KR102636303B1 - Apparatus and method of measuring quantum correlation in single qubit metrology in quantum information integrated processing system - Google Patents

Abstract

The present invention relates to a quantum correlation measurement device and method in single qubit metrology, and in single qubit metrology experiments such as a deterministic quantum computation model using a single qubit. It relates to a quantum correlation measurement technology that investigates the use of metrological resources of a quantum correlation probe. In single qubit measurement according to an embodiment of the present invention, a quantum correlation measurement device is a Hadamard ( Hadamard ) based control qubit in a mixed quantum state. an input unit that inputs a quantum state by applying the Hadamard operation, an interaction unit that performs unitary evolution on the input quantum state based on controlled-unitary, and a unitary evolution according to the performed unitary evolution. It may include a measurement unit that generates a normalized trace of a single operator and measures quantum correlation using the generated normalized trace.

Description

Quantum correlation measurement device and method in single qubit measurement in quantum information processing integrated system {APPARATUS AND METHOD OF MEASURING QUANTUM CORRELATION IN SINGLE QUBIT METROLOGY IN QUANTUM INFORMATION INTEGRATED PROCESSING SYSTEM}

The present invention relates to a quantum correlation measurement device and method in single qubit metrology, and in single qubit metrology experiments such as a deterministic quantum computation model using a single qubit. It concerns quantum correlation measurement techniques that investigate the metrological resource utilization of quantum correlation probes. The quantum correlation measurement device and method can be applied to the quantum correlation measurement processing unit of the quantum observation and analysis platform in the quantum information technology integrated system.

Quantum correlation forms the basis of core quantum technologies, and entanglement between layers of existing correlations is critical for advancements in the fields of quantum metrology, quantum cryptography, and quantum information processing.

In other words, traditional quantum correlations, namely entanglement and discord, are key elements that ensure quantum advantage in quantum information processing tasks.

Entanglement has the problem of not being able to fully explain the role of mixed states due to the separability issue of mixed states. Additionally, nonlocality can exist even without entanglement.

Although not all states are entangled, almost all quantum states can be disturbed and exhibit nonclassical effects such as discord.

Quantum mismatch plays an important role in quantum acceleration and mixed-state metrology.

Meanwhile, quantum metrology is a high-accuracy measurement field that utilizes quantum correlation as a resource.

Quantum metrology is generally a four-step process, where a quantum probe is prepared in a bipartite state.

Meanwhile, quantum Fisher information (QFI) is an acceptable quantifier of the accuracy achievable through arbitrary quantum probes through instrumentation.

On the other hand, because separable quantum states exhibit non-annihilating QFI, entanglement has the problem of not being able to fully explain the role of quantum correlation in metrology.

Additionally, there is a problem that the quantum correlation of separable states limits the applicability of entanglement to capture the minimum sensitivity exhibited by the probe.

For example, quantum correlations induced by disturbances during measurement, i.e. measurement-induced disturbance (MID), are useful.

However, quantum correlation under a single operation, i.e. Local Non-Effective Unitary Operation (LNU), does not provide a relative advantage because it asymptotically disappears as the system size increases.

This could lead to speculation that quantum algorithms may exist that do not exploit quantum entanglement as a resource and have exponential or polynomial advantages over their classical counterparts.

Recently, a new bimolecular quantum correlation was discovered that depends on how strongly the observer perturbs the unobserved system.

These quantum correlations are called measurement-based quantum correlations (MbQC), and this unique property distinguishes MbQC from traditional quantum correlations such as entanglement and discordance.

A quantum information integration system is needed that can support not only quantum correlation measurement processing, but also various quantum information processing to support the implementation and research of related technologies.

Korean Patent No. 10-2211060, “Quantum measurement method using perturbation sensitivity and quantum system using the same” Korean Patent No. 10-2120663, “Measurement-based quantum correlation measurement method in mixed-state quantum measurement”

The purpose of the present invention is to investigate the use of metrological resources of quantum correlation probes in single qubit metrology experiments, such as deterministic quantum computation models using single qubits. .

The purpose of the present invention is to provide functions related to detection of non-functional qubit probes in measurement experiments, quantum clock synchronization for quantum communication, and efficient quantum algorithms for quantum computation.

The purpose of the present invention is to identify mixed resource utilization states for noisy qubit measurements, identify metrological resources of various random unitary operators, and provide an improved metrological method for quantum parameter estimation. Do it as

The purpose of the present invention is to provide an efficient computational framework for noisy NISQ (Noisy Intermediate-Scale Quantum) devices as an improved metrological device for quantum sensing devices.

In single qubit measurement according to an embodiment of the present invention, the quantum correlation measurement device includes an input unit that inputs the quantum state by applying Hadamard operation based on a controlled qubit to the mixed quantum state, and a controlled-unitary ) based on the interaction unit that performs unitary evolution on the input quantum state and a normalized trace of a single operator according to the performed unitary evolution, and using the generated normalized trace It may include a measuring unit that measures quantum correlation.

The input unit may input the quantum state based on Equation 2 below.

[Equation 2]

The interaction unit may perform single evolution on the input quantum state as shown in Equation 3 below.

[Equation 3]

The controlled-unitary can be given as Equation 4 below.

[Equation 4]

The control qubit can be given as Equation 5 below.

[Equation 5]

The measurement unit measures the quantum correlation in which the unitary operator is a random Hermitian unitary operator, and discord and entanglement disappear as the generated normalized trace converges to 1. You can.

The quantum correlation measurement method in single qubit measurement according to an embodiment of the present invention includes the steps of applying a Hadamard operation based on a control qubit to a mixed quantum state in an input unit to input a quantum state, and an interaction unit. In the step of performing unitary evolution for the input quantum state based on controlled-unitary and the measurement unit, generating a normalized trace of a unitary operator according to the performed unitary evolution, , may include measuring quantum correlation using the generated normalized trace.

The step of inputting a quantum state by applying a Hadamard operation based on a control qubit to the mixed quantum state may include inputting the quantum state based on Equation 2 below.

[Equation 2]

In the step of performing unitary evolution on the input quantum state based on the controlled-unitary, unitary evolution on the input quantum state can be performed as shown in Equation 3 below: .

[Equation 3]

The step of generating a normalized trace of a single operator according to the performed unitary evolution and measuring quantum correlation using the generated normalized trace is where the single operator is a random Hermitian unitary operator. , measuring the quantum correlation such that discord and entanglement disappear as the generated normalized trace converges to 1.

The present invention can investigate the use of metrological resources of quantum correlation probes in single qubit metrology experiments, such as deterministic quantum computation models using single qubits.

The present invention can provide functions related to detection of non-functional qubit probes in measurement experiments, quantum clock synchronization for quantum communication, and efficient quantum algorithms for quantum computation.

The present invention can provide an improved metrological method for identifying mixed resource utilization states for noisy qubit measurements, identifying metrological resources of various random unitary operators, and quantum parameter estimation. .

The present invention is an improved metrological device for quantum sensing devices and can provide an efficient computational framework for noisy intermediate-scale quantum (NISQ) devices.

Additionally, according to embodiments of the present invention, it is possible to support not only quantum correlation measurement processing but also various quantum information processing to support implementation and research of related technologies.

1 is a diagram illustrating a quantum system according to an embodiment of the present invention.
Figure 2 is a diagram illustrating a deterministic quantum calculation protocol using a single qubit according to an embodiment of the present invention.
Figure 3 is a diagram illustrating a quantum correlation measurement device in single qubit measurement according to an embodiment of the present invention.
Figure 4 is a diagram illustrating a quantum correlation measurement method in single qubit measurement according to an embodiment of the present invention.
Figure 5 is a diagram illustrating quantum correlation measurement results in a deterministic quantum calculation protocol using a single qubit according to an embodiment of the present invention.
Figure 6 is a diagram for explaining a counterfactual entanglement distribution model according to an embodiment of the present invention.
Figure 7 is a diagram showing the concept of controlled quantum teleportation according to an embodiment of the present invention.
Figure 8 is a flowchart showing the operation of a controlled counterfactual quantum teleportation support device according to an embodiment of the present invention.
Figure 9 is a diagram explaining a mechanism for performing QST in qubit transmission.
Figure 10 shows the Log infidelity of the reconstructed state and the actual state for pure states (a) and mixed states (b).
Figure 11 is a block diagram of a tomographic imaging device according to an embodiment of the present invention.
Figure 12 is a diagram showing a quantum information technology integrated system according to an embodiment of the present invention.

Hereinafter, various embodiments of this document are described with reference to the attached drawings.

The embodiments and terms used herein are not intended to limit the technology described in this document to a specific embodiment, and should be understood to include various changes, equivalents, and/or substitutes for the embodiments.

In the following description of various embodiments, if a detailed description of a related known function or configuration is judged to unnecessarily obscure the gist of the invention, the detailed description will be omitted.

The terms described below are terms defined in consideration of functions in various embodiments, and may vary depending on the intention or custom of the user or operator. Therefore, the definition should be made based on the contents throughout this specification.

In connection with the description of the drawings, similar reference numbers may be used for similar components.

Singular expressions may include plural expressions, unless the context clearly indicates otherwise.

In this document, expressions such as “A or B” or “at least one of A and/or B” may include all possible combinations of the items listed together.

Expressions such as “first,” “second,” “first,” or “second,” can modify the corresponding components regardless of order or importance and are used to distinguish one component from another. It is only used and does not limit the corresponding components.

When a component (e.g. a first) component is said to be "connected (functionally or communicatively)" or "connected" to another (e.g. a second) component, it means that the component is connected to the other component. It may be connected directly to a component or may be connected through another component (e.g., a third component).

In this specification, “configured to” means “suitable for,” “having the ability to,” or “changed to,” depending on the situation, for example, in terms of hardware or software. ," can be used interchangeably with "made to," "capable of," or "designed to."

In some contexts, the expression “a device configured to” may mean that the device is “capable of” working with other devices or components.

For example, the phrase "processor configured (or set) to perform A, B, and C" refers to a processor dedicated to performing the operations (e.g., an embedded processor), or by executing one or more software programs stored on a memory device. , may refer to a general-purpose processor (e.g., CPU or application processor) capable of performing the corresponding operations.

Additionally, the term 'or' means 'inclusive or' rather than 'exclusive or'.

That is, unless otherwise stated or clear from the context, the expression 'x uses a or b' means any of the natural inclusive permutations.

Terms such as '..unit' and '..unit' used hereinafter refer to a unit that processes at least one function or operation, and may be implemented as hardware, software, or a combination of hardware and software.

1 is a diagram illustrating a quantum system according to an embodiment of the present invention.

Figure 1 illustrates a quantum system comprised of a plurality of quantum devices according to an embodiment of the present invention. The device shown in FIG. 1 can be applied to the quantum correlation measurement processing unit 2035 of FIG. 11.

Referring to FIG. 1, a quantum system 100 according to an embodiment of the present invention includes a first quantum device 110, a second quantum device 120, and a third quantum device 300. Here, the number of quantum devices can be added or decreased.

For example, the first quantum device 110 may be a transmitting device, and the second quantum device 120 may be a receiving device, but they are not limited to this and may be interchanged.

For example, when the second quantum device 120 transmits to the first quantum device 110, the first quantum device 110 and the second quantum device 120 can operate as a receiving device and a transmitting device, respectively. there is.

According to one embodiment of the present invention, the third quantum device 130 may be a quantum device that controls the first quantum device 110 and the second quantum device 120.

For example, the third quantum device 130 may be a device that measures quantum correlation between the first quantum device 110 and the second quantum device 120.

Quantum correlation of quantum system 100 can be considered a resource for quantum computation and quantum metrology.

However, in general, not all quantum correlations can be useful for deterministic quantum computation with single qubit (DQC1) protocols.

For example, each of the first to third quantum devices 110 to 130 includes an interface unit, a control unit, and a memory unit.

For example, the interface unit may be a quantum interface unit that uses a quantum channel, or it may be a general interface unit that uses a classical channel.

The memory may be a quantum memory that stores quantum information, or it may be a general memory that stores general information.

In traditional quantum correlation, entanglement and discord may be key elements to ensure quantum advantage in quantum information processing tasks.

Quantum information processing protocols that estimate the normalized trace of a single operator more efficiently than conventional operators can be utilized.

The accuracy achievable in the Deterministic Quantum Computation with Single qubit (DQC1) protocol, which will be described in Figure 2, can be related to the quantum coherence present in the control qubit.

For example, quantum correlations induced by perturbations during measurement, i.e. measurement-induced disturbances (MID), are useful, but quantum correlations under a single operation, i.e. Local Non-Effective Unitary Operations (LNUs), asymptotically decrease as system size increases. Because it disappears, it may not provide a relative advantage.

This could lead to speculation that quantum algorithms may exist that do not exploit quantum entanglement as a resource and have exponential or polynomial advantages over their classical counterparts.

Figure 2 is a diagram illustrating a deterministic quantum calculation protocol using a single qubit according to an embodiment of the present invention.

Figure 2 illustrates a configuration for measuring quantum correlation in quantum metrology based on a deterministic quantum computing protocol using a single qubit according to an embodiment of the present invention.

Referring to FIG. 2, the deterministic quantum computation with single qubit (DQC1) protocol 200 according to an embodiment of the present invention efficiently tracks a unitary operator. It is estimated that

As an example, the deterministic quantum computation protocol 200 using a single qubit measures quantum correlation in a single qubit metrology experiment.

According to one embodiment of the present invention, the deterministic quantum computing protocol 200 using a single qubit is designed to efficiently estimate the normalized trace of a unitary matrix Un given by τ = tr(Un) /2n.

As an example, the deterministic quantum computing protocol 200 using a single qubit consists of a control qubit system 210 and an auxiliary system 220.

Quantum metrology generally involves four stages: preparation stage, encoding stage, measurement stage, and estimation stage.

In the preparation stage, a very mixed initial state is presented as shown in Equation 1 below.

[Equation 1]

In Equation 1, p may be associated with a quantum state, p ^CA may represent a mixed state, I may represent an input operator, and Z may represent a Pauli operator.

According to one embodiment of the present invention, in the deterministic quantum calculation protocol 200 using a single qubit, the input value is a control qubit (C), and a Hadamard (H) gate is applied to the control qubit to create a synthetic structure. indicates.

The deterministic quantum calculation protocol 200 using a single qubit inputs an initial state as shown in Equation 2 below after Hadamard operation on the control qubit.

[Equation 2]

In Equation 2, p _i ^CA may represent a Hadamard-operated mixed state based on the control qubit.

As an example, the deterministic quantum computation protocol 200 using a single qubit undergoes this input quantum state through a controlled single evolution based on the auxiliary system 220 to calculate Equation 3 below.

[Equation 3]

In Equation 3, p _f ^CA can represent a quantum state in which controlled single evolution has occurred.

For example, controlled-unitary can be given as Equation 4 below.

[Equation 4]

In Equation 4, U _n may represent a single matrix. Meanwhile, before measurement, the control qubit can be given as in Equation 5 below.

[Equation 5]

In Equation 5, α determines the purity of a single control qubit system given by (1+α ² )/2, and finally, the observable You can.

[Equation 6]

In Equation 6, Xp _f ^C may represent the normalized trace for the X measurement, and Yp _f ^C may represent the normalized trace for the Y measurement.

Sufficient execution of the deterministic quantum computation protocol 200 using a single qubit can yield a normalized trace estimate of the single operator used.

The quantum computational advantage of the deterministic quantum computing protocol 200 using a single qubit depends on the presence of quantum correlation in the final state ρCAf for the partition control qubit system (C)|auxiliary system (A) .

The presence of quantum correlations may depend on the normalized trace of the single operator used.

Accordingly, the quantum state according to the quantum correlation measured according to the deterministic quantum computation protocol 200 using a single qubit can be maintained as a separable state for a single operator.

In other words, the deterministic quantum computation protocol 200 using a single qubit involves applying a control qubit to a Hadamard gate in the control system 210, followed by controlled single evolution to produce a final state in the auxiliary system 220. After processing, observable measurements on the evolved control qubit can yield an estimate of the normalized trace of a single operator. Here, the calculated estimate can be used to measure quantum correlation.

Therefore, the present invention can investigate the use of metrological resources of quantum correlation probes in single qubit metrology experiments, such as deterministic quantum computation models using single qubits. .

Figure 3 is a diagram illustrating a quantum correlation measurement device in single qubit measurement according to an embodiment of the present invention.

Figure 3 illustrates the components of a quantum correlation measurement device in single qubit measurement according to an embodiment of the present invention.

Referring to FIG. 3, in single qubit measurement according to an embodiment of the present invention, the quantum correlation measurement device 300 includes an input unit 310, an interaction unit 320, and a measurement unit 330.

For example, the input unit 310 may perform a similar role as the control qubit system 210 described in FIG. 2.

For example, the interaction unit 320 may perform a similar role as the auxiliary system 220 described in FIG. 2 .

According to one embodiment of the present invention, the input unit 310 inputs a quantum state by applying a Hadamard operation based on a control qubit to the mixed quantum state.

For example, the input unit 310 is a prepared step and inputs the initial quantum state by converting the initial quantum state into a highly mixed initial quantum state through Hadamard transformation.

According to one embodiment of the present invention, the interaction unit 320 performs unitary evolution on the input quantum state based on controlled-unitary.

That is, the interaction unit 320 performs controlled single evolution on the quantum state input by the input unit 310.

For example, the interaction unit 320 performs an encoding step of quantum measurement.

According to one embodiment of the present invention, the measurement unit 330 can generate a normalized trace of a single operator according to the single evolution performed by the interaction unit 320, and measure quantum correlation using the normalized trace. there is.

For example, the measurement unit 330 uses quantum correlation where the unitary operator is a random Hermitian unitary operator and discord and entanglement disappear as the generated normalized trace converges to 1. It can be measured.

0을 갖는 Haar 랜덤 단일 연산자에 대해 불일치가 존재하지만, 정규화된 추적 tr(U2n)/2n

1을 갖는 랜덤 에르미트 단일 연산자에 대해서는 불화 및 얽힘이 사라질 수 있다.For example, normalized trace tr(U2n)/2n

A discrepancy exists for the Haar random unary operator with 0, but the normalized trace tr(U2n)/2n

For a random Hermitian unitary operator with 1, discord and entanglement can disappear.

Therefore, the present invention can provide an efficient computational framework for noisy intermediate-scale quantum (NISQ) devices as an improved metrological device for quantum sensing devices.

Figure 4 is a diagram illustrating a quantum correlation measurement method in single qubit measurement according to an embodiment of the present invention.

Figure 4 illustrates a quantum correlation measurement method in single qubit measurement, such as a deterministic quantum computation model using a single qubit, according to an embodiment of the present invention.

Referring to FIG. 4, the quantum correlation measurement method in single qubit measurement according to an embodiment of the present invention inputs the quantum state by applying Hadamard operation based on the control qubit in step 401.

That is, the quantum correlation measurement method in single qubit measurement is based on the Hadamard operation according to the control qubit of the control system and the Hadamard gate, and inputs a highly mixed initial quantum state into the auxiliary system.

In step 402, the quantum correlation measurement method in single qubit measurement according to an embodiment of the present invention performs a single evolution of the quantum state based on the control unit.

That is, the quantum correlation measurement method in single qubit measurement performs unity evolution on the input value of Equation 2 and outputs Equation 3, the control single is given as Equation 4, and the control qubit is given by Equation 5 and can be given together.

In step 403, the method for measuring quantum correlation in single qubit measurement according to an embodiment of the present invention generates a normalized trace of a single operator according to single evolution, and measures quantum correlation using the normalized trace.

In other words, the quantum correlation measurement method in single qubit measurement is that the unitary operator is a random Hermitian unitary operator, and as the generated normalized trace converges to 1, discord and entanglement disappear. Quantum correlation can be measured.

Figure 5 is a diagram illustrating quantum correlation measurement results in a deterministic quantum calculation protocol using a single qubit according to an embodiment of the present invention.

Referring to FIG. 5, a graph 500 represents the change in α, which determines the purity of a single control qubit, on the horizontal axis, and the change in quantum correlation measurement value on the vertical axis.

The graph line 501 represents measurement-based quantum correlations (MbQC) of a random Hermitian single operator.

Graph line 502 represents the discord of any Harr single operator.

The graph line 503 represents the MbQC of an arbitrary Harr single operator.

Graph line 504 represents a discord of a random Hermitian unitary operator.

Graph line 505 represents entanglement.

Graph lines 501 to 503 show an increase according to α, which determines the purity of the control qubit.

However, graph line 504 indicates that the random Hermitian unitary operator discord disappears.

Graph lines 501 and 503 indicate that there are no exceptions according to control qubits in MbQC.

That is, the graph 500 represents a numerical comparative analysis of quantum correlations in a deterministic quantum computing protocol model using a single qubit, and suggests that MbQC expels entanglement and inconsistency in terms of resource relevance and generality.

The existence of quantum correlation may vary depending on the normalized trace of the single operator used, and can be supplementally explained through Equation 7 below.

[Equation 7]

In equation 7, Q(p ^CA ) can represent quantum correlation, can represent the characterization of the local von Neumann measurement, may represent the normalization factor, can represent an effective distance matrix between the condition states of the pre-measurement and post-measurement of the auxiliary system.

MbQC can be proven to be a simpler and analytically calculable measurement figure of merit in single-qubit measurements.

The quantum advantage in a deterministic quantum computing protocol using a single qubit can depend on the single operator used as well as the presence of quantum correlations in the control qubit system (C)|auxiliary system (A).

In a deterministic quantum computing protocol using a single qubit, MbQC, disparity and entanglement values are obtained for bipartition and the average values can be calculated for numerous Harr and Hermitian unitary operators.

Therefore, the present invention can provide functions related to non-functional qubit probe detection in measurement experiments, quantum clock synchronization for quantum communication, and efficient quantum algorithms for quantum computation.

In addition, the present invention provides an improved metrological method for identifying mixed resource utilization states for noisy qubit measurements, identifying metrological resources of various random unitary operators, and quantum parameter estimation. You can.

Figure 6 is a diagram for explaining a counterfactual entanglement distribution model according to an embodiment of the present invention.

Referring to FIG. 6, the first terminal (Alice) 610, the second terminal (Bob) 620, and the third terminal (Charlie) 630 can each operate as a transmitting terminal, a receiving terminal, and a control terminal. there is. For example, when Alice (610) sends quantum information to Bob (620), Charlie (630) acts as a control terminal, and conversely, it is also possible for Bob (620) to send quantum information to Alice (630). do. As such, the controlled quantum teleportation protocol according to an embodiment of the present invention assumes a three-party quantum network.

Meanwhile, in FIG. 6, reference numeral 640 'SW' refers to a quantum channel including a switchable mirror and a switchable rotor, QAO is an abbreviation for quantum absorptive object, and V-CQZ is a vertical ( Vertical) indicates that it contains a CQZ gate, and H-CQZ indicates that it contains a horizontal (Horizontal) CQZ gate.

At least one of the first terminal (Alice) 610, the second terminal (Bob) 620, and the third terminal (Charlie) 630 shown in FIG. 6 is the counterfactual quantum communication processor 2013 of FIG. 12. can be applied to

[Step 1]

In the first step, each terminal counterfactually generates 2N GHZ states.

To safely perform controlled counterfactual quantum teleportation, Alice, Bob, and Charlie use a CQZ (Counterfactual quantum Zeno) gate to counterfactually generate 2N GHZ states. Here, N represents a natural number, and each terminal generates a 2N tripartie GHZ state with one qubit of the 2N GHZ state.

Alice, assumed to be a transmitting terminal, has her qubit operate as a quantum absorptive object (QAO) in each round of GHZ state creation in order to counterfactually distribute entanglement. Prepare the qubit in this state. At this time, Bob and Charlie each and Prepare the qubit in the in state.

Bob and Charlie alternately connect with Alice through quantum channels using CQZ gates in each round of GHZ state generation, as shown in Figure 6.

At the end of each round, the initial states of Alice, Bob, and Charlie are converted as shown in Equation 8.

[Equation 8]

,

.

Here, A, B, and C are indices representing Alice, Bob, and Charlie, respectively, and i refers to the index representing the ith round of GHZ state creation.

[Second stage]

In the second step, Alice, Bob, and Charlie perform qubit measurements and announce the results.

First, for security check, Alice selects a random subset of N GHZ states among 2N GHZ states and announces the location of the selected subset.

Bob and Charlie randomly select a measurement basis from the {x, y} set, perform measurements on each qubit, and report the measurement results and the selected basis set.

Alice compares or matches the measurement results of Bob and Charlie by referring to the measurement result table after Bob and Charlie's measurements as shown in [Table 1]. Through this, you can check whether the measurement results are consistent.

[Table 1]

If the consistency of the measurement results is reliable up to a preset acceptable threshold, Alice, Bob, and Charlie can perform the next steps. That is, Alice, Bob, and Charlie can transmit arbitrary unknown qubits in a tripartite quantum network using N GHZ states other than the subset selected in the next step.

Meanwhile, considering that each party can operate as a transmitting terminal, receiving terminal, and control terminal, the GHZ state subscripts will be expressed as s, r, and c in the following description. Here, s, r, and c represent qubits provided (or owned by the corresponding terminal) in the transmitting terminal, receiving terminal, and control terminal, respectively.

[Step 3]

In the third step, the transmitting terminal teleports the unknown quantum state by performing a bell-based measurement on the qubit it owns. More specifically, the unknown quantum state defined as Equation 9: Assume that the transmitting terminal transmits.

[Equation 9]

,

.

The combination state of the unknown quantum state and the GHZ state can be expressed as Equation 10.

[Equation 10]

,

.

here, and represents the bell state, and j represents the jth round of controlled quantum teleportation.

In the third step, the transmitting terminal performs Bell basis measurements on qubits q and s _j , and records and maintains the measurement results.

[Step 4]

In the fourth step, the control terminal measures the qubit based on the Hadamard basis and notifies the receiving terminal of the measurement result to reconstruct the unknown quantum state.

More specifically, the control terminal performs the Hadamard gate H on qubit c _j and measures the qubit as a computational basis. The control terminal notifies the measurement result through 1 bit of classical information m to approve the controlled counterfactual quantum teleportation.

When the control terminal approves communication between the sending terminal and the receiving terminal, the sending terminal is in the ring state. corresponding to The measurement result is notified to the control terminal and the receiving terminal through 2 bits of m1m2, which is classical information from the set.

Figure 7 is a diagram showing the concept of controlled quantum teleportation according to an embodiment of the present invention.

Referring to FIG. 7, it can be seen that the transmitting terminal performs controlled quantum teleportation by notifying the bell measurement result with the approval of the control terminal.

[Step 5]

In the fifth step, the receiving terminal reconstructs the quantum state.

More specifically, the receiving terminal performs a unitary operation defined as Equation 11 on the qubit r _j based on the measurement results of the transmitting terminal and the control terminal to reconstruct the quantum state.

[Equation 11]

Here, X and Z refer to the Pauli x and z operators, respectively.

The method described through steps 1 to 5 can reconstruct an unknown quantum state sequence under the approval of a control terminal performing control. Therefore, it is safer against various attack strategies and can be used in a multi-party communication system.

If a terminal attempting eavesdropping attempts to participate in the entanglement distribution process, it may be in an entanglement relationship, but the terminal attempting eavesdropping may be detected during the security check process of the second stage.

Additionally, if a terminal attempting eavesdropping adds invisible photons to steal Alice's unknown quantum state, more errors may occur in the security check step due to the invisible photons.

Figure 8 is a flowchart showing the operation of a controlled counterfactual quantum teleportation support device according to an embodiment of the present invention.

Here, the controlled counterfactual quantum teleportation support device may be any one of a transmitting terminal, a receiving terminal, and a control terminal, and Figure 8 corresponds to the case where the controlled counterfactual quantum teleportation support device is a transmitting terminal.

Referring to FIG. 8, the transmitting terminal counterfactually generates 2N GHZ (Greenberger-Horne-Zeilinger) states in step 810.

In step 820, the transmitting terminal selects a random subset of N GHZ states and notifies the location of the selected subset.

In step 830, the transmitting terminal checks whether the measurement results are consistent.

If the consistency of the measurement result is reliable up to a preset acceptable threshold, the transmitting terminal performs bell-based measurement on the owned qubit in step 840.

Thereafter, if communication is approved by the control terminal, in step 850, the transmitting terminal notifies the measurement result using the classic 2 bits corresponding to the bell state.

Hereinafter, a quantum state tomography processing unit that can be applied to the quantum experiment platform in the quantum information technology integrated system of the present invention will be described.

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.

The quantum state tomography processing unit according to an embodiment of the present invention transmits the quantum state to be estimated to the quantum switch, performs quantum state tomography in the state output from the quantum switch, and calculates the output of the quantum state tomography and algebra. Based on this, the initial state can be estimated.

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 before measurement. Additionally, in this specification, we examine the impact of setting the quantum channel to an 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 12 below.

[Equation 12]

The qubit density operator can be parameterized as in Equation 13 below.

[Equation 13]

here, and the measurement set These basis It can be assumed that it can be parameterized as shown in Equation 14 below.

[Equation 14]

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 in Equation 15.

[Equation 15]

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

From Equation 15 above, the linear regression form for QST may be as shown in Equation 16 below.

[Equation 16]

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 17.

[Equation 17]

here, is the estimation error represents the observed frequencies with . Next, the linear regression equation in Equation 16 may be as shown in Equation 18 below.

[Equation 18]

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 18 above is to minimize Equation 19: It can be.

[Equation 19]

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 20 below.

[Equation 20]

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 21 below.

[Equation 21]

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 22 below.

[Equation 22]

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 23 below.

[Equation 23]

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 24 below, which does not depend on .

[Equation 24]

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

control qubit (here, ), if the channel is set up in an infinite causal sequence, the output of the channel can be as shown in Equation 25 below.

[Equation 25]

When measuring the control qubit in basis, the output of the channel can be converted as shown in Equation 26 below.

[Equation 26]

here, and are respectively and can represent the probability of obtaining a measurement result. If the channel is maximally noisy, from equation 26: 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 26, which can be recovered as shown in Equation 27 below.

[Equation 27]

If Equation 27 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 semidefinite 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 26 is measured, then the state for The probability of obtaining the measurement result of can be defined as Equation 28 below.

[Equation 28]

situation In the case of , the certainty can be defined as follows in Equation 29.

[Equation 29]

For the maximum noise channel (q=0), may be equal to the following equation 30:

[Equation 30]

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 20, 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 33 through the following process from Equation 31 to Equation 32.

[Equation 31]

[Equation 32]

[Equation 33]

here, ego, and ego, and am.

Here, equation 33 is It can be expressed as is a contradiction. therefore, It can be done.

From Equation 24, 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.

FIG. 9 is a diagram illustrating a mechanism for performing QST in a quantum depolarization channel with maximum noise by employing a quantum switch 1010 in qubit transmission.

Referring to FIG. 9, the tomography device 1000 according to an embodiment configured during simulation includes a quantum switch 1010, a control qubit measurement device 1020, and , may include a quantum state tomography device 1030.

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 10.

Figure 10 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 34 below.

[Equation 34]

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 10, 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 1100 according to the embodiment of FIG. 11 may perform the above-described operation. For example, the output passing through the quantum switch unit 1110 of the tomography apparatus 1100 may be expressed as Equation 25. And/or, the quantum switch unit 1110 of the tomography device 1100 The output of the channel from the control qubit in the basis can be converted as shown in Equation 26. And/or, the tomography unit 1120 and/or the estimation unit 1130 of the tomography apparatus 1100 may estimate the quantum state based on a linear regression approach, as described above for quantum state tomography. . For example, the tomography unit 1120 and/or the estimation unit 1130 of the tomography apparatus 1100 may perform the calculations of Equations 12 to 20. And/or, the tomography unit 1120 and/or the estimation unit 1130 of the tomography apparatus 1100 based on Equation 26 can also be reconstructed.

And/or, the tomography unit 1120 and/or the estimation unit 1130 of the tomography apparatus 1100 is calculated by Equation 23: to or can be determined and/or estimated.

The operation of the tomography apparatus described with reference to FIG. 11 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. 11 above.

The tomography apparatus of FIG. 11 can be applied to the quantum state tomography processing unit 2031 of FIG. 12.

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

Referring to FIG. 11, the tomography apparatus 1100 according to an embodiment of the present invention includes a quantum switch unit 1110 that is controlled by a control qubit and receives a quantum state to be estimated, and a quantum switch unit 1110. It includes a tomography unit 1120 that performs quantum state tomography on the output state, and an estimation unit 1130 that estimates the initial state based on the output of the tomography unit 1120 and algebra, even in the maximum noise channel. Quantum state imaging can be performed.

And/or, the quantum switch unit 1110 may combine quantum channels.

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

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.

According to another embodiment, the tomography device 1100 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 1100 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. 11 are not limited to this, and all or some of the operations described in this specification may be performed by the tomography apparatus.

Figure 12 is a diagram showing a quantum information technology integrated system according to an embodiment of the present invention.

Referring to FIG. 12, it includes a quantum communication platform (2010), a quantum computing platform (2020), a quantum observation and analysis platform (2030), a platform control unit (2040), and a platform interface unit (2050).

Quantum Communication Platform (2010) is a platform that provides implementation and experimentation of element technologies related to quantum communication and quantum information communication.

Quantum communication platform (2010) is a quantum key distribution processor (2011) that performs secure communication through quantum key distribution (QKD), particle-free information transmission in which no particles exist in the channel. and a counterfactual quantum communication processing unit (2013) that performs a new form of full-duplex information transmission and blind tele-computation, and radio resource management when applying quantum communication to next-generation mobile communication. It may include a radio resource management processing unit 2015 that performs and a network interference control processing unit 2017 that performs network interference control. For example, the counterfactual quantum communication processing unit 2013 may perform the role of at least one of the first terminal (Alice), the second terminal (Bob), and the third terminal (Charlie).

The Quantum Computing Platform (2020) is a platform that provides implementation and experiments of element technologies for the practical use of quantum computers capable of exponentially fast parallel calculations based on quantum algorithms.

The quantum computing platform (2020) includes a variant quantum algorithm processor (2021) that provides a variational quantum algorithm (VQA) specifically designed for noisy intermediate-scale quantum (NISQ) devices, the Grover search algorithm or Quantum search algorithm processing unit (2023) that provides a quantum search algorithm (QSA) such as a quantum counting algorithm and quantum machine learning such as quantum reinforcement learning of a multi-body Hamiltonian It may include a quantum machine learning processing unit (2025) that provides learning: QML) technology.

The Quantum Observation and Analysis Platform (2030) supports quantum measurements and quantum simulations and directly utilizes the quantum properties of real particles, such as quantum coherence and entanglement, solving problems that are difficult to simulate on classical computers. It is a platform for doing this.

The quantum observation and analysis platform (2030) includes a quantum state tomography processor (2031) that provides quantum state tomography, a quantum state measurement processor (2033) that supports quantum state measurements, and quantum correlation measurements for qubit measurements and experiments. It includes a quantum correlation measurement processing unit 2035 and a tensor network experiment processing unit 2037 supporting tensor network and Hamiltonian-based quantum simulation for simulation.

The platform control unit 240 stores the history and measurement results performed on the quantum communication platform (2010), the quantum computing platform (2020), and the quantum observation and analysis platform (2030), and controls the components of the other platform from any one platform. When you need to use it, you can perform an action to connect it.

For example, if a Bell measurement is required in the counterfactual quantum communication processing unit (2013) of the quantum communication platform (2010), the Bell measurement is performed in the quantum state measurement processing unit (2033) or the quantum correlation measurement processing unit (2033) of the quantum observation and analysis platform (2030). 2035).

The platform interface unit 2050 may be a computer device that allows a user or experimenter to access each platform. For example, a user can access the quantum observation and analysis platform 2030 through the platform interface unit 2050 and run a desired quantum simulation.

For example, a quantum information technology integrated system according to an embodiment of the present invention performs the operations described in Figures 9 to 11 and a quantum communication platform including a counterfactual quantum communication processor that performs the first to fifth steps. It may include a quantum observation and analysis platform including a quantum state tomography processing unit.

Quantum information technology is converging with the fields of communication, algorithms, and simulation technologies, and research is ongoing to overcome the limitations of next-generation mobile communications, such as network security, wireless resource management, channel estimation, and network interference management. Accordingly, the components included in each of the quantum communication platform (2010), the quantum computing platform (2020), and the quantum observation and analysis platform (2030) may be replaced or improved according to the development and emergence of each technology.

For example, the quantum key distribution processor 2011 can be improved to provide technology for integration into the security layer of the next-generation mobile communication system.

For example, each component of the quantum communication platform (2010) can be improved for use in interference management of next-generation mobile communication systems that require intelligent network functions.

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.

300: Quantum correlation measurement device
310: input unit 320: interaction unit
330: Measuring unit

Claims (10)

An input unit that inputs a quantum state by applying a Hadamard operation based on a control qubit to the mixed quantum state;
an interaction unit that performs unitary evolution on the input quantum state based on controlled-unitary; and
Generate a normalized trace of a single matrix, which is a single operator according to the performed single evolution, and measure quantum correlation in quantum metrology using the generated normalized trace, wherein the generated normalized trace is set to “1”. When convergence occurs, when the single operator is a random Hermitian unitary operator, discord and entanglement disappear for the single operator, characterized in that it includes a measuring unit for measuring the quantum correlation.
A quantum correlation measurement device in single-qubit metrology. According to paragraph 1,
The input unit inputs the quantum state based on Equation 2 below,
In Equation 2 below, p _i ^CA represents the Hadamard-operated mixed state based on the control qubit, α determines the purity of the single control qubit system given by (1+α ² )/2, and I is Characterized in that it represents the input operator, and Z represents the Pauli operator.
A quantum correlation measurement device in single-qubit metrology.
[Equation 2]
According to paragraph 2,
The interaction unit performs a single evolution on the input quantum state as shown in Equation 3 below,
In Equation 3 below, p _f ^CA represents the quantum state in which controlled single evolution has occurred, p ^CA represents the mixed state, and α determines the purity of the single controlled qubit system given by (1+α ² )/2 , I represents the input operator, and U _n represents a single matrix.
A quantum correlation measurement device in single-qubit metrology.
[Equation 3]
According to paragraph 1,
The controlled-unitary is given by Equation 4 below,
In Equation 4 below, I represents an input operator, and U _n represents a single matrix.
A quantum correlation measurement device in single-qubit metrology.
[Equation 4]
According to clause 4,
The control qubit is given by Equation 5 below,
In Equation 5 below, p _f ^C represents the control qubit, p _f ^CA represents the quantum state in which controlled single evolution has occurred, and α is the purity of the single control qubit system given by (1+α ² )/2. Determine , tr may represent a normalized trace, and U _n may represent a single matrix.
A quantum correlation measurement device in single-qubit metrology.
[Equation 5]
delete At the input unit, inputting a quantum state by applying a Hadamard operation based on a control qubit to the mixed quantum state;
In the interaction unit, performing unitary evolution on the input quantum state based on controlled-unitary; and
In the measurement unit, a normalized trace of a single matrix, which is a single operator according to the performed single evolution, is generated, and quantum correlation in quantum measurement is measured using the generated normalized trace, where the generated normalized trace is When converging to "1", discord and entanglement disappear for the unitary operator in the case where the unitary operator is a random Hermitian unitary operator, comprising measuring the quantum correlation. characterized by
Quantum correlation measurement method in single-qubit instrumentation. In clause 7,
The step of inputting a quantum state by applying a Hadamard operation based on a control qubit to the mixed quantum state is,
Including the step of inputting the quantum state based on Equation 2 below,
In Equation 2 below, p _i ^CA represents the Hadamard-operated mixed state based on the control qubit, α determines the purity of the single control qubit system given by (1+α ² )/2, and I is represents the input operator, and Z represents the Pauli operator.
[Equation 2]

The step of performing unitary evolution on the input quantum state based on the controlled-unitary,
It includes performing a single evolution on the input quantum state as shown in Equation 3 below,
In Equation 3 below, p _f ^CA represents the quantum state in which controlled single evolution has occurred, p ^CA represents the mixed state, and α determines the purity of the single controlled qubit system given by (1+α ² )/2 , I represents the input operator, and U _n represents a single matrix.
Quantum correlation measurement method in single-qubit instrumentation.
[Equation 3]
In clause 7,
The controlled-unitary is given by Equation 4 below,
In Equation 4 below, I represents an input operator, U _n represents a single matrix,
[Equation 4]

The control qubit is given by Equation 5 below,
In Equation 5 below, p _f ^C represents the control qubit, p _f ^CA represents the quantum state in which controlled single evolution has occurred, and α is the purity of the single control qubit system given by (1+α ² )/2. Determine , tr may represent a normalized trace, and U _n may represent a single matrix.
Quantum correlation measurement method in single-qubit instrumentation.
[Equation 5]
delete