set used in this work is This is the Stokes measurement set. Based on Born's rule, if multiple copies of an identically prepared quantum system are measured, The probability of obtaining the measurement result may be as follows:

here, is a column vector is the transpose of am.

From Equation 4 above, the linear regression form for QST may be as shown in Equation 5 below.

here, ego, am.

each About status It can be assumed that the quantum system is measured N times. Measurement results The probability of obtaining can be estimated by the following equation 6.

here, is the estimation error represents the observed frequencies with . Next, the linear regression equation in Equation 5 may be as shown in Equation 7 below.

here, ego, am. For large N, is the mean 0 and the change It can converge to a distribution for a normal distribution with . If more measurements are performed, measurements of the qubit state can become more accurate. The solution to the linear regression problem in Equation 7 above is to minimize Equation 8: It can be.

measurement set This information is complete or overflowing. It can be assumed that it has a station. estimate is an unbiased estimate with a mean squared error, and asymptotically may be equivalent to Equation 9 below.

here, am.

Quantum Depolarizing Channel

Quantum information can be transmitted through quantum channels. A quantum channel can be mathematically defined as a complete positive trace that preserves the map over the density operator. While transmitted through the channel, the qubit can evolve into a fully mixed state with probability (1-q) and remain with probability q. This type of quantum channel is called quantum depolarization channel It can be called . Evolution of qubits is defined as in Equation 10 below.

Quantum Switch

As described above, a quantum system transmitted through two quantum channels evolves in a clear causal sequence, with the second channel acting on the output of the first channel. Because quantum systems can evolve even in ICOs, it is impossible to determine which process occurs before the other. This is possible because of quantum switches. Quantum switches combine quantum channels, qubits called control qubits. You can control the relative order by . control qubit When in the state, the quantum switch is We will go through quantum systems in order. control qubit If in a state, the quantum state is It passes through quantum channels in order. control qubit together with and If there is a superposition of , the quantum switch changes the channel to and It can be placed in a nested order. As a result of the quantum switch, the Kraus operator of the quantum channel can be defined as Equation 11 below.

Here, channel and The Kraus operators are respectively and It can be expressed as

The quantum switch for one qubit system can be given as Equation 12 below.

Tomography methods, simulations and results

Quantum state to infer It can be assumed that is transmitted to the measurement system through a noisy depolarizing channel in the classical trajectory. If the channel is at maximum noise, e.g. When , from Equation 10, the qubit is It can be converted to the maximum mixed state of Equation 13 below, which does not depend on .

as a result, If QST is performed in It is impossible to reconstruct.

control qubit (here, ), if the channel is set in an infinite causal sequence with ), the output of the channel can be as shown in Equation 14 below.

When measuring the control qubit in basis, the output of the channel can be converted as shown in Equation 15 below.

here, and are respectively and can represent the probability of obtaining a measurement result. If the channel is maximally noisy, from Equation 15, If you perform QST on is the state meaning that it is possible to reconstruct can depend on

If you measure not You will get an estimate of

Whether the probability of a quantum state remains intact is Can be determined by quantum process tomography from Equation 15, which can be recovered as shown in Equation 16 below.

When Equation 16 is applied to the above, There may be some cases where is a non-physical matrix.

For this reason, can be projected onto the physical matrix space and expressed as a space of unit traces and positive semi-definite matrices. A simple projection algorithm can be adopted, the idea of which may be to transform all the eigenvalues of the density matrix into positive values and their sum.

<MSE analysis>

If the quantum state in Equation 15 is measured, then the state for The probability of obtaining the measurement result of can be defined as in Equation 17 below.

situation In the case of , the certainty can be defined as follows in Equation 18.

For the maximum noise channel (q=0), may be equal to the following equation 19:

Here, the control qubit is in the state In the case that applies, that is, In this case, better results can be obtained.

From equation 9, X can obtain the same value for both states obtained after measuring the control qubit. Therefore, there is a need to identify the differences caused by P.

To assume a contradiction, Can be expressed as Equation 22 through the following process from Equation 20 to Equation 21.

here, ego, and ego, and am.

Here, equation 22 is It can be expressed as is a contradiction. therefore, It can be done.

From Equation 13, if the quantum state is transmitted through a classical arrangement, The probability of obtaining a measurement result of And, accordingly, and A larger MSE will be given.

<Numerical Simulation>

Below, numerical simulations are performed on a single qubit system using the above-described scheme. QST is performed in pure and mixed states to ensure that the system performs well in both states.

Figure 1 is a diagram illustrating a mechanism for performing QST in a quantum depolarization channel with maximum noise by employing a quantum switch 110 in qubit transmission.

Referring to FIG. 1, the tomography device 100 according to an embodiment configured during simulation includes a quantum switch 110, a control qubit measurement device 120, and , may include a quantum state tomography device 130.

uniformly distributed according to the Haar measure and according to the Hilbert-Schmidt measure or Bures measure for mixed states. Random quantum states can be created. It is possible to model qubit transport in depolarizing channels and ICOs with maximum noise assembled from classical orbits prior to measurement. Measurements are made on each qubit. It can be performed every time. To quantify how close the actual state is to the estimate, both states The fidelity of all can be calculated.

next, Average log accuracy of quantum states can be calculated. If the states are transmitted in finite causal order and ICO, the Log infidelity of the actual pure states with this estimate can be represented as in Figure 2.

Figure 2 shows the reconstruction states for pure states (a) and mixed states (b) and actual state Indicates the log infidelity.

When real states are transmitted over channels placed in an ICO array, the log infidelity is and Measurement results can be displayed by red and green lines. Additionally, the average accuracy of the estimated state can be calculated as shown in Equation 23 below.

here, and is the measurement of the control qubit and In the case corresponding to , it may be the probability and accuracy of the estimated state.

From Figure 2, it can be seen that for the quantum depolarization channel with maximum noise, the ICO arrangement helps the QST estimation performance for both pure and mixed states. Measurement results The QST on the corresponding states is consistent with the previously described MSE analysis. A better estimate can be obtained than the corresponding states.

For example, the tomographic imaging device 300 according to the embodiment of FIG. 3 may perform the above-described operation. For example, the output passing through the quantum switch unit 310 of the tomography apparatus 300 may be expressed as Equation 14. And/or, the quantum switch unit 310 of the tomography apparatus 300 The output of the channel from the control qubit in the basis can be converted as shown in Equation 15. And/or, the tomography unit 320 and/or the estimation unit 330 of the tomography apparatus 300 may estimate the quantum state based on a linear regression approach, as described above for quantum state tomography. . For example, the tomography unit 320 and/or the estimation unit 330 of the tomography apparatus 300 may perform the calculations of Equations 1 to 9. And/or, the tomography unit 320 and/or the estimation unit 330 of the tomography apparatus 300 based on Equation 15 can also be reconstructed.

And/or, the tomography unit 320 and/or the estimation unit 330 of the tomography apparatus 300 is calculated by Equation 16: to or can be determined and/or estimated.

The operation of the tomography apparatus described with reference to FIG. 3 is not limited to the above-described method, and the operations described herein can be performed in various ways. Additionally, the tomographic imaging device may perform all or some of the operations described in this specification in addition to the operations of the tomographic imaging device described with reference to FIG. 3 above.

In this specification, we look at QST in a depolarizing channel with maximum noise. In this specification, a quantum switch was applied to enable QST in the maximum noise channel. As described herein, transmitting qubits in a quantum switching array provides more accurate quantum state estimates than transmitting them via classical orbitals. In other words, the present invention can perform accurate quantum state tomography in a depolarized quantum channel with maximum silencing.

Figure 3 is a block diagram of a tomographic imaging device according to an embodiment of the present invention.

Referring to FIG. 3, the tomography apparatus 300 according to an embodiment of the present invention includes a quantum switch unit 310 that is controlled by a control qubit and receives a quantum state to be estimated, and a quantum switch unit 310. It includes a tomography unit 320 that performs quantum state tomography on the output state, and an estimation unit 330 that estimates the initial state based on the output of the tomography unit 320 and algebra, even in the maximum noise channel. Quantum state imaging can be performed.

And/or, the quantum switch unit 310 may combine quantum channels.

And/or, tomography may be based on linear regression estimation.

And/or, the maximum noise channel may be set to Indefinite Causal Order (ICO).

And/or, the maximum noise channel may be a quantum depolarization channel.

According to another embodiment, the tomography device 300 transmits the quantum state to be estimated to the quantum switch, performs quantum state tomography in the state output from the quantum switch, and uses algebra in predicting the state output from the quantum switch. By performing , the quantum state tomography accuracy can be improved in the maximum noise depolarization quantum channel by estimating the initial state.

According to another embodiment, the tomography device 300 transmits the quantum state to be estimated to the quantum switch, performs quantum state tomography in the state output from the quantum switch, and uses algebra in predicting the state output from the quantum switch. By performing to estimate the initial state, quantum state tomography can be performed even when the channel noise is maximum.

And/or, when the quantum state is transmitted in a maximally noisy depolarizing quantum channel, it transforms into a maximally mixed state, making it impossible to perform quantum state tomography. Even when the channel noise is maximum, the quantum state tomography device according to one embodiment first transmits the quantum state to be estimated to the quantum switch, performs quantum state tomography in the state output from the quantum switch, and outputs from the quantum switch. The initial state can be estimated by performing algebra on the prediction of the resulting state.

The operations of the tomography apparatus described with reference to FIG. 3 are not limited to this, and all or some of the operations described in this specification may be performed by the tomography apparatus.

Figure 4 is a flowchart of a tomography method according to an embodiment of the present invention.

For example, the tomography method of FIG. 4 may be performed by the tomography apparatus of FIG. 3 .

Referring to FIG. 4, the tomography apparatus according to one embodiment may transmit the quantum state to be estimated to the quantum switch in step S401.

And/or, the tomography apparatus according to one embodiment may perform quantum state tomography in a state output from the quantum switch in step S402.

And/or, the tomography apparatus according to one embodiment may estimate the initial state based on the output and algebra of quantum state tomography in step S403.

Accordingly, the tomographic imaging device according to one embodiment can perform quantum state imaging even in the maximum noise channel.

And/or, quantum switches can combine quantum channels and be controlled based on control qubits.

And/or, the maximum noise channel can be set to Indefinite Causal Order (ICO).

And/or, the maximum noise channel may be a quantum depolarization channel.

And/or, tomography may be based on linear regression estimation.

According to another embodiment, the tomography method transmits the quantum state to be estimated to a quantum switch, performs quantum state tomography on the state output from the quantum switch, and performs algebra on the prediction of the state output from the quantum switch. By including operations to estimate the initial state, the accuracy of quantum state tomography in maximum noise depolarizing quantum channels can be improved.

According to another embodiment, the tomography method transmits the quantum state to be estimated to a quantum switch, performs quantum state tomography on the state output from the quantum switch, and performs algebra on the prediction of the state output from the quantum switch. Quantum state tomography can be performed even when channel noise is maximum, including the operation of estimating the initial state.

And/or, when the quantum state is transmitted in a maximally noisy depolarizing quantum channel, it transforms into a maximally mixed state, making it impossible to perform quantum state tomography. Even when the channel noise is maximum, the quantum state tomography method according to one embodiment first transmits the quantum state to be estimated to the quantum switch, performs quantum state tomography in the state output from the quantum switch, and outputs from the quantum switch. It may include an operation of estimating the initial state by performing algebra in the prediction of the state.

The tomography method described with reference to FIG. 4 is not limited to this, and all or some of the operations described in this specification may be included in the tomography method.

Although the embodiments have been described with limited drawings as described above, various modifications and variations can be made by those skilled in the art from the above description. For example, the described techniques are performed in a different order than the described method, and/or components of the described system, structure, device, circuit, etc. are combined or combined in a different form than the described method, or other components are used. Alternatively, appropriate results may be achieved even if substituted or substituted by an equivalent.

Therefore, other implementations, other embodiments, and equivalents of the claims also fall within the scope of the claims described below.

Claims (8)

A quantum switch controlled by a control qubit and receiving a quantum state to be estimated, comprising:
The quantum switch combines quantum channels and qubits called control qubits. The relative order is controlled by, and the control qubit is When in the state, the quantum switch is Passing through the quantum system in order, the control qubits If in a state, the quantum state is According to the order, the quantum switch passes through the quantum channel, and the control qubit is together with and If there is a superposition of the quantum switch, the channel and A quantum switch unit characterized in that it is placed in a superposition of orders;
a tomography unit that performs quantum state tomography on the state output from the quantum switch unit; and
An estimation unit that estimates the initial state based on the output of the tomography unit and algebra,
The switch part Convert the output of the channel in the control qubit at the basis,
The tomography unit or the estimation unit estimates the quantum state based on a linear regression approach as described in the description of quantum state tomography above,
The tomography unit or estimation unit is in a quantum state Reconstruct,
The tomography unit or estimation unit can be projected onto a physical matrix space and is characterized in that it newly estimates the reconstructed quantum state so that it can be expressed in a space of unit traces and positive semi-definite matrices. A tomographic device in which quantum state imaging is performed even in the highest noise channel. According to paragraph 1,
The quantum switch unit is a tomography device that combines quantum channels. According to paragraph 1,
A tomography device in which the maximum noise channel is set to an infinite causal order (ICO). According to paragraph 1,
A tomography device wherein the maximum noise channel is a quantum depolarization channel. transmitting the quantum state to be estimated to a quantum switch;
performing quantum state tomography in a state output from the quantum switch; and
estimating an initial state based on the output of the quantum state tomography and algebra,
Quantum state imaging is performed even in the maximum noise channel,
The quantum switch combines quantum channels and qubits called control qubits. The relative order is controlled by, and the control qubit is When in the state, the quantum switch is Passing through the quantum system in order, the control qubits If in a state, the quantum state is According to the order, the quantum switch passes through the quantum channel, and the control qubit is together with and If there is a superposition of the quantum switch, the channel and Characterized by being placed in a nested order,
The quantum switch is Convert the output of the channel in the control qubit at the basis,
The step of performing the quantum state tomography includes estimating the quantum state based on a linear regression approach as described in the description of the quantum state tomography above,
the quantum state Reconstruct,
The step of estimating the initial state based on the output and algebra of the quantum state tomography is,
The tomography or estimation can be projected onto a physical matrix space and is characterized by newly estimating the reconstructed quantum state so that it can be expressed as a space of unit traces and positive semi-definite matrices. How to shoot. According to clause 5,
A tomography method in which the quantum switch combines quantum channels and is controlled based on a control qubit. According to clause 5,
A tomographic method in which the maximum noise channel is set to an infinite causal order (ICO). According to clause 5,
A tomography method wherein the maximum noise channel is a quantum depolarization channel.