Method for introducing magic state in quantum computer and quantum computer therefor

Abstract

The present embodiments provide a method for introducing a magic state in a quantum computer and a quantum computer therefor, the method comprising the steps of: constructing a logical qubit by using an error correction code of a heavy hexagonal structure that connects two data qubits and one syndrome qubit from both sides by introducing two or more flag qubits between four data qubits and one syndrome qubit; and performing a logical operation on the logical qubit, wherein the error correction code performs a magic state injection process that maps a magic state implemented in physical qubits to a logical state.

Description

Method for introducing a magic state in a quantum computer and the quantum computer

[0001] The present embodiments relate to a method for introducing a magic state of a quantum computer and the quantum computer.

[0002] Currently, various types of quantum computers are being manufactured, and many quantum computations are being performed using these devices. However, so far, 10 -3 No quantum computer has achieved a 2-qubit gate error rate of less than 10. To execute quantum algorithms such as Shor's algorithm, which are expected to outperform classical computers, a gate error rate of approximately 10 -7 ~10 -12 It must be reduced to.

[0003] It is nearly impossible to achieve such a low error rate by improving hardware alone. Therefore, long and complex algorithms must be executed by constructing logical qubits using quantum error correction and performing logical operations on these qubits.

[0004] For fault-tolerant quantum computing, a quantum error correction code with a high threshold error rate that can be implemented in a quantum computer is first required. Once this condition is met, errors can be detected and corrected by changing the logical state to maintain the intended logical state. However, to perform quantum computing, logical operators corresponding to arbitrary unit operations that transform a prepared logical state into another desired logical state are also required. Since it is impossible to find the physical operations for all unit operators required during computation, arbitrary unit gates must be implemented by combining specific gates whose corresponding physical operations are known. A set of gates capable of expressing arbitrary unit operations is called a universal gate set. While a universal gate set can be expressed through various gate combinations, every universal gate set must include at least one non-Clifford gate. It is well known that logical Clifford gates can be implemented directly in data qubits using transversal gates, but it is also known that logical non-Clifford gates cannot be expressed using transversal gates. Therefore, a different method is required to implement logical non-Clifford gates.

[0005] The present embodiments provide a method for introducing a magic state of a quantum computer that satisfies a quantum error correction (QEC) code with a high threshold error rate by introducing a magic state, and the quantum computer.

[0006] In one aspect, the embodiments may provide a method for introducing a magic state in a quantum computer, comprising the steps of configuring a logical qubit using a heavy hexagonal structure error correction code that connects two data qubits and one syndrome qubit on both sides by introducing two or more flag qubits between four data qubits and one syndrome qubit, and performing a logical operation on the logical qubit, wherein the error correction code performs a magic state injection process that maps a magic state implemented in a physical qubit to a logical state.

[0007] In another aspect, the embodiments may provide a quantum computer comprising a quantum processor that performs quantum processing, a control device that controls the quantum processor, and a measuring device that measures the result of the quantum processor performing quantum processing, wherein the quantum processor includes a logical qubit using a heavy hexagonal structure error correction code that connects two data qubits and one syndrome qubit on both sides by introducing two or more flag qubits between four data qubits and one syndrome qubit, performs logical operations on the logical qubit, and performs a magic state injection process in which the error correction code maps a magic state implemented in a physical qubit to a logical state.

[0008] According to the method for introducing a magic state of a quantum computer and the quantum computer according to the embodiments of the present invention, a quantum error correction (QEC) code of a threshold error rate can be satisfied by introducing a magic state.

[0009] FIG. 1 is a flowchart of a method for introducing a magic state of a quantum computer according to one embodiment.

[0010] FIG. 2 is a block diagram of a quantum computer according to another embodiment.

[0011] Figure 3 illustrates a logical quantum circuit for implementing a logical T gate. In Figure 3, the state |m > L = |0 > L+e iπ / 4When |1 > L is ready, this circuit allows applying a logical T gate to any state using only logical Clifford gates. Since the magic state collapses due to measurements during the process, a new magic state is required for each logical T gate application.

[0012] Figures 4 and 5 illustrate the surface code of a heavy hexagonal structure and the surface code of a lattice structure, respectively. In Figures 4 and 5, data qubits are shown in gray, X syndrome qubits in red, Z syndrome qubits in blue, and flag qubits in green. The areas shaded in red and blue correspond to stabilization operators. The flag qubit enables stabilization measurements by indirectly connecting data qubits and syndrome qubits that cannot be directly connected to a 2-qubit gate in the heavy hexagonal structure. Except for the green flag qubit, the surface code of the heavy hexagonal structure is identical to the surface code of the lattice structure.

[0013] Figures 6 and 7 illustrate an XZZX code with a heavy hexagonal structure. In Figures 6 and 7, the XZZX code detects and corrects logical state errors through a stabilizer composed of Pauli X and Pauli Z operators. There are two ways to arrange the Pauli X and Pauli Z in the stabilizer. In Figure 6, the XZZX type arranges the X operator vertically and the Z operator horizontally relative to the syndrome qubit. In Figure 7, the ZXXZ type arranges the Z operator vertically and the X operator horizontally. In these two cases, the shapes of the logical X and Z operators are reversed. In Figures 6 and 7, the logical X operator is highlighted in red and the logical Z operator is highlighted in blue. Generally, this structural difference does not affect the performance of the error correction code in the grid structure. However, in a heavy hexagonal structure using flag qubits, the number of flag qubits required to connect the vertical and horizontal directions differs, which can lead to differences in error correction performance depending on the stabilizer configuration.

[0014] Figure 8 illustrates a type of ballast measurement. In Figure 8, (a) the data qubit is prepared in the eigenstate of the ballast. Error detection is possible without changing the state. (b) The data qubit is not in the eigenstate of the ballast. The quantum state collapses into the eigenstate during the measurement process. Errors occurring during this process are not detected in the syndrome qubit because the state collapses during measurement.

[0015] Figure 9 illustrates the error propagation circuit of an X stabilizer in a heavy hexagonal structure surface code. In Figure 9, (a) errors occurring in the data qubit are propagated to the syndrome qubit through the flag qubit. Since all errors propagated through the flag qubit are Z errors, they do not affect the flag qubit measurement process, and the errors are detected only in the syndrome qubit measurement. (b) illustrates the error propagation circuit of an XZZX type stabilizer in an XZZX code of a heavy hexagonal structure. When an X error occurs in the flag qubit, it is propagated to the data qubit as an X or Z error depending on the stabilizer through the CNOT gate. These errors are not detected in the syndrome qubit measurement but can be identified through the flag qubit measurement. The additional error in the flag qubit is an important feature of the heavy hexagonal structure.

[0016] Figure 10 illustrates an example of additional errors caused by an X error in a flag qubit. In Figure 10, additional X or Z errors occur depending on the position of the data qubit in the surface code. Different stabilizer types in the XZZX code result in more X errors in the XZZX type and more Z errors in the ZXXZ type for all data qubits. These additional errors introduce a bias beyond what is provided by the error model.

[0017] Figure 11 illustrates four types of qubit initialization. Figure 11 shows (a) the right triangle method, (b) the bottom triangle method, (c) the right square method, and (d) the bottom square method. "Bottom" indicates that regions I and III (blue regions) have a larger number of data qubits, and "Right" indicates that regions II and IV (red regions) have a larger number of data qubits. The data qubits in each region are initialized to the same state, namely |0 > or |+ >. Blind qubits are shown in purple, and syndrome qubits capable of detecting errors during the initialization process are shown in red and blue. It can be seen that the bottom and right methods are symmetric with respect to the diagonal extending from the top left to the bottom right. The square method divides the initialization region using a straight line from top to bottom or from left to right, whereas the triangle method divides the region using a diagonal from the top left to the bottom right. Syndrome qubits capable of detecting errors are shown in dark red or blue. Regardless of the method used, there is one data qubit (blind qubit, purple) for which errors cannot be detected during the initialization process. The location and initialization state of the blind qubit affect the logical error rate. Since the location of the blind qubit is the same in the bottom square and right triangle method and the right square and bottom triangle method, these pairs of initialization methods exhibit similar trends in logical error rates.

[0018] Figure 12 illustrates a comparison of logical error rates for heavy hexagonal and grid structures with biases of 0.5 and 100 (without distance expansion) (top) and a comparison of logical error rates for heavy hexagonal and grid structures with biases of 0.5 and 100 (without distance expansion) (bottom). In Figure 12, the logical error rate in the grid structure is nearly identical regardless of the initialization method, without bias or distance expansion. On the other hand, for the heavy hexagonal structure, the logical error rate depends on the location of the qubit where the error cannot be detected. As the bias increases, the XZZX code has a lower logical error rate compared to the surface code. While the two types of XZZX codes exhibit the same logical error rate for equivalent initialization in the grid structure, the ZXXZ type has a lower error rate in the heavy hexagonal structure. Additionally, triangle-based initialization methods tend to have lower error rates overall when distance expanded.

[0019] Figure 13 is a graph showing how the logical error rate changes according to various biases. In Figure 13, the left graph shows the change in the logical error rate according to various physical error rates for surface codes (XZZX and ZXXZ types) at various biases. The right graph shows the change in the logical error rate according to various biases for each error correction code at specific physical error rates from the left graph. For both the heavy hexagonal structure and the lattice structure, the total logical error rate and the logical X error rate decrease as the bias increases. However, the logical Z error decreases in the heavy hexagonal structure but increases in the lattice structure. This may be because the X error in the flag qubit decreases as the bias increases.

[0020] Figure 14 illustrates the logical error rate when extending a logical state prepared with a distance-3 error correction code to a distance-9 code. Figure 14 shows the results for Down triangle initialization and ZXXZ error correction code. In a heavy hexagonal structure, the error rate becomes nearly the same as the physical error rate decreases. In a grid structure, the logical error rate decreases as the physical error rate increases in the low error region, but increases in the high error region.

[0021] Hereinafter, some embodiments of the present disclosure will be described in detail with reference to the exemplary drawings. In assigning reference numerals to the components of each drawing, the same components may have the same reference numeral as much as possible, even if they are shown in different drawings. Furthermore, in describing the embodiments, if it is determined that a detailed description of related known components or functions may obscure the essence of the technical concept, such detailed description may be omitted. Where terms such as "comprising," "having," or "consisting of" are used in this specification, other parts may be added unless "only" is used. Where a component is expressed in the singular, it may include a plural unless otherwise specified.

[0022] Additionally, terms such as first, second, A, B, (a), (b), etc., may be used to describe the components of the present disclosure. These terms are used merely to distinguish the components from other components, and the nature, order, sequence, or number of the components are not limited by such terms.

[0023] In describing the positional relationship of components, where it is stated that two or more components are "connected," "combined," or "joined," it should be understood that while the two or more components may be directly "connected," "combined," or "joined," they may also be "connected," "combined," or "joined" with other components "intervened." Here, the other components may be included in one or more of the two or more components that are "connected," "combined," or "joined" with one another.

[0024] In describing the temporal flow relationship regarding components, methods of operation, or methods of production, for example, when the temporal or sequential relationship is described using "after," "following," "next," or "before," it may include cases where the relationship is not continuous unless "immediately" or "directly" is used.

[0025] Meanwhile, where numerical values ​​or corresponding information regarding a component (e.g., levels, etc.) are mentioned, even without separate explicit notation, the numerical values ​​or corresponding information may be interpreted as including a range of error that may occur due to various factors (e.g., process factors, internal or external shocks, noise, etc.).

[0026] The present embodiment will be described below with reference to the drawings.

[0027] FIG. 1 is a flowchart of a method for introducing a magic state of a quantum computer according to one embodiment.

[0028] Referring to FIG. 1, a method (10) for introducing a magic state of a quantum computer according to one embodiment includes the step (S12) of configuring a logical qubit using a heavy hexagonal structure error correction code (Quantum Error correction (QEC)) that connects two data qubits and one syndrome qubit by introducing two or more flag qubits between four data qubits and one syndrome qubit, and the step (S14) of performing a logical operation on the logical qubit.

[0029] Error Correction Code (QEC) performs a magic state injection process that maps the magic state implemented in the physical qubit to a logical state.

[0030] The syndrome qubit may be an X syndrome qubit or a Z syndrome qubit as described below with reference to FIGS. 4 and FIGS. 5.

[0031] The heavy hexagonal structure may have an XZZX code containing both an X operator and a Z operator, as described below with reference to FIGS. 6 and FIGS. 7 through 10. The XZZX code may include an XZZX type in which the X operator is vertically positioned and the Z operator is horizontally positioned for the syndrome qubit, or a ZXXZ type in which the Z operator is vertically positioned and the X operator is horizontally positioned.

[0032] Error Correction Code (QEC) can initialize the remaining data qubits, excluding the physical qubit in which the magic state is implemented and the data qubits in the same row and column as that physical qubit.

[0033] In each round, the above flag qubit can be measured to detect errors and reset the state.

[0034] Error Correction Code (QEC) can initialize data qubits according to a triangle method, which divides the initialization area using a diagonal line from the top left to the bottom right to distinguish the area where data qubits are initialized to |0〉 and |+〉, and a square method, which divides the area using a straight line from top to bottom or from left to right, as described below with reference to FIG. 11.

[0035] The triangle method is one of the right triangle method and the bottom triangle method, and the square method can be one of the right square method and the bottom square method.

[0036] FIG. 2 is a block diagram of a quantum computer according to another embodiment.

[0037] Referring to FIG. 2, a quantum computer (100) according to another embodiment includes a control device (110) that controls a quantum processor (120), a quantum processor (120) that performs quantum processing, and a measuring device (130).

[0038] The control unit (110) converts a quantum circuit diagram or a quantum algorithm into an input pulse train and provides it to the quantum processor (120). Here, "quantum circuit diagram" or "quantum algorithm" does not represent physical components like actual electronic circuits, but rather represents an abstract design of quantum operations to be executed by the quantum processor (120). In other words, the quantum circuit diagram is a visual representation of a series of operators (gates) that process and transform quantum states. Based on this, commands can be transmitted to the quantum computer (100).

[0039] The control device (110) provides a control signal to control the quantum processor (120) as an input pulse train, thereby enabling the quantum processor (120) to perform quantum processing.

[0040] The quantum processor (120) performs operations in units of qubits rather than bits. A qubit can have a state in which 0 and 1 are simultaneously superpositioned, and if there are M qubits, 2^M states can be represented simultaneously.

[0041] A quantum processor (120) can use various types of quantum gates (e.g., Pauli / Rotation / Hadamard / CNOT / SWAP / Toffoli) that receive one or more qubits to perform quantum operations and perform specified operations, and can combine quantum gates to form a quantum circuit that performs special functions.

[0042] A quantum processor (120) includes two or more qubits and performs quantum processing according to the quantum phenomena of the qubits according to the input pulse train of the control device (110). A measuring device (130) measures the state of the qubits of the quantum processor (120). The qubits of the quantum processor (120) have their quantum states collapsed by the measurement of the measuring device (130) and are determined to be in a single state. The measuring device (130) can verify the determined state of the qubit.

[0043] A quantum processor (120) includes a logical qubit using a heavy hexagonal structure error correction code that connects two data qubits and one syndrome qubit by introducing two or more flag qubits between four data qubits and one syndrome qubit, and performs logical operations on the logical qubit.

[0044] Error Correction Code (QEC) performs a magic state injection process that maps the magic state implemented in the physical qubit to a logical state.

[0045] The syndrome qubit may be an X syndrome qubit or a Z syndrome qubit as described below with reference to FIGS. 4 and FIGS. 5.

[0046] The heavy hexagonal structure may have an XZZX code containing both an X operator and a Z operator, as described below with reference to FIGS. 6 and FIGS. 7 through 10. The XZZX code may include an XZZX type in which the X operator is vertically positioned and the Z operator is horizontally positioned for the syndrome qubit, or a ZXXZ type in which the Z operator is vertically positioned and the X operator is horizontally positioned.

[0047] Error Correction Code (QEC) can initialize the remaining data qubits, excluding the physical qubit in which the magic state is implemented and the data qubits in the same row and column as that physical qubit.

[0048] In each round, the above flag qubit can be measured to detect errors and reset the state.

[0049] Error Correction Code (QEC) can initialize data qubits according to a triangle method, which divides the initialization area using a diagonal line from the top left to the bottom right to distinguish the area where data qubits are initialized to |0〉 and |+〉, and a square method, which divides the area using a straight line from top to bottom or from left to right, as described below with reference to FIG. 11.

[0050] The triangle method is one of the right triangle method and the bottom triangle method, and the square method can be one of the right square method and the bottom square method.

[0051] With reference to FIGS. 1 and 2, a method for introducing a magic state in a quantum computer according to embodiments and the quantum computer said have been described above. Below, a method for introducing a magic state in a quantum computer and experimental examples thereof will be described in detail with reference to FIGS. 3 to 14.

[0052] The magic state injection process is essential for building fault-tolerant quantum computing. Most existing research has focused on square lattice structures where each qubit can be directly connected to four other qubits via a two-qubit gate. However, hardware that does not adhere to lattice structures, such as IBM's heavy hexagonal structure, is also being developed. For such hardware, many Quantum Error Correction (QEC) codes designed for lattice structures cannot be applied directly; therefore, QEC codes must be corrected using additional qubits, such as flag qubits. However, using flag qubits alters several properties of the QEC code, introducing various variables into the magic state injection process.

[0053] As previously mentioned, for fault-tolerant quantum computing, a Quantum Error Correction (QEC) code with a high threshold error rate that can be implemented in a quantum computer is first required. Once this condition is met, errors can be detected and corrected by changing the logical state to maintain the intended logical state. However, to perform quantum computing, logical operators corresponding to arbitrary unit operations that transform a prepared logical state into another desired logical state are also required. Since it is impossible to find physical operations for all unit operators required during computation, arbitrary unit gates must be implemented by combining specific gates that know the corresponding physical operations. A set of gates capable of expressing arbitrary unit operations is called a set of general-purpose gates. While a set of general-purpose gates can be expressed through various gate combinations, every set of general-purpose gates must include at least one non-Clifford gate. It is well known that logical Clifford gates can be directly implemented in data qubits using transversal gates, but it is also known that logical non-Clifford gates cannot be expressed using transversal gates. Therefore, a different method is required to implement logical non-Clifford gates. For this purpose, the present embodiment proposes a method using a magic state. Magic states are special quantum states that can indirectly implement logical non-Clifford gates and can only be created using gates that include non-Clifford gates.

[0054] Depending on the non-Clifford gate to be implemented, there is a corresponding magic state. If a logical qubit is ready in a magic state, applying only the Clifford gate between this magic state qubit and the logical qubit to be subjected to the non-Clifford gate will yield the same result as applying the non-Clifford gate directly to the target logical qubit.

[0055] For example, to implement the logical T gate, a typical non-Clifford gate, through magic state injection, the magic state |m > L = |0 > L + e iπ / 4 |1 〉 L It must be prepared. For the circuit shown in Fig. 3, the state |ψ > to which the logical qubit prepared in this magic state and the logical T gate are to be applied must be prepared. L If applied to other logical qubits of TL|ψ 〉 L A state corresponding to can be obtained. Here, T L represents a logical T gate. To implement a logical non-Clifford gate in this way, a logical magic state must be prepared. Additionally, since one magic state is required for each operation, an efficient method for generating magic states is necessary. However, logical states that can be easily obtained using only logical Clifford gates (e.g., the logical 0 state (|0 > ) L It is impossible to prepare a magic state in )). Therefore, to implement a desired logical magic state, one must prepare the magic state in a physical qubit system capable of implementing non-Clifford gates, and then generate the logical magic state by encoding that quantum state into a logical state. This process of encoding a physical magic state into a logical magic state is called magic state injection.

[0056] The magic state injection process can vary depending on the quantum error correction method used to implement the logical non-Clifford gate. The optimal quantum error correction code for a given hardware depends on the characteristics of the hardware. Several quantum error correction (QEC) codes, including surface codes, are designed with structures that allow direct connection to four surrounding qubits. For example, Google has developed hardware based on this structure and is conducting research on error correction methods. However, not all hardware meets these conditions. For instance, IBM's hardware uses a heavy hexagonal structure, which allows one qubit to be directly connected to only two or three other qubits through a two-qubit gate.

[0057] Due to these limitations, surface codes requiring up to four adjacent qubits cannot be directly applied to 2-qubit gates. Consequently, implementing such codes requires the use of additional qubits, such as flag qubits. Furthermore, it is known that most qubits in IBM hardware exhibit a Z bias, where the T1 time is longer than the T2 time, making Z errors more likely to occur than X errors. It is well known that in the presence of such error bias, other quantum error correction codes, such as XZZX codes, leverage these characteristics to achieve better performance. Therefore, it is essential to evaluate the hardware characteristics to determine the best error correction code to use and to verify that the magic state injection process can operate efficiently with the selected code. In the embodiments described below, the magic state injection process was implemented using surface codes and XZZX codes in both lattice and heavy hexagonal structures. In the experimental examples described below, the logical error rates of the generated logical magic states were compared while varying various parameters such as bias, distance, and initialization.

[0058] In this process, the present embodiment and the present experimental example observed several characteristics unique to the magic state injection process in a heavy hexagonal structure using flag qubits, and based on these observations, explored the magic state injection method most suitable for heavy hexagonal structure hardware.

[0059] 1. Example: Magic State Injection Method in a Heavy Hexagonal Structure

[0060] 1) QEC code in heavy hexagonal structure

[0061] Surface codes typically require each qubit to be connected to four neighboring qubits. However, IBM's heavy hexagonal structure restricts qubits to being connected to two or three neighboring qubits. In previous research, this embodiment enabled stabilization measurements even with such limited connectivity by indirectly connecting the data qubit and the syndrome qubit using an additional qubit called a flag qubit. Using this approach, an error correction code was constructed in the heavy hexagonal structure as shown in FIGS. 4 and 5.

[0062] As shown in Figures 4 and 5, the surface code of the heavy hexagonal structure, excluding the flag qubit, follows the same layout as the surface code of the lattice structure. The surface code used in the experiment includes two types of syndrome qubits.

[0063] The X syndrome qubit (red) measures a stabilizer of the form XAXBXCXD by applying the Pauli X operator to four adjacent data qubits. The Z syndrome qubit (blue) measures a stabilizer of the form ZAZBZCZD by applying the Pauli Z operator to four adjacent data qubits. The boundary stabilizer contains three data qubits that take the form of XAXCXD or ZAZBZD.

[0064] Logical X and Z operators correspond to error chains extending from one boundary to the opposite boundary. The Z error chain from the top boundary to the bottom boundary defines logical Z errors, and the X error chain from the left boundary to the right boundary defines logical X errors. Since the stabilizer operators remain unchanged, logical operations on surface codes for heavy hexagonal structures are identical to those for lattice structures. By introducing flag qubits, existing surface codes requiring 4-qubit connections can be applied to heavy hexagonal structures.

[0065] 2) XZZX code

[0066] The XZZX code is an error correction code that detects both X and Z errors using a single type of stabilizer, constructed using both the Pauli X and Pauli Z operators. The stabilizer of the XZZX code is of the form XAZBZCXD, where Pauli X is applied to the data qubits above and below the syndrome qubit, and Pauli Z is applied to the data qubits to the left and right.

[0067] For boundaries, similar to surface codes, the stabilizer operates on only three data qubits using Pauli operators (see Figs. 6 and 7). In this stabilizer structure, X and Z errors propagate vertically, allowing the relative position of the detected syndrome to distinguish which type of error has occurred. Since a single syndrome qubit detects both types of errors, if the error is biased and the information is known in advance, it is possible to determine whether the detected error is more likely to be an X error or a Z error. Using this additional information increases the probability of finding the correct correction operator during the decoding process. Consequently, several studies have shown that the XZZX code achieves better performance in biased error models. Similar to the method described previously, the XZZX code can also be implemented in heavy hexagonal structures by appropriately positioning the data, syndrome, and flag qubits.

[0068] However, there are additional considerations when applying the XZZX code to a heavy hexagonal structure. In the previously described XZZX code, Pauli X is applied to the data qubits above and below the syndrome qubit, while Pauli Z is applied to the left and right data qubits. However, the stabilizer can be configured differently. The stabilizer of the form ZAXBXCZD, which applies Pauli Z above and below the syndrome qubit and Pauli X to the left and right qubits, can detect and correct errors in the same way as the XAZBZCXD form. In a general grid structure, since the two stabilizer forms are symmetrical, this difference does not have a significant impact. However, in a heavy hexagonal structure, performance differs depending on whether the stabilizer is in the XZZX or ZXXZ form because the number of flag qubits required to link data qubits and syndrome qubits differs in the vertical and horizontal directions. Therefore, experiments were conducted by dividing the XZZX code into two types: XZZX and ZXXZ.

[0069] The logical operators of the XZZX code are defined in the same way as the logical operators of the surface code. However, since the stabilizer forms of the two types of XZZX codes are inverted, the corresponding logical operators are also changed. For example, the logical operators of the XZZX type are identical to the logical operators of the surface code. However, in the ZXXZ type, the forms are inverted. Specifically, in the ZXXZ type, the X error chain from the upper boundary data qubit to the lower boundary data qubit corresponds to a logical X error, and the Z error chain from the left boundary data qubit to the right boundary data qubit corresponds to a logical Z error (see Figs. 6 and 7).

[0070] 3) Magic State Injection Process

[0071] The magic state injection process aims to map a desired magic state implemented in a physical qubit to the logical state of the corresponding error correction code. To achieve this, a state must be created that is the eigenstate of all stabilizer operators used in the error correction code, and the expectation of the logical Pauli operator for the logical state must be equal to the expectation of the physical Pauli operator for the physical qubit. The state can be made into the eigenstate of the stabilizer through a projection process that includes stabilizer measurements. At the same time, the physical state prepared for a specific physical qubit must be encoded into a logical state. Since the logical Pauli operator is commutative with all stabilizer operators, the expectation of the logical Pauli operator can be made equal to the expectation of the physical Pauli operator for the physical state when the state is prepared in advance.

[0072] For example, assume that we want to inject an arbitrary magic state |M > = α|0 > + β|1 > into a logical state. In this case, the Z error for all data qubits in the first row is the logical Z error (Z L It corresponds to ). If a physical magic state is prepared for one of the qubits in the first row and the other qubits in the same row are initialized to the 0 state, the Z for this state L The expected value of is Z for the physical magic state L It matches the expected value of Z L Since is commutative with all stabilizers, this relationship remains valid even after performing stabilizer measurements. This initialized state can be expressed as follows: where |φ〉 else is Z L It represents the state of a data qubit that is not related to, and d is the distance of the logical state being encoded.

[0073] [Mathematical Formula 1]

[0074]

[0075] Likewise, by preparing all qubits in the same row as the physical magic state from the + state, XL The expected value of will match the expected value of X for the physical magic state:

[0076] [Mathematical Formula 2]

[0077]

[0078] After performing a ballast measurement in this initialized state, this embodiment can project the state onto the eigenstate of the ballast. When all syndrome qubits are measured as 0, the logical quantum state |M〉 L It can be expressed as follows. Here, S i represents the i-th element of the set of stabilizer operators.

[0079] [Mathematical Formula 3]

[0080]

[0081] As can be seen in the process above, there are various methods to initialize the remaining data qubits |φ〉, excluding the physical qubit in which the magic state is implemented and the data qubits in the same row and column as that physical qubit. However, since various errors can occur during this process, it is essential to prepare the state of the data qubits in a manner that allows for error detection during stabilizer measurements to minimize errors.

[0082] There are two distinct scenarios when performing a stabilization measurement on a quantum state (Fig. 8). The first case occurs when the quantum state is already an eigenstate of the stabilization. If no error occurs during the stabilization measurement, the quantum state does not change and the syndrome qubit is measured as 0. If an error occurs, the syndrome qubit is measured as 1, indicating that there is an error. In the second case, when the quantum state is not an eigenstate of the stabilization, performing a stabilization measurement reduces the state to one of the eigenstates of the stabilization operator. Even if an error occurs during this process, it cannot be detected because the state is reduced during the measurement.

[0083] To detect errors during magic state injection, as many data qubits as possible must be applied to the error-detecting stabilizer measurement. Specifically, since the error-detecting stabilizer measurement must be performed on data qubits in the same row and column as the physical qubit prepared for the magic state, surrounding data qubits must be prepared for the same state to detect errors. Therefore, errors can be verified using the stabilizer measurement by initializing the remaining data qubits to |0〉 or |+〉 depending on the state of the surrounding qubits. Since preparing qubits to the same state as the surroundings aids in error detection, the data qubits must be divided into two regions: one prepared for |0〉 and the other for |+〉. There are still various methods for determining the boundary between these two regions. Selecting the initialization method during magic state injection is important for minimizing the logical error rate of the final logical magic state.

[0084] By following this procedure, physical magic states can be encoded into logical states while simultaneously detecting errors through stabilizer measurements. However, errors occurring during this process cannot be corrected because the initially prepared state is not a logical state. Therefore, errors are identified through post-selection. Post-selection is a process of discarding states where errors are detected and retaining states considered to be error-free. One problem with this method is that when using a larger number of physical qubits to create logical states of greater distance, the probability of obtaining an error-free state becomes very small. To address this issue, this embodiment adopts a method of extending the state injected into a smaller distance code to a larger distance code. Using this method, since the logical state of the smaller distance has already been obtained, error correction becomes possible during the distance extension process. An additional error correction round must be performed to correct errors that occur during this process. In this way, this embodiment was able to obtain a magic state of a larger distance and maintain a reasonable success probability for the magic state.

[0085] 4) Error Model

[0086] The types of physical errors occurring in actual quantum computers are highly diverse, making it difficult to mathematically express all of them accurately. Therefore, it is necessary to use an error model capable of effectively simulating the errors of actual quantum computers. In this experimental example, simulations were performed using a depolarization error model and a Z-bias Pauli error model because they closely represent the types of errors commonly occurring in actual quantum computers.

[0087] The depolarization error model is the most commonly used error model for evaluating the performance of error correction technology. It represents a situation where errors occur with equal probability p / 3 for Pauli X, Y, and Z, and the prepared state becomes a maximum mixed state with probability p. Additionally, an error model was developed to reproduce the Z-bias error present in actual quantum computers. In this model, as the bias η increases, the Z error is more likely to occur than the X and Y errors. Since this embodiment considers the error model at the gate level, the error model for both single-qubit and two-qubit gates must be adjusted according to the bias. First, for a single-qubit gate, the error probability is defined as follows.

[0088] [Mathematical Formula 4]

[0089]

[0090] In this error model, when the bias η = 0.5, P X = P Y = P Z = P single It corresponds to the depolarization error model with a value of 3. As η increases to infinity, only the Z error occurs with probability p.

[0091] In the case of a 2-qubit gate, the depolarization error model Each error in the set results in an error occurring with probability p / 15. In the biased case, the probability of an error occurring in a single or two qubits (specifically ZZ, ZI, and IZ) increases by adjusting η as follows.

[0092] [Mathematical Formula 5]

[0093]

[0094] [Mathematical Formula 6]

[0095]

[0096] When η = 0.5, this model has all errors with probability P doubleIt becomes a depolarization error model caused by / 15. As η approaches infinity, P ZZ , P ZI , P IZ is P double It approaches / 3, and the probability of all other errors becomes 0.

[0097] This error model represents a situation where the X and Y error rates decrease as the bias increases, while the Z error rate increases. In the surface code and XZZX code used in this embodiment, Y errors are detected simultaneously by X and Z syndromes. That is, in the error correction code, Y errors correspond to a situation where X and Z errors occur together. Consequently, since both X and Y errors decrease, the probability of logical X errors decreases, but the probability of logical Z errors increases less because the probability of Y errors also decreases. Therefore, because Y errors decrease, the overall logical error rate tends to decrease as the bias increases.

[0098] To evaluate the error rate of the magic state obtained in this embodiment, the logical error rate was calculated in the following experimental example. The experimental example measured logical qubits based on X or Z to verify whether a logical error occurred in the prepared state. Logical X and Z error rates were obtained by repeating the magic state injection and measuring based on X or Z, and the total logical error rate can be calculated using the following formula. Here, the total logical error rate (E total ) means the probability that neither logical X error nor Z error occurs at 1.

[0099] [Mathematical Formula 7]

[0100]

[0101] 2. Experimental Example: Performance Evaluation of Quantum Error Correction Codes

[0102] Experiments were conducted to investigate various characteristics in order to identify the optimal magic state injection strategy in hexagonal structures. This experiment explored how the logical error rate changes as the physical error rate varies in lattice and hexagonal structures for surface codes and two types of XZZX codes. Additionally, this experiment investigated how the performance of each error correction code changes by varying the bias of the error model, the distance of the error correction code, and the initialization method.

[0103] 1) Characteristics of a flag qubit

[0104] Quantum error correction codes typically perform stabilization measurements using syndrome qubits, each syndrome qubit having a specified number of data qubits (i.e., weights) that must be connected. However, depending on the hardware used, the connection requirements of the error correction code may not be met. In such cases, it may be impossible to apply the error correction code directly. To solve this problem, flag qubits were used. Using flag qubits allows data qubits and syndrome qubits to be connected indirectly even if hardware connections are insufficient.

[0105] FIG. 9 shows a portion of the ballast measurement circuit for a heavy hexagonal structure error correction code as illustrated in FIGS. 4 through 7. These figures show that errors occurring in data qubits are propagated through flag qubits and measured in syndrome qubits. However, using flag qubits may introduce additional physical qubits, which may result in additional errors that need to be detected and corrected.

[0106] The flag qubit is connected to the neighboring syndrome qubit, data qubit, and other flag qubits through a CNOT gate, where the qubit closest to the syndrome qubit acts as the control qubit and the qubit closest to the data qubit acts as the target qubit. The CNOT gate propagates X errors from the control qubit to the target qubit and Z errors from the target qubit to the control qubit. Therefore, if an X error occurs in the flag qubit, it is propagated to the data qubit, and if a Z error occurs, it is propagated to the syndrome qubit. Consequently, the X error in the flag qubit acts as an additional error in the data qubit, while the Z error acts as an additional measurement error in the syndrome qubit.

[0107] In this experimental example, the flag qubit is measured in each round to detect errors and reset the state. Since the flag qubit is prepared to the 0 state after the Z measurement, the Z error propagated through the flag qubit does not affect the flag qubit measurement result. Therefore, the Z error occurring in the flag qubit affects only the measurement result of the syndrome qubit and can be considered as a read error of the syndrome qubit. On the other hand, the X error of the flag qubit is an error that depends on the stabilizer operator and is propagated to the data qubit. Therefore, it is possible to check whether an X error has occurred in the flag qubit by measuring the flag qubit. Additionally, the CNOT gate used with the flag qubit is applied symmetrically once more to cancel out errors propagated to other flag qubits, leaving only the error detected in the original flag qubit. Therefore, it is possible to determine which flag qubit experienced the X error through the measurement results.

[0108] Consequently, the error rate of the data qubit increases due to additional X errors in the flag qubit. The impact of these errors depends on the stabilization form and the position of the data qubit (see Fig. 10). For this reason, in a heavy hexagonal structure, the error of each data qubit appears with a different bias. Since these errors are caused by X errors in the flag qubit, increasing the Z bias in a heavy hexagonal structure results in X errors in the flag qubit. Therefore, the logical error rate decreases as the bias increases. This effect further reduces the error rate beyond the reduction described by the error model.

[0109] 2) Qubit Initialization

[0110] The logical error rate of the resulting magic state may vary depending on how each physical qubit is prepared from its physical state during magic state injection. This experimental example investigated the case where the physical magic state is prepared in the top-left qubit. In this situation, the data qubits in the top row and the leftmost column must be initialized to |0〉 or |+〉 according to the logical operator determined by the given error correction code. The remaining data qubits must be prepared in a manner that minimizes initialization errors so that errors can be effectively detected.

[0111] If the data qubit corresponding to Pauli Z of a given stabilizer is initialized to |0〉 and the data qubit corresponding to Pauli X is initialized to the + state, errors occurring during the initialization process can be detected. However, since it is impossible to initialize data qubits in this manner for all stabilizers, syndrome qubits capable of detecting errors must be selected. To reduce logical errors during magic state injection, the number of syndrome qubits capable of detecting errors must be maximized. To achieve this, the region where data qubits are prepared must be appropriately divided into |0〉 or |+〉. These regions can be divided in various ways.

[0112] Two main methods were devised for initializing data qubits. The triangle method divides the initialization area using a diagonal line from the top-left to the bottom-right to distinguish the regions where data qubits are initialized to |0〉 and |+〉. The square method divides the area using a straight line from top to bottom or from left to right. In the case of the square method, dividing the area using a single straight line reduces the number of stabilizer measurements capable of detecting errors; therefore, this was resolved by adjusting the boundaries around the physical qubits where the magic state (injected qubit) is prepared. Although there are various ways to form the boundaries, this experimental example decided to configure the initialization using these two schemes. This is because forming the boundaries in the simplest straight-line shape maximizes the number of stabilizer measurements capable of detecting errors. Each scheme is divided into two methods depending on which of the two regions is larger. If there are more data qubits in the right region, it is classified as "right," and if there are more data qubits in the bottom region, it is classified as "bottom." Accordingly, this experimental example performed simulations for four initialization methods (right square, bottom square, right triangle, and bottom triangle) (see Fig. 11).

[0113] When the physical magic state is placed in the top-left qubit, there is always one data qubit that cannot detect errors; this is called the blind qubit. If an error propagates to this blind qubit during initialization and projection measurements, a logical error may occur. Therefore, errors occurring in the blind qubit must be carefully investigated. Errors occurring in the blind qubit depend on its relative position and physical state. The blind qubit's relative position changes when "right" and "bottom" are swapped. Additionally, the physical state in which the blind qubit is prepared may vary depending on the error correction code used. The XZZX code has been divided into two types: XZZX and ZXXZ. When using a ZXXZ type stabilizer, the logical operators differ, requiring a different physical state for preparation.

[0114] Consequently, for the ZXXZ type, the data qubit types prepared in each region also change between the |0〉 and |+〉 states. However, the XZZX and ZXXZ types are symmetric with respect to the diagonal extending from the top left to the bottom right. Since the four initialization methods introduced earlier also possess a symmetric structure, if the XZZX and ZXXZ type codes select a symmetric initialization method, the two become equivalent. To make a fair comparison, the right initialization of the XZZX type must be compared with the bottom initialization of the ZXXZ type, and the bottom initialization of the XZZX type must be compared with the right initialization of the ZXXZ type.

[0115] There are two cases where errors in blind qubits go undetected and affect logical errors. One occurs while preparing the physical state of the blind qubit, and the other occurs when errors propagate during the first stabilizer measurement process. Errors in preparing data qubits during the initialization process are essentially single-qubit errors caused by Hadamard gates, and generally, the error rate is higher when preparing |+〉 compared to |0〉. However, this experimental example assumed that 2-qubit errors are 20 times larger than single-qubit errors, and in reality, it was observed that single-qubit gate errors are much less frequent than 2-qubit gate errors. Therefore, this effect is much less significant than errors in the first stabilizer measurement using 2-qubit gates. Errors occurring in stabilizer measurements where errors cannot be detected during the first projection measurement do not affect logical errors. However, errors occurring in stabilizer measurements where errors can be detected can affect logical errors. In particular, errors in syndrome or flag qubits can propagate to data qubits, potentially causing additional errors in addition to errors in the initialization process. Therefore, the relative position of the syndrome qubit to the blind qubit performing the error detection stabilizer measurement can affect the logical error rate.

[0116] If an error in a syndrome or flag qubit propagates to a blind qubit via a stabilizer measurement, these errors go undetected and lead directly to logical errors. To understand the impact of this, we must examine how errors propagate to blind qubits during initialization. If a blind qubit is positioned below a qubit ready for a magic state (e.g., the right triangle or downward square method), the error is more likely to propagate through the syndrome qubit located to the right. Conversely, if a blind qubit is positioned to the right of a qubit ready for a magic state (e.g., the downward triangle or right square method), the error is more likely to propagate through the syndrome qubit located below it. In a grid structure, errors propagating from blind qubits do not show significant differences depending on the direction because the horizontal and vertical directions of the stabilizer are identical. Therefore, the difference in the single-qubit gate error rate due to the initialization state is more pronounced. However, in a heavy hexagon structure using more flag qubits in the vertical direction, blind qubits located to the right (down triangle or right square) experience greater error propagation, so the position of the blind qubit has a greater impact.

[0117] Figure 12 shows the experimental results using various initialization methods. This experimental example compared four initialization methods and three error correction code types to find the optimal combination with the lowest error rate in a heavy hexagonal structure. To verify this, this experimental example observed changes in the logical error rate during magic state injection with various biases and distance extensions. Initially, logical errors were verified in a grid structure without bias or distance extension.

[0118] All initialization methods exhibited similar error rates, but there were slight differences due to the additional Hadamard gate used when the blind qubit was initialized to |+〉. In the heavy hexagonal structure, the experimental example observed two different error rates depending on whether the blind qubit was to the right or below the magic state-ready qubit. When the blind qubit is to the right, errors propagate through the syndrome qubit, resulting in a higher error rate than when it is below. As the bias increases, the XZZX code shows a lower logical error rate due to the bias. In the lattice structure, the XZZX and ZXXZ codes using symmetric initialization exhibit similar error rates, differing only in the initialization state of the blind qubit. When the blind qubit is initialized to |0〉, X errors lead to logical errors, and when initialized to |+〉, Z errors lead to logical errors. Therefore, as the Z bias increases, the error rate decreases when the blind qubit is at |0〉. This trend is maintained in the heavy hexagonal structure as well, but in this case, the ZXXZ type has a lower error rate than the XZZX type as the bias increases.

[0119] If distance expansion is not applied, the difference in logical error rates according to the initialization method is small due to similar area sizes. However, as the distance increases, the difference in the number of data qubits in each area becomes more significant, leading to greater variations depending on the boundary configuration. Methods with larger qubit areas (e.g., surface codes or right-hand methods of the XZZX type) require additional Hadamard gates, resulting in more errors. Additionally, square methods often involve connecting data qubits prepared in the same physical state in a line from end to end, increasing the likelihood of consecutive errors of the same type and resulting in a relatively high logical error rate.

[0120] This effect is further emphasized with distance expansion, where the right square (or bottom square for the ZXXZ type) exhibits the highest error rate, and the bottom triangle (or right triangle for the ZXXZ type) exhibits the lowest error rate in the grid structure. In the heavy hexagon structure, the error rate varies for each initialization method and error correction code combination due to the influence of the flag qubit. Because the flag qubit exists, in addition to the influence of the various initialization methods described earlier, the number of flag qubits propagating errors can vary depending on the relative position of the blind qubit. Due to these variations, the error rate for each initialization method in the heavy hexagon structure differs. Based on the results using bias and distance expansion, the most suitable error correction code and initialization method for the heavy hexagon structure is the ZXXZ type using the right triangle method.

[0121] 3) Effect of bias

[0122] Previous experiments demonstrated that lattice structures and heavy hexagonal structures exhibit different responses to changes in bias. Generally, due to the characteristics of the error model used in this experimental example, the logical error rate decreases as the bias increases. Furthermore, in the heavy hexagonal structure, if the X error of the flag qubit decreases, the logical error rate decreases more significantly as the bias increases. Additionally, the number of flag qubits required for connection in the physical qubit stabilization patch used in this experimental example depends on whether the data qubits are positioned vertically or horizontally relative to the syndrome qubits. Therefore, in the heavy hexagonal structure, the error rate depends on the relative position of the data qubits. The influence of the flag qubits causes different types of errors depending on the position of the data qubits. In the surface code, each data qubit acquires additional X or Z errors depending on the type of syndrome qubit above and below it.

[0123] For example, for a specific data qubit, if the syndrome qubit in the vertical direction corresponds to an X stabilizer and the syndrome qubit in the horizontal direction corresponds to a Z stabilizer, that data qubit is more likely to experience an X error. In surface codes, data qubits with higher X errors and data qubits with higher Z errors exist in similar proportions, so no additional bias is generated for all data qubits. However, in XZZX codes, the operators for the vertical and horizontal directions of the stabilizers are different, and only one type of stabilizer is used; therefore, errors corresponding to the vertical operator of the stabilizer propagate more frequently to all data qubits. Consequently, in XZZX codes, using an XZZX type stabilizer results in additional X errors for the data qubits, and using a ZXXZ type results in additional Z errors, generating additional bias beyond the bias set in the error model. (See Fig. 13)

[0124] This effect of the flag qubit results in different performance for the XZZX and ZXXZ types in heavy hexagonal structures. In particular, the ZXXZ type exhibits better performance when a Z bias is present, as it reinforces the Z bias. Therefore, when using the flag qubit, if there is an additional bias error in the data qubit, different characteristics appear compared to the lattice structure, leading to different results for magic state injection in heavy hexagonal structures.

[0125] In this experimental example, magic state injection experiments were performed using a biased error model in both lattice and heavy hexagonal structures. For XZZX, ZXXZ types, and surface codes, as expected from the characteristics of the error model, both the lattice and heavy hexagonal structures showed a trend of decreasing logical error rates as the bias increased. Additionally, it was confirmed that the XZZX and ZXXZ types had lower logical error rates compared to standard surface codes when the Z bias was high. In the lattice structure, the difference in error rates between error correction codes was relatively small when there was no bias (η = 0.5).

[0126] However, in heavy hexagonal structures, even without bias, the logical error rate varies depending on the type of error correction code due to the presence of errors propagating from the flag qubit. As the bias changes, the total logical error rate and the logical X and Z error rates change differently in lattice and heavy hexagonal structures. For surface codes in lattice structures, increasing the Z bias decreases the logical X error rate and increases the logical Z error rate, which is generally expected considering the variation in physical qubit error rates. On the other hand, for XZZX and ZXXZ types in lattice structures, logical Z errors increase or decrease more slowly compared to surface codes when the Z bias increases. This phenomenon occurs because XZZX and ZXXZ types can detect errors more effectively with additional bias information. Conversely, in heavy hexagonal structures, increasing the bias tends to decrease both the logical X and Z error rates for all error correction codes, which differs from lattice structures.

[0127] This behavior stems from the error characteristics of the flag qubit. In heavy hexagonal structures, X errors in the flag qubit propagate to the data qubit, adding an additional error to the data qubit. As the Z bias increases, the X error in the flag qubit relatively decreases, lowering the error rate of the data qubit and consequently the logical error rate. Furthermore, the difference between the XZZX and ZXXZ types becomes more pronounced as the bias increases due to the difference in the types of errors propagating from the flag qubit to the data qubit. In the XZZX type, X errors propagate more widely through the three flag qubits, whereas in the ZXXZ type, Z errors propagate more extensively. As a result, each code experiences a different effective bias, leading to different total error rates. The ZXXZ type strengthens the Z bias by introducing additional Z errors, while the XZZX type reduces the Z bias by introducing X errors. Consequently, the ZXXZ type XZZX code exhibits the lowest logical error rate as the Z bias increases.

[0128] 4) Extended error

[0129] The logical error rate after performing magic state injection can be attributed to two main types of errors.

[0130] The first type arises from errors during the initialization and projection measurement of logical states. To mitigate these errors, a post-selection process is used to discard all states with detected errors. However, certain combinations of errors go undetected, which can lead to logical errors in prepared magic states. Additionally, errors occurring in specific data qubits, such as those in physical magic states or when preparing blind qubits, go undetected, resulting in logical errors in these qubits.

[0131] The second type of error occurs during the process of extending the distance and performing additional rounds, which can lead to logical errors. For the first type of error, the performance of the error correction code is significantly affected by the initialization method and the number of physical qubits used. On the other hand, the second type of error can be corrected if the error correction code is effective at detecting and correcting errors. Therefore, if the physical error rate is sufficiently low, the error correction code can handle the second type of error occurring during the distance extension process. However, since this error reduction does not eliminate the first type of error during the initialization process, reducing only the physical error rate cannot indefinitely reduce errors occurring during the magic state injection process.

[0132] We examined how the logical error rate of the magic state changes as the distance increases. We conducted experiments to observe the logical error rate when expanding from a distance-3 code (d1=3) to a larger distance (d2 increasing by 2 each time). For comparison, we also performed an experiment with d1=d2, where no distance expansion was applied. Each experiment involved injecting the state into an error correction code at distance d1 and then expanding the distance to d2. We compared the logical error rates of hexagonal and grid structures. We observed that when the error rate is low, increasing the distance maintains a similar logical error rate. In the grid structure, the logical error rate decreases when expanding the distance below a certain error rate, whereas in the heavy hexagonal structure, it remains nearly constant regardless of distance expansion in the low error region.

[0133] However, if the physical error rate is not sufficiently low, increasing the distance significantly increases the error rate. This suggests that using sufficiently good hardware allows most errors occurring during distance expansion to be corrected, and that a similar level of logical error rate can be achieved even in logical magic states with larger distances. Nevertheless, since some errors are inevitably present during the initial process, this indicates that a distillation process is still necessary to further reduce errors.

[0134] Through these experiments, it was confirmed that the presence of flag qubits in heavy hexagonal structures introduces various phenomena during magic state injection that are not present in lattice structures. Because flag qubits are present in heavy hexagonal structures, additional bias errors are introduced into data qubits depending on the stabilizer, resulting in two types of stabilizers that exhibit different performance for XZZX codes. Furthermore, the choice of initialization method, the location of blind qubits, and the orientation of the boundary lines influence the logical error rate of the magic state injection process. Considering these factors, the most suitable magic state injection method for heavy hexagonal structures is to use a ZXXZ type XZZX code utilizing a right-hand triangle initialization method. This study highlights the considerations necessary to identify suitable magic state injection methods in hardware with flag qubits, taking into account connection constraints. These findings are expected to aid in the implementation of fault-tolerant quantum computing by lowering the error rate for non-Clifford gates in such hardware.

[0135] 5) Magic State Injection Process

[0136] Magic state injection is the process of encoding a physical magic state prepared in a physical qubit into a logical magic state in a logical qubit. To achieve this, one must first determine the error correction code to be used to construct the logical qubit and determine the number of physical qubits required to achieve a logical qubit of a given distance. The magic state injection process consists of two main steps.

[0137] In Step 1, a magic state is injected into a logical qubit at distance d1. Then, all errors occurring during this process are detected, and a logical magic state where no errors are detected is obtained using post-selection. In Step 2, an error correction code for a larger distance d2 containing the physical qubit from Step 1 is constructed. Then, additional rounds are performed to detect and correct errors occurring in this logical qubit at a larger distance and to measure the data qubit to evaluate the performance of the magic state injection process.

[0138] In this experiment, the present example set d1 to 3, varied d2 to 3, 5, 7, and 9, and set the bias η to 0.5, 1, 5, 10, and 100. The physical error rate variable used in this example represents both the 2-qubit error rate and the read error rate, and it is assumed that both error rates are equal. The probability of a 2-qubit gate error was changed from 0.05% to 1%, and for simulation purposes, the probability of a single-qubit gate error was assumed to be 1 / 20 of the 2-qubit gate error probability. The simulation was performed using Stim code, and decoding was performed using the Pymatching algorithm. This example uses four types of gates: CZ, CNOT, Hadamard, and computational measures. Sampling is 1×10⁻⁶ 7 performed the meeting.

[0139] (1) Step 1

[0140] ● As the first step of Stage 1, you must determine the location of the physical qubit to prepare the magic state.

[0141] The selection of the location affects the subsequent region configuration. In this experiment, this qubit is set as a data qubit. It is located in the top-left corner.

[0142] ● To initialize data qubits, it must be determined how to divide the regions. The regions are generally divided into four. Regions 1 and 2 are distinguished based on whether the data qubits in the distance d1 region are initialized to |0〉 or |+〉. Regions 3 and 4 are distinguished based on whether the data qubits in the distance d2 region, which is not included in regions 1 and 2, are initialized to |0〉 or |+〉. The regions are configured so that regions 1 and 3 are initialized to the same quantum state, and regions 2 and 4 are initialized to different identical quantum states. In the special case where d1 = d2, regions 3 and 4 are undefined. Initializing data qubits to |0〉 or |+〉 depends on how the logical X and Z operators for the error correction code are defined. For example, in a surface code or an XZZX code of type XZZX, the data qubits in regions I and III are initialized to |+〉, and the data qubits in regions II and IV are initialized to |0〉. Conversely, for the XZZX code of type ZXXZ, the data qubits of regions I and III are initialized to |0 > and the data qubits of regions II and IV are initialized to |+ >.

[0143] ● Then, a ballast measurement is performed to project the prepared state onto the ballast's eigenstate. The ballast measurement includes data qubits already prepared in the ballast operator's eigenstate and can detect errors that occurred during data qubit initialization. Through these specific ballast measurement results, errors occurring during the initialization process can be detected. However, due to issues such as readout errors that may occur during measurement, performing the ballast measurement only once cannot verify whether the projected state is actually a logical state.

[0144] ● Therefore, the stabilizer measurement is performed one more time. By performing the stabilizer measurement twice, it is possible to check whether an error occurred during the process of encoding the physical state into a logical state. If an error is detected in one of the two rounds, the state is discarded through a post-selection process. Errors occurring in the flag qubit can be identified by measuring the flag qubit and comparing the parity. After each stabilizer measurement step, the syndrome and flag qubit are re-initialized for the next round. Using this method, a d1 distance logical state can be obtained probabilistically in step 1.

[0145] (2)2nd stage

[0146] ● The first step of the second stage is to initialize the data qubits of regions III and IV to achieve the distance d2 magic state. Initialization is performed in a manner similar to how regions I and II were prepared so that region III is initialized to the same state as region I and region IV is initialized to the same state as region II.

[0147] ● Then, a ballast measurement is performed across the entire d2 distance area. Since the state in the d1 distance area is already prepared as the ballast's eigenstate, errors can be detected through the measurement results of the syndrome qubit. The ballast measurements in areas III and IV project the prepared data qubit state onto the d2 error correction code. Since the state of the data qubit in the d1 distance area is already the ballast's eigenstate, not only is the occurrence of an error identified, but the location and type of the error can also be identified, allowing the process to detect and correct the error.

[0148] ● After the projection measurement, the d2 distance logical magic state was obtained. However, additional rounds are required to detect and correct additional errors that may occur during the measurement process. As the distance increases, more physical qubits are used, so more rounds are required to check for errors in these qubits. To detect errors that increase with distance, a stabilization measurement of round d2 is performed for each logical qubit. If d1 = d2, the measurement of round d1 is added without performing additional qubit initialization, and the error rate is checked for a fair comparison.

[0149] ● To verify the error rate of the d2-distance logical magic state, the obtained state is measured. Possible logical errors include logical X, logical Y, and logical Z errors. Since a logical Y error can be considered a simultaneous occurrence of logical X and Z errors, the focus is on detecting logical X and Z errors to determine the probability of logical errors. To detect logical X and Z errors, all data qubits are measured against the X or Z reference. When a data qubit is measured, the logical state is reduced to |+ >, |1 > or |0 >, |1 > depending on the reference used. Using the measurement results, the MWPM algorithm is used to identify errors that occurred during the magic state injection process and to calculate the appropriate correction operator. After applying the correction operator, the parity of the measurement result of the data qubit corresponding to the logical operator is checked. If the parity of the logical X operator is 0 when measured against the X reference, the logical state is considered to have been reduced to the + state. If the parity of the logical Z operator is 1 when measured against the Z reference, the logical state is considered to have been reduced to the 1 state. By repeating this measurement process several times, the expected values ​​for Pauli X and Z can be calculated. By comparing these expected values ​​with the expected values ​​for the desired state, the probabilities of logical X and Z errors can be calculated. The total logical error rate is calculated as shown in Equation 7 above.

[0150] Through this process, the experimental example can obtain |M>L of distance d2 and implement non-Clifford logical operators in heavy hexagonal structures.

[0151] According to the method for introducing a magic state of a quantum computer and the quantum computer according to the embodiments described above, a quantum error correction (QEC) code of a threshold error rate can be satisfied by introducing a magic state.

[0152] A method for introducing a magic state of a quantum computer according to embodiments and an experimental example have been described with reference to the drawings above, but are not limited thereto.

[0153] The foregoing description is merely an illustrative explanation of the technical concept of the present disclosure, and those skilled in the art to which the present disclosure pertains may make various modifications and variations within the scope of the essential characteristics of the technical concept. Furthermore, since these embodiments are intended to explain, not limit, the scope of the technical concept is not limited by these embodiments. The scope of protection of the present disclosure shall be interpreted by the claims below, and all technical concepts within an equivalent scope shall be interpreted as being included within the scope of rights of the present disclosure.

[0154]

[0155] CROSS-REFERENCE TO RELATED APPLICATION

[0156] This patent application claims priority pursuant to Section 119(a) of the U.S. Patent Act (35 USC §119(a)) to Korean Patent Application No. 10-2025-0021579 filed on February 19, 2025, the entire contents of which are incorporated by reference into this patent application. Additionally, this patent application claims priority in countries other than the United States for the same reasons as above, and the entire contents of which are incorporated by reference into this patent application.

Claims

1. A step of constructing a logical qubit using a heavy hexagonal structure error correction code that connects two data qubits and one syndrome qubit on both sides by introducing two or more flag qubits between four data qubits and one syndrome qubit; and It includes the step of performing a logical operation on the above logical qubit, The above error correction code is a method for introducing a magic state in a quantum computer, wherein a magic state injection process is performed to map a magic state implemented in a physical qubit to a logical state.

2. In Paragraph 1, A method for introducing a magic state in a quantum computer, wherein the above syndrome qubit is an X syndrome qubit or a Z syndrome qubit.

3. In Paragraph 1, The above heavy hexagonal structure is a method for introducing a magic state of a quantum computer having an XZZX code that includes both X and Z operators.

4. In Paragraph 3, A method for introducing a magic state of a quantum computer, wherein the above XZZX code includes an XZZX type in which an X operator is vertically arranged and a Z operator is horizontally arranged for the above syndrome qubit, or a ZXXZ type in which a Z operator is vertically arranged and an X operator is horizontally arranged.

5. In Paragraph 1, The above error correction code is a method for introducing a magic state in a quantum computer, which initializes the remaining data qubits excluding the physical qubit in which the magic state is implemented and the data qubits in the same row and column as the physical qubit.

6. In Paragraph 5, A method for introducing a magic state in a quantum computer, which detects an error and resets the state by measuring the flag qubit in each round.

7. In Paragraph 5, The above error correction code is a method for introducing a magic state in a quantum computer, which initializes data qubits according to a triangle method that divides the initialization area using a diagonal line from the top left to the bottom right to distinguish the area where data qubits are initialized to |0 > and |+ >, and a square method that divides the area using a straight line from top to bottom or from left to right.

8. In Paragraph 7, A method for introducing a magic state in a quantum computer, wherein the above triangle method is one of the right triangle method and the bottom triangle method, and the above square method is one of the right square method and the bottom square method.

9. A quantum processor that performs quantum processing; A control device for controlling the above quantum processor; and The above quantum processor includes a measuring device for measuring the result of performing quantum processing, and The above quantum processor is, A logical qubit is included using a heavy hexagonal structure error correction code that connects two data qubits and one syndrome qubit from both sides by introducing two or more flag qubits between four data qubits and one syndrome qubit, and performs logical operations on said logical qubit. The above error correction code is a quantum computer in which a magic state injection process is performed to map a magic state implemented in a physical qubit to a logical state.

10. In Paragraph 9, The above syndrome qubit is a quantum computer in which the syndrome qubit is an X syndrome qubit or a Z syndrome qubit.

11. In Paragraph 9, The above heavy hexagonal structure is a quantum computer having an XZZX code that includes both X and Z operators.

12. In Paragraph 11, A quantum computer comprising the above XZZX code, which includes an XZZX type in which the X operator is vertically positioned and the Z operator is horizontally positioned for the above syndrome qubit, or a ZXXZ type in which the Z operator is vertically positioned and the X operator is horizontally positioned.

13. In Paragraph 9, The above error correction code is a quantum computer that initializes the remaining data qubits, excluding the physical qubit in which the above magic state is implemented and the data qubits in the same row and column as the physical qubit.

14. In Paragraph 13, A quantum computer that detects errors and resets the state by measuring the flag qubit in each round.

15. In Paragraph 13, The above error correction code is a quantum computer that initializes data qubits according to a triangle method, which divides the initialization region using a diagonal line from top left to bottom right to distinguish the region where data qubits are initialized to |0〉 and |+〉, and a square method, which divides the region using a straight line from top to bottom or from left to right.

16. In Paragraph 15, A quantum computer in which the above triangle method is one of the right triangle method and the bottom triangle method, and the above square method is one of the right square method and the bottom square method.

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