Quantum computer system and control method therefor
A classical computer system compresses data using tensor networks to expand the qubit capacity of quantum computers, facilitating complex computations.
Patent Information
- Application Number
- JP2024047006
- Authority / Receiving Office
- JP · JP
- Patent Type
- Applications
- Current Assignee / Owner
- Filing Date
- 2024-03-22
- Publication Date
- 2025-10-03
- Estimated Expiration
- 2044-03-22
AI Technical Summary
Quantum computers are limited by the number of qubits they can process, making it difficult to handle complex computations requiring more quantum bits.
A computer system comprising a classical computer and a quantum computer, where the classical computer compresses data using tensor networks and singular value decomposition to reduce the number of quantum bits, enabling the quantum computer to perform computations beyond its qubit capacity.
Enables quantum computers to process data requiring more quantum bits than their native capacity by compressing data through tensor networks, allowing for complex computations.
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Figure 2025146310000001_ABST
Abstract
Description
[Technical Field]
[0001] The present invention relates to a quantum computer system including a quantum computer and a control method thereof. [Background technology]
[0002] A quantum computer is a computer that applies the principles of quantum mechanics to calculations, while conventional binary digital computers are called classical computers.
[0003] Quantum computers are said to be capable of parallel computation on a scale that is impossible with classical computers. However, even the most recent quantum computers can process only a maximum of 100 qubits, making it difficult to process data with a large number of parameters. [Prior art documents] [Patent documents]
[0004] [Patent Document 1] Special Publication No. 2023-510706 Summary of the Invention [Problem to be solved by the invention]
[0005] Meanwhile, as a method for simulating quantum circuits on a classical computer, a method using a tensor network, as described in Patent Document 1, has recently attracted attention. However, in the case of the method using the tensor network, the amount of calculation increases significantly as the quantum computation becomes more complex.
[0006] An object of one aspect of the present invention is to enable, using a quantum computer, processing that requires more quantum bits than the number of quantum bits that can be processed by the quantum computer. [Means for solving the problem]
[0007] In order to solve the above problem, a computer system according to one embodiment of the present invention comprises a classical computer and a quantum computer, wherein the classical computer comprises an acquisition unit that acquires a plurality of first quantum bits, a compression unit that converts a tensor including the plurality of first quantum bits into a tensor network and performs information compression, thereby compressing the plurality of first quantum bits into a plurality of second quantum bits having a smaller number of quantum bits than the plurality of first quantum bits, and an output unit that outputs the plurality of second quantum bits to the quantum computer, and the quantum computer performs quantum computation on the plurality of second quantum bits.
[0008] A control method for a computer system according to another aspect of the present invention is a control method for a computer system comprising a classical computer and a quantum computer, and includes the steps of: the classical computer acquiring a plurality of first quantum bits; the classical computer compressing information by converting a tensor including the plurality of first quantum bits into a tensor network, thereby compressing the plurality of first quantum bits into a plurality of second quantum bits having a smaller number of quantum bits than the plurality of first quantum bits; the classical computer outputting the plurality of second quantum bits to the quantum computer; and the quantum computer performing quantum computation on the plurality of second quantum bits. [Effects of the Invention]
[0009] According to one aspect of the present invention, a quantum computer can be used to perform processing that requires more quantum bits than the number of quantum bits that can be processed by the quantum computer. [Brief explanation of the drawings]
[0010] [Figure 1] 1 is a block diagram showing an overview of a network system according to an embodiment of the present invention; [Figure 2] FIG. 2 is a block diagram showing a schematic configuration of an information compression device in the network system. [Figure 3]4 is a flowchart showing a flow of information compression processing in the information compression device. [Figure 4] FIG. 10 is a block diagram showing a schematic configuration of an information compression device in a network system according to another embodiment of the present invention. [Figure 5] 4 is a flowchart showing a flow of information compression processing in the information compression device. [Figure 6] FIG. 10 is a diagram illustrating an overview of a distributed network system according to yet another embodiment of the present invention. [Figure 7] FIG. 2 is a block diagram showing a schematic configuration of an edge server in the distributed network system. [Figure 8] 10 is a flowchart showing the flow of information compression processing in the edge server. [Figure 9] 1 is a block diagram showing a schematic configuration of a distributed network system according to an embodiment of the present invention; DETAILED DESCRIPTION OF THE INVENTION
[0011] Hereinafter, embodiments of the present invention will be described in detail. For the sake of convenience, the same reference numerals will be used to designate components having the same functions as those in the embodiments, and the description thereof will be omitted where appropriate.
[0012] [Embodiment 1] An embodiment of the present invention will be described with reference to FIGS.
[0013] (Network System) 1 is a block diagram showing an outline of a network system according to this embodiment. As shown in FIG. 1, the network system 1 includes a quantum computer system 2, a communication network 3, and a plurality of terminals 4.
[0014] The quantum computer system 2 includes a quantum computer 5 and an information compression device 6. The quantum computer 5 is a NISQ (Noisy Intermediate-Scale Quantum) machine that has been used recently. Currently, NISQ machines are capable of quantum calculations of approximately 100 to 1000 qubits. The quantum computer 5 is communicably connected to the information compression device 6. Details of the quantum computer 5 are publicly known, and therefore will not be described here.
[0015] The information compression device 6 is a classical computer, and is communicatively connected to a plurality of terminals 4 via a communication network 3 such as the Internet. Examples of the terminals 4 include drones, surveillance cameras, automobiles, mobile terminals, and IoT (Internet of Things) devices. The information compression device 6 collects various data from the plurality of terminals 4. Examples of the data include location data of the terminals 4, detection data from sensors in the terminals 4, video data from cameras in the terminals 4, and various parameters of the terminals 4.
[0016] The data from each terminal 4 is a vector including a plurality of values. Therefore, the plurality of data from the plurality of terminals 4 is a tensor including a plurality of vectors.
[0017] Furthermore, each of the multiple values contained in the tensor is normalized to a value between 0 and 1. This value corresponds to the state of one quantum bit. The state of one quantum bit can be expressed as a superposition of two defined vectors |0> and |1>. Therefore, the multiple values contained in the tensor correspond to multiple quantum bits (first quantum bits).
[0018] The data compression device 6 converts the tensor into a tensor network, which is a network of tensors with a dimension smaller than that of the tensor. Specifically, the data compression device 6 converts the tensor into a tensor network of a matrix product state or a tensor product state using singular value decomposition or Schmidt decomposition. The data compression device 6 also performs low-rank approximation on the tensor network using singular value decomposition or Schmidt decomposition.
[0019] The information compression device 6 then performs a reduction calculation on the low-rank approximated tensor network. This reduction calculation may be performed sequentially starting from certain data, or may be performed for each piece of data from the terminal 4. If there are multiple tensor networks, the reduction calculation may be performed for each of the multiple tensor networks, and then the multiple tensor networks may be combined into a single tensor. Alternatively, a tensor network may be created for each type of terminal 4, and the reduction calculation may be performed for each of the multiple tensor networks created, and then the multiple tensor networks may be combined into a single tensor.
[0020] This allows the plurality of pieces of data to be compressed into a smaller number of pieces of data than the plurality of pieces of data. Note that the information compression device 6 can perform any processing that allows information compression, such as mean field approximation, in addition to low-rank approximation.
[0021] The multiple values included in the information-compressed data correspond to multiple quantum bits (second quantum bits). The number of second quantum bits becomes smaller than the number of first quantum bits due to the information compression. Information compression device 6 outputs the multiple second quantum bits included in the information-compressed data to quantum computer 5.
[0022] For example, suppose the number of the second quantum bits is 1 / 10,000 of the number of the first quantum bits. In this case, if the number of the first quantum bits is 1,000,000, the information compression device 6 compresses the information into second quantum bits, the number of which is 100. This enables the quantum computer 5 to perform quantum calculations on the second quantum bits. As a result, the network system 1 configured as described above becomes capable of processing that requires a larger number of first quantum bits (1,000,000 in the above example) than the number of second quantum bits that can be processed by the quantum computer 5 (100 in the above example).
[0023] (information compression device) 2 is a block diagram showing a schematic configuration of the information compression device 6. As shown in FIG. 2, the information compression device 6 includes a control unit 10, a communication unit 11, and an output unit 12.
[0024] The control unit 10 comprehensively controls the operations of various components of the information compression device 6, and is configured by, for example, a computer including a GPU (Graphics Processing Unit) and memory. The operations of the various components are controlled by causing the computer to execute a control program. The control unit 10 will be described in detail later.
[0025] The communication unit 11 transmits and receives information to and from an external communication device. The communication unit 11 includes a communication device such as a transmission / reception circuit. The output unit 12 outputs the quantum bits generated by the control unit 10 to the quantum computer 5.
[0026] In this embodiment, the control unit 10 is configured to include an acquisition unit 20, a normalization unit 21, a tensor network conversion unit 22 (compression unit), and an information compression unit 23 (compression unit). The acquisition unit 20 acquires tensors, which are multiple pieces of data, from multiple terminals 4 via the communication network 3 and the communication unit 11. The acquisition unit 20 sends the acquired tensors to the normalization unit 21.
[0027] The normalization unit 21 normalizes the multiple values included in the tensor from the acquisition unit 20 to values between 0 and 1 to generate multiple first quantum bits. The normalization unit 21 sends the tensor including the multiple first quantum bits to the tensor network conversion unit 22.
[0028] The tensor network conversion unit 22 converts the tensor from the normalization unit 21 into the above-mentioned tensor network. The tensor network conversion unit 22 sends the converted tensor network to the information compression unit 23.
[0029] The information compression unit 23 performs a contraction calculation by performing low-rank approximation using singular value decomposition on the tensor network from the tensor network conversion unit 22. This compresses the information of the multiple first quantum bits included in the tensor into multiple second quantum bits, which have a smaller number of quantum bits than the first quantum bits. The information compression unit 23 outputs the compressed information of the multiple second quantum bits to the quantum computer 5 via the output unit 12.
[0030] (information compression processing) FIG. 3 is a flowchart showing the flow of information compression processing in the information compression device 6 of this embodiment. As shown in FIG. 3, first, the acquisition unit 20 acquires tensors, which are multiple pieces of data, from multiple terminals 4 (S10). Next, the normalization unit 21 normalizes multiple values included in the tensor to values between 0 and 1 to generate multiple first quantum bits (S11). Next, the tensor network conversion unit 22 converts the tensor including multiple first quantum bits into a tensor network (S12). Next, the information compression unit 23 performs a contraction calculation by performing low-rank approximation using singular value decomposition on the tensor network, thereby compressing the multiple first quantum bits included in the tensor into multiple second quantum bits having a smaller number of quantum bits than the first quantum bits (S13). Then, the information compression unit 23 outputs the compressed multiple second quantum bits to the quantum computer 5 via the output unit 12 (S14). Thereafter, the information compression processing ends.
[0031] [Embodiment 2] Another embodiment of the present invention will be described with reference to Figures 4 and 5. The network system 1 according to this embodiment is different from the network system 1 shown in Figures 1 to 3 in the configuration of the information compression device 6, but the other configurations are the same.
[0032] Fig. 4 is a block diagram showing a schematic configuration of the information compression device 6 of this embodiment. The information compression device 6 shown in Fig. 4 differs from the information compression device 6 shown in Fig. 2 in that an information compression unit 24 (compression unit) is provided instead of the information compression unit 23 and in that a quantum computing unit 25 is added, but the other configurations are the same.
[0033] In this embodiment, information compression unit 24 compresses the plurality of first quantum bits included in the tensor from normalization unit 21 into a plurality of second quantum bits and a plurality of third quantum bits, each of which has a smaller number of quantum bits than the first quantum bits. Information compression unit 23 outputs the plurality of second quantum bits to quantum computer 5 via output unit 12, and sends the plurality of third quantum bits to quantum computing unit 25.
[0034] A quantum circuit in quantum computer 5 can be represented by a tensor network, and quantum computation of the quantum circuit corresponds to contracting the tensor network. Therefore, quantum computation unit 25 includes a tensor network corresponding to a certain quantum circuit, inputs the third quantum bit from information compression unit 24 to the tensor network, and contracts the tensor network to execute quantum computation of the quantum circuit.
[0035] This allows the number of qubits used in quantum computation to be increased to the sum of the number of second qubits and the number of third qubits. Furthermore, by having the quantum computer 5 perform complex quantum computations and the data compression device 6 perform simple quantum computations, it is possible to suppress an increase in the amount of computation required by the data compression device 6, which is a classical computer.
[0036] Fig. 5 is a flowchart showing the flow of information compression processing in the information compression device 6 of this embodiment. The information compression processing shown in Fig. 5 differs from the information compression processing shown in Fig. 3 in that it includes step S20 instead of step S13 and in that step S21 is added after step S14, but the other steps are the same.
[0037] In step S20, information compression unit 24 compresses the plurality of first qubits included in the tensor from normalization unit 21 into a plurality of second qubits and a plurality of third qubits, which have fewer qubits than the first qubits.
[0038] In step S21, quantum computing unit 25 inputs the third quantum bit to a tensor network corresponding to a quantum circuit and contracts the tensor network, thereby performing quantum computation for the quantum circuit. Then, the information compression process ends. Note that steps S14 and S21 may be performed either first or simultaneously.
[0039] [Embodiment 3] Still another embodiment of the present invention will be described with reference to FIGS.
[0040] Fig. 6 is a diagram showing an overview of a distributed network system 1a according to this embodiment. The distributed network system 1a shown in Fig. 6 differs from the network system 1 shown in Figs. 1 to 3 in that an edge server 7 and a base station 8 are provided between the communication network 3 and the terminal 4, but the other configurations are the same.
[0041] The information compression device 6 is communicably connected to a plurality of edge servers 7 via the communication network 3. Therefore, the information compression device 6 of this embodiment acquires tensors, which are a plurality of pieces of data, from the plurality of edge servers 7.
[0042] Each of the edge servers 7 is connected to a plurality of base stations 8 via wire or wirelessly (radio frequency) so as to be able to communicate with them. Each of the base stations 8 is connected to a plurality of terminals 4 via wireless so as to be able to communicate with them. Note that each of the edge servers 7 may also be connected to a terminal 4 via wire so as to be able to communicate with them.
[0043] The edge server 7 is a classical computer that collects various data from the terminal 4. Data processing that was previously performed by the cloud is now performed by multiple edge servers 7 that are closer to the terminal 4 than the cloud. This allows the edge server 7 to achieve low-latency, low-load data processing.
[0044] In this embodiment, similar to the information compression device 6, the edge server 7 acquires tensors, which are data from multiple terminals 4, and normalizes multiple values included in the acquired tensors to values between 0 and 1 to generate multiple quantum bits (fourth quantum bits). Next, the edge server 7 converts the tensor including the multiple fourth quantum bits into a tensor network, and performs low-rank approximation using singular value decomposition on the tensor network to perform contraction calculations. This allows the multiple pieces of data to be compressed into data that is fewer in number than the multiple pieces of data.
[0045] The multiple values included in the information-compressed data correspond to multiple quantum bits (fifth quantum bits). The number of fifth quantum bits becomes smaller than the number of fourth quantum bits due to the information compression. The edge server 7 transmits the multiple fifth quantum bits included in the information-compressed data to the information compression device via the communication network 3.
[0046] For example, suppose there are 100 edge servers 7, and the number of the fifth qubits is 1 / 100 of the number of the fourth qubits. In this case, if the number of the fourth qubits is 1 million, each edge server 7 compresses the information into 10,000 fifth qubits, which is 10,000 qubits, and transmits the information to the information compression device 6. The information compression device 6 receives 10,000 x 100 fifth qubits, i.e., 1 million first qubits, from the 100 edge servers 7 and compresses the information into 100 second qubits. This allows the quantum computer 5 to perform quantum calculations on the second qubits. In other words, the distributed network system 1a configured as described above is capable of processing requiring 1 million x 100 = 100 million qubits (fourth qubits).
[0047] Furthermore, because the data transmitted by the edge server 7 is a plurality of compressed fifth qubits, it is difficult to infer the actual data, resulting in improved security. By distributing the edge servers 7 throughout the country, even if the cloud becomes inoperable due to a disaster or other reason, the edge servers 7 can continue to operate, thereby maintaining the network between the edge servers 7 and the terminals 4 communicatively connected to the edge servers 7. As a result, fault tolerance is improved. Furthermore, the amount of long-distance communication between the edge servers 7 and the cloud can be reduced, thereby reducing communication costs.
[0048] Fig. 7 is a block diagram showing a schematic configuration of the edge server 7 of this embodiment. As shown in Fig. 7, the edge server 7 is configured to include a control unit 30 and a communication unit 31 (transmission unit). Note that the control unit 30 and the communication unit 31 shown in Fig. 7 have the same hardware configuration as the control unit 10 and the communication unit 11 shown in Fig. 2, and therefore a description thereof will be omitted.
[0049] In this embodiment, the control unit 30 is configured to include an acquisition unit 40, a normalization unit 41, a tensor network conversion unit 42 (compression unit), and an information compression unit 43 (compression unit). The acquisition unit 40 acquires tensors, which are multiple pieces of data, from multiple terminals 4 via the base station 8 and the communication unit 31. The acquisition unit 40 sends the acquired tensors to the normalization unit 41.
[0050] Normalization unit 41 and tensor network conversion unit 42 are similar to normalization unit 21 and tensor network conversion unit 22 shown in Fig. 2. Information compression unit 43 differs from information compression unit 23 shown in Fig. 2 in that it transmits compressed data (plurality of fifth quantum bits) to information compression device 6 via communication unit 31 and communication network 3, but is otherwise similar. As a result, acquisition unit 20 of information compression device 6 acquires tensors, which are multiple pieces of data from multiple edge servers 7. Furthermore, since the multiple values included in this tensor have already been normalized, they are treated as multiple first quantum bits, and normalization unit 21 is omitted.
[0051] FIG. 8 is a flowchart showing the flow of information compression processing in the edge server 7 of this embodiment. As shown in FIG. 8, first, the acquisition unit 40 acquires tensors, which are multiple pieces of data, from multiple terminals 4 (S30). Next, the normalization unit 41 normalizes multiple values included in the tensor to values between 0 and 1 to generate multiple fourth quantum bits (S31). Next, the tensor network conversion unit 42 converts the tensor including multiple fourth quantum bits into a tensor network (S32). Next, the information compression unit 43 performs a contraction calculation by performing low-rank approximation using singular value decomposition on the tensor network, thereby compressing the multiple fourth quantum bits included in the tensor into multiple fifth quantum bits, which have a smaller number of quantum bits than the fourth quantum bits (S33). Then, the information compression unit 23 transmits the compressed multiple fifth quantum bits to the information compression device 6 via the communication unit 31 and the communication network 3 (S34). Thereafter, the information compression processing ends.
[0052] (Additional notes) The edge server 7 may perform quantum computation using a tensor network, similar to the information compression device 6 shown in Fig. 4. In this case, the number of quantum bits for performing quantum computation can be further increased.
[0053] (Example) An embodiment of the distributed network system 1a shown in Figures 6 to 8 will be described with reference to Figure 9. This embodiment is a system for identifying abnormalities in financial transactions.
[0054] Fig. 9 is a block diagram showing a schematic configuration of a distributed network system 1a of this embodiment. As shown in Fig. 9, an information compression device 6 is communicatively connected to edge servers 7 of financial institutions A, B, and C via a communication network 3. The edge servers 7 of each company are communicatively connected to terminals 4, such as ATMs (Automated Teller Machines) and smartphones, via a communication network 9.
[0055] Financial transaction data is accumulated in the edge server 7 of each company. Therefore, the edge server 7 of each company can identify anomalies in financial transactions within its own company by performing machine learning using the financial transaction data. However, it is difficult to identify anomalies in financial transactions between different companies.
[0056] Therefore, in this embodiment, each company's edge server 7 compresses the accumulated financial transaction data (fourth quantum bit) and transmits it to the information compression device 6 via the communication network 3. The information compression device 6 further compresses the data (plurality of first quantum bits) from each company's edge server 7 and outputs the compressed plurality of second quantum bits to the quantum computer 5. The quantum computer 5 performs machine learning using the second quantum bits to generate a learning model that identifies anomalies in financial transactions. Note that the quantum computer 5 may perform machine learning together with the information compression device 6, which is a classical computer.
[0057] The learning model generated by the quantum computer 5 is transmitted from the information compression device 6 to each company's edge server 7 via the communication network 3. By using the learning model, each company's edge server 7 can identify anomalies not only in financial transactions within its own company, but also in financial transactions between different companies.
[0058] [Software implementation example] The functions of the information compression device 6 and the edge server 7 (hereinafter referred to as the "devices") can be realized by a program that causes a computer to function as the device, and a program that causes a computer to function as each control block of the device (particularly each part included in the control units 10 and 30).
[0059] In this case, the device includes a computer having at least one control device (e.g., a processor) and at least one storage device (e.g., a memory) as hardware for executing the program. The control device and storage device execute the program, thereby realizing the functions described in each of the above embodiments.
[0060] The program may be non-transitory and may be recorded on one or more computer-readable recording media. The recording media may or may not be included in the device. In the latter case, the program may be supplied to the device via any wired or wireless transmission medium.
[0061] Furthermore, some or all of the functions of the control blocks can be realized by logic circuits. For example, an integrated circuit in which a logic circuit that functions as each of the control blocks is formed is also included in the scope of the present invention. In addition, the functions of the control blocks can also be realized by, for example, a quantum computer.
[0062] Furthermore, each process described in each of the above embodiments may be executed by AI (Artificial Intelligence). In this case, the AI may run on the control device or on another device (for example, an edge computer or a cloud server).
[0063] The present invention is not limited to the above-described embodiments, and various modifications are possible within the scope of the claims. Embodiments obtained by appropriately combining the technical means disclosed in different embodiments are also included in the technical scope of the present invention. [Explanation of symbols]
[0064] 1 Network System 1a Distributed Network System 2. Quantum Computer System 3, 9 Communication Network 4. Terminal 5. Quantum Computers 6. Information Compression Device (Classical Computer) 7 Edge Server 8 base station 10, 30 Control unit 11 Communications Department 12 Output section 20, 40 Acquisition Department 21, 41 Normalization part 22, 42 Tensor network conversion part (compression part) 23, 24, 43 Information compression section (compression section) 25 Quantum calculation section 30 Control Unit 31 Communication unit (transmission unit)
Claims
1. Equipped with classical and quantum computers, The classical computer is an acquisition unit that acquires a plurality of first quantum bits; A compression unit that compresses information of the first quantum bits into a plurality of second quantum bits having a smaller number of quantum bits than the first quantum bits by converting a tensor including the first quantum bits into a tensor network and performing information compression; an output unit that outputs the plurality of second quantum bits to the quantum computer; The quantum computer performs quantum computation on the plurality of second quantum bits. Quantum computer system.
2. 2. The quantum computer system according to claim 1, wherein the compression unit of the classical computer converts the tensor into a tensor network using singular value decomposition or Schmidt decomposition, and compresses the information by performing a contraction calculation on the converted tensor network by performing low-rank approximation using singular value decomposition or Schmidt decomposition.
3. the compression unit of the classical computer compresses information of the plurality of first quantum bits into a plurality of second quantum bits and a plurality of third quantum bits, the number of which is smaller than the number of first quantum bits; The quantum computer system according to claim 1 , wherein the classical computer performs quantum computation using a tensor network on the plurality of third quantum bits.
4. Further comprising a plurality of edge computers communicatively connected to the classical computer, Each of the plurality of edge computers is communicatively connected to a plurality of terminals, an acquisition unit that acquires tensors, which are multiple pieces of data, from multiple terminals; a normalization unit that normalizes a value included in the tensor and converts it into a tensor including a plurality of fourth quantum bits; A compression unit that compresses the information of the plurality of fourth quantum bits into a plurality of fifth quantum bits having a smaller number of quantum bits than the plurality of fourth quantum bits by converting a tensor including the plurality of fourth quantum bits into a tensor network and performing information compression; a transmitter that transmits the plurality of fifth qubits to the classical computer; 2. The quantum computer system according to claim 1, wherein the acquisition unit of the classical computer acquires the plurality of fifth qubits from each of the plurality of edge computers as the first qubit.
5. The quantum computer comprises:
5. The quantum computer system according to claim 4, wherein machine learning is performed using the plurality of second quantum bits, and the machine-learned learning model is transmitted to at least one of the plurality of edge computers via the classical computer.
6. 2. A control program for causing a computer to function as the quantum computer system according to claim 1, the control program causing the computer to function as the compression unit.
7. A control method for a quantum computer system including a classical computer and a quantum computer, obtaining a plurality of first qubits by the classical computer; The classical computer converts a tensor including the plurality of first quantum bits into a tensor network and performs information compression, thereby compressing the plurality of first quantum bits into a plurality of second quantum bits having a smaller number of quantum bits than the plurality of first quantum bits; the classical computer outputting the second plurality of qubits to the quantum computer; and a step of causing the quantum computer to perform quantum computation on the plurality of second quantum bits.
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