Techniques for preparing fault-tolerant cluster states

Abstract

Quantum systems and techniques are described for generating fault-tolerant cluster states for use in quantum computing, quantum networking, and other applications. The systems and techniques include generating an initial resource state by initializing a state with a first qubit and performing a first Pauli product measurement on a set of X-type and / or Z-type qubits of the first qubit, the initial resource state including a qubit cluster state including at least three qubits. A final cluster state can then be generated by fusing two or more initial resource states, the fusing including performing a second Pauli product measurement between the qubits of the two or more initial resource states.

Description

[Technical field]

[0001] REFERENCE TO RELATED APPLICATIONS This application claims the benefit under 35 USC §119(e) of U.S. Provisional Patent Application No. 63 / 286,362, filed on December 6, 2021, and entitled "Technique for preparing a fault-tolerant cluster state," and U.S. Provisional Patent Application No. 63 / 292,868, filed on December 22, 2021, and entitled "TECHNIQUE FOR PREPARING A FAULT-TOLERANT CLUSTER STATE," each of which is incorporated by reference herein in its entirety.

[0002] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH This invention was made with Government support under OMA-2137740 awarded by the National Science Foundation. The Government has certain rights in this invention. [Background technology]

[0003] background Quantum information processing techniques perform computations by manipulating one or more quantum objects. These techniques are sometimes referred to as "quantum computing." To perform computations, quantum information processors utilize quantum objects to reliably store and retrieve information. Some quantum information processing approaches have developed a quantum analog to classical computing "bits" (equal to 1 or 0), called quantum bits or "qubits." Qubits can consist of any quantum system that has two distinct states (which may be thought of as 1 and 0 states), but also has the special property that the system can be placed in a quantum superposition, thereby existing in both of these states at once. Summary of the Invention

[0004] Quick Overview Some embodiments relate to a quantum system. The quantum system includes at least one controller and at least one non-transitory computer-readable medium storing computer-readable instructions, where the at least one controller is configured to create an XZZX cluster state. Generating the XZZX cluster state includes: initializing a state with a first qubit, where the first qubit includes an X-type and a Z-type qubit; generating an initial resource state by performing a first Pauli product measurement on a set of X-type and / or Z-type qubits of the first qubit, where the initial resource state includes a qubit cluster state including at least three qubits; and generating the XZZX cluster state by fusing two or more initial resource states, where the fusing includes performing a second Pauli product measurement between the qubits of the two or more initial resource states.

[0005] Some embodiments relate to methods of generating an XZZX cluster state, the generating an XZZX cluster state including: initializing a state with a first qubit, where the first qubit includes an X-type and a Z-type qubit; generating an initial resource state by performing a first Pauli product measurement on a set of X and / or Z qubits of the first qubit, where the initial resource state includes a qubit cluster state including at least three qubits; and generating an XZZX cluster state by fusing two or more initial resource states, where the fusing includes performing a second Pauli product measurement between the qubits of the two or more initial resource states.

[0006] In some embodiments, generating the initial resource state includes generating a first three-qubit cluster state by initializing states of two X-type qubits and one Z-type qubit; and generating the first three-qubit cluster state by performing a three-qubit Z measurement on the initialized qubits.

[0007] In some embodiments, generating the initial resource state includes generating a five-qubit cluster state by fusing each of two X-type qubits of the first three-qubit cluster state with a Z-type qubit of the second and third three-qubit cluster states.

[0008] In some embodiments, fusing each of the two X-type qubits of the first three-qubit cluster state with the Z-type qubit of the second and third three-qubit cluster states comprises: performing a two-qubit Z measurement between each of the two X-type qubits and the Z-type qubit; and performing a two-qubit X measurement between each of the two X-type qubits and the Z-type qubit.

[0009] In some embodiments, generating the initial resource state includes: initializing a state with three Z-type qubits and one X-type qubit; and generating a four-qubit cluster state by performing two-qubit Z measurements between each of the three Z-type qubits and the one X-type qubit.

[0010] In some embodiments, generating the initial resource state includes: initializing the state with an additional X-type qubit; and generating a 5-qubit cluster state by performing a CZ gate between the additional X-type qubit and one of the X-type qubits of the 4-qubit cluster state.

[0011] In some embodiments, generating the initial resource state includes generating a four-qubit cluster state by: initializing a state with four X-type qubits; and generating a four-qubit cluster state by performing a two-qubit X measurement between pairs of qubits of the initialized four X-type qubits. In some embodiments, generating the four-qubit cluster state further includes performing three Z measurements of the initialized four X-type qubits.

[0012] In some embodiments, generating the initial resource state includes: generating three 3-qubit cluster states; and generating a 6-qubit cluster state by fusing qubits of the three 3-qubit cluster states to generate a 6-qubit cluster state including two Z-type qubits and four X-type qubits.

[0013] In some embodiments, generating the three three-qubit cluster states includes: initializing a state with six X-type qubits and three Z-type qubits; and performing at least four two-qubit X and / or Z measurements for each of the three three-qubit cluster states to generate the three three-qubit cluster states.

[0014] In some aspects, fusing the two or more initial resource states includes performing a Bell measurement between a first qubit of the first initial resource state and a second qubit of the second initial resource state.

[0015] In some embodiments, initializing the state with the first qubit includes generating the photonic qubit using a single photon source.

[0016] In some embodiments, initializing the state of the first qubit comprises: generating one or more optical or microwave signals using one or more optical or microwave sources; and communicating the generated one or more optical or microwave signals to the first qubit to initialize the state.

[0017] Some embodiments relate to a quantum system including: a plurality of physical qubits; at least one computer-readable medium storing a plurality of drive waveforms; and at least one controller configured to initialize an alternating grid of X-start and Z-start cluster states at physical qubits of the plurality of physical qubits; initialize at least one physical qubit of the plurality of physical qubits to be an X-type qubit; and measure the at least one physical qubit of the plurality of physical qubits at an X reference to measure an XZZX stabilizer of a corresponding cluster state.

[0018] Some embodiments relate to a method of constructing fault-tolerant cluster states using a quantum system including a plurality of physical qubits, the method including: initializing an alternating grid of X-start and Z-start cluster states at physical qubits of the plurality of physical qubits; initializing at least one physical qubit of the plurality of physical qubits to be an X-type qubit; and measuring the at least one physical qubit of the plurality of physical qubits in an X basis to measure an XZZX stabilizer of a corresponding cluster state.

[0019] In some embodiments, initializing the alternating grid of X-start and Z-start cluster states comprises applying one or more CX and / or CZ gates to one or more of the plurality of physical qubits.

[0020] In some embodiments, the method further comprises teleporting the logical information to other physical qubits of the plurality of physical qubits by measuring an X-type physical qubit in an X reference; and measuring a Z-type physical qubit in a Z reference.

[0021] In some embodiments, the method further comprises detecting an error in one of the plurality of physical qubits by measuring one of a Z error on the X-type physical qubit or an X error on the Z-type physical qubit.

[0022] In some embodiments, detecting the error includes: detecting a flipped stabilizer by measuring at least one X-type qubit. [Brief description of the drawings]

[0023] BRIEF DESCRIPTION OF THE DRAWINGS Various aspects and embodiments are described with reference to the following drawings. The figures are not necessarily drawn to scale. For clarity, not all components may be labeled in every figure. In the drawings: [Figure 1] 1A and 1B are schematic diagrams illustrating a Raussendorf-Harrington-Goyal (RHG) surface code and an XZZX surface code, according to some embodiments described herein. [Diagram 2] FIG. 2 is a schematic diagram of an example quantum system suitable for initializing RHG and / or XZZX surface codes according to certain embodiments described herein. [Diagram 3] FIG. 3 is a schematic diagram of another example quantum system suitable for initializing RHG and / or XZZX surface codes according to certain embodiments described herein. [Figure 4-1]4A and 4B are schematic diagrams illustrating schematic representations of coupling between two qubits initialized in |+> states and coupling between a qubit initialized in |+> states and an arbitrary state |ψ>, where the coupling is generated by a CZ gate, according to some embodiments described herein. FIG. 4C is a schematic diagram illustrating teleportation of one-dimensional cluster states using logical operators and stabilizer measurements, according to some embodiments described herein. FIG. 4D is a schematic diagram illustrating teleportation of one-dimensional cluster states using only logical operators, according to some embodiments described herein. [Figure 4-2] Fig. 4E is a schematic diagram illustrating the formation of an RHG cluster state by coupling an ancillary qubit to a one-dimensional cluster state so that a multi-qubit logical operator is measured during teleportation according to some embodiments described herein. Fig. 4F is a schematic diagram showing an RHG cluster state according to some embodiments described herein. Fig. 4G is a schematic diagram of X and Z errors in an RHG cluster state according to some embodiments described herein. [Figure 5-1] 5A, 5B, and 5C are schematic diagrams illustrating schematic representations of coupling between (i) a Z-type qubit initialized in the |0> state and an X-type qubit initialized in the |+> state, (ii) a Z-type qubit initialized in the |0> state and an X-type qubit initialized in an arbitrary state |ψ>, and (iii) a Z-type qubit initialized in an arbitrary state |ψ> and an X-type qubit initialized in the |+> state, where the coupling is generated by a CX gate, according to some embodiments described herein. FIG. 5D is a schematic diagram illustrating teleportation of one-dimensional cluster states initiated with an X-type qubit using logical operators and stabilizer measurements, according to some embodiments described herein. FIG. 5E is a schematic diagram illustrating teleportation of one-dimensional cluster states initiated with an X-type qubit using only logical operators, according to some embodiments described herein. [Figure 5-2] 5F is a schematic diagram illustrating teleportation of a one-dimensional cluster state initiated with a Z-type qubit using logical operators and stabilizer measurements, according to some embodiments described herein. FIG. 5G is a schematic diagram illustrating teleportation of a one-dimensional cluster state initiated with a Z-type qubit using only logical operators, according to some embodiments described herein. FIG. 5H is a schematic diagram illustrating the formation of an XZZX cluster state by coupling an ancilla qubit to a one-dimensional cluster state such that a multi-qubit logical operator is measured during teleportation, according to some embodiments described herein. [Figure 5-3] 5I is a schematic diagram of a unit cell of an XZZX surface code according to some embodiments described herein. FIG. 5J is a schematic diagram of the effect of X and Z errors in an XZZX surface code according to some embodiments described herein. [Figure 6] FIG. 6 is a graph illustrating threshold error rates as a function of bias for XZZX and RHG cluster states according to certain embodiments described herein. [Figure 7-1] 7A and 7B are schematic diagrams of exemplary fusion measurements on a pair of X- and Z-type qubits according to some embodiments described herein. Figures 7C and 7D are schematic diagrams of exemplary three- and four-qubit cluster states according to some embodiments described herein. [Figure 7-2] Figure 7E is a schematic diagram illustrating the generation of a larger 5-qubit cluster state by fusing 3-qubit cluster states according to some embodiments described herein. Figure 7F is a schematic diagram illustrating the generation of another 5-qubit cluster state according to some embodiments described herein. Figure 7G is a schematic diagram illustrating the generation of a portion of an XZZX cluster state using a 5-qubit cluster state according to some embodiments described herein. [Figure 7-3]FIG. 7H is a schematic diagram illustrating the alignment of a five-qubit cluster state according to some embodiments described herein. [Figure 8-1] 8A and 8B are schematic diagrams illustrating alternative four-qubit cluster states that can be used to generate an XZZX cluster state according to some embodiments described herein. [Figure 8-2] FIG. 8C is a schematic diagram of an example photonic circuit for generating the four-qubit cluster state of FIGS. 8A and 8B according to certain embodiments described herein. [Figure 9-1] Figure 9A is a schematic diagram illustrating a six-qubit cluster state according to some embodiments described herein. Figures 9B and 9C are schematic diagrams illustrating three-qubit cluster states that can be used to generate the six-qubit cluster state of Figure 9A according to some embodiments described herein. Figure 9D is a schematic diagram illustrating a fusion process used to generate the six-qubit cluster state of Figure 9A according to some embodiments described herein. [Figure 9-2] FIG. 9E is a schematic diagram of an example photonic circuit for generating the three-qubit cluster state of FIGS. 9B and 9C according to certain embodiments described herein. [Figure 9-3] Figure 9F is a schematic diagram of an example photonic circuit for generating the six-qubit cluster state of Figure 9A according to some embodiments described herein. Figures 9G and 9H are schematic diagrams illustrating fusion operations on a pair of Z-type and X-type qubits according to some embodiments described herein. [Figure 9-4] FIG. 9I is a schematic diagram illustrating a fusion pattern used to link six-qubit cluster states to form an XZZX cluster state, according to some embodiments described herein. [Figure 10]FIG. 10 is a plot of the probability of loss per photon as a function of the probability of failure of a fusion operation for the generation of an XZZX cluster state using 4-qubit and 6-qubit cluster states according to some embodiments described herein. [Figure 11] FIG. 11 is a flow chart illustrating a process of generating an XZZX cluster state using fusion measurements according to certain embodiments described herein. [Figure 12] FIG. 12 is a schematic diagram of an exemplary conventional computer system according to certain embodiments described herein. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0024] Detailed Description There are two main approaches to building quantum computers. In circuit model quantum computing, gates are applied to qubits that remain fixed throughout the computation. In contrast, measurement-based quantum computing (MBQC) proceeds by preparing qubits in entangled resource states, including many-body entangled states known as "cluster states". The cluster states can then be used to perform computations by measuring the qubits in a specific basis. Because photons are measured immediately after being prepared, MBQC is particularly suitable for photonic quantum computing platforms, but is also applicable to cavity quantum electrodynamics (cQED) systems, trapped neutral atom quantum computing systems, and trapped ion quantum computing systems. MBQC is also appropriate when physically available gates and other operations are limited in the physical hardware, for example when only destructive single or multi-qubit measurements are available at the physical level. Apart from MBQC, cluster states are also useful for robust quantum communication and networking.

[0025] A major difficulty in building quantum technology is correcting errors in physical quantum hardware. Quantum error-correcting codes allow errors to be detected and corrected in hardware by redundantly sorting information, assuming the probability of error is below some threshold. Error-correcting codes with high error thresholds are desirable because they can tolerate noisier hardware.

[0026] The inventors recognize and understand that error-correcting codes can be embedded in cluster states. A quantum error-correcting code in a circuit model can be converted into an error-correcting cluster state for MBQC using a method known as foliation. A widely used standard cluster state is called the Raussendorf-Harrington-Goyal (RHG) lattice. This cluster state is generated by foliating a standard surface code shown in FIG. 1A. The two-dimensional surface code 100 features a mesh of qubits 102 aligned in alternating plaquette 104 and 106 with X and Z stabilizers. The RHG lattice is constructed by preparing a standard surface code 100 with qubits in the |+> state and then entangling the pair of qubits using a controlled-phase (CZ) gate.

[0027] The inventors further recognize that some qubit hardware exhibits asymmetric or biased noise (e.g., where one type of error, such as phase flips, dominates over others, such as bit flips), and that this feature can be exploited for more efficient error correction. The inventors recognize and understand that the process of foliation converts high probability errors into low probability errors, effectively making the noise channel symmetric; i.e., the process is not bias-preserving. Novel classes of surface codes, such as XZZX codes, have recently been discovered for effective error correction of biased noise in circuit-based approaches. The XZZX surface code 110 is shown diagrammatically in FIG. 1B and includes a meshwork of qubits 112 with identical X and Z stabilizers on their respective plaquette 114.

[0028] The inventors recognize that standard foliation of codes such as the XZZX surface code does not yield high-threshold cluster states even when the underlying qubits have biased noise due to the noise symmetrization effect mentioned above. Therefore, the inventors have developed an error-correcting code for the MBQC system that is bias-preserving. The error-correcting code includes a generalized cluster state as a tool for fault tolerance in a measurement-based model. When noise is dominated by phase flips, a generalized cluster state is constructed by preparing qubits in |0> and |+> states and then entangling pairs of qubits using both controlled-phase (CZ) and controlled-not (CX) gates. Using the generalized cluster state, a foliation protocol that preserves noise bias can be constructed. This property enables the use of the generalized cluster state to construct a high-threshold foliated version of a stabilizer code designed for biased noise in a circuit-based approach. While this method can be applied to any stabilizer code, the description herein explicitly focuses on foliating the XZZX surface code so that it can be efficiently decoded (i.e., errors can be located) by a simple adaptive decoder. Under biased circuit-level noise, this novel cluster state, the XZZX cluster state, is shown to have a noise threshold approximately two times higher than the standard RHG cluster state (i.e., 2.2% vs. <1.0%).

[0029] According to some embodiments, a generalized cluster state is first constructed using entangled linear chains of qubits ("linear cluster states") that are stabilizer states of two sets of stabilizer operators: one set containing products of one type of Pauli operator (e.g., Pauli Z products) and the other set containing products of another type of Pauli operator (e.g., Pauli X products). Such linear cluster states can then be foliated with other stabilizer codes using standard techniques.

[0030] The foliation procedure described herein extends existing foliation procedures using generalized cluster states to generate error-correcting states that can exploit biased noise. Existing foliation procedures do not respect noise bias and convert Z-errors into a mixture of valid Z-errors and valid X-errors. This does not allow for exploitation of circuit model codes that have a high threshold for biased noise. Using generalized cluster states, a foliation procedure that preserves biased noise is developed, resulting in error-correcting cluster states that have a high threshold for biased noise.

[0031] Several experimental platforms have the capability to realize the fundamental operations for generating generalized cluster states. To realize the benefits of XZZX cluster states, high-quality qubits that naturally lend themselves to MBQC and whose errors are biased are desirable. As an example, generalized cluster states can be realized in dual-rail photonic platforms. Furthermore, generalized cluster states can also be realized in circuit-QED, trapped neutral atom systems and optomechanical architectures.

[0032] This application relates to improved quantum error correction techniques for correcting errors in the state of a quantum system. An "error" in this context refers to a change in the state of a quantum system, which may be caused, for example, by a qubit loss, a qubit gain, dephasing, time evolution, etc., of the system, that changes the state of the system such that the information stored in the system is changed.

[0033] I. Exemplary Hardware Implementation FIG. 2 illustrates an exemplary quantum system suitable for implementing aspects of the present application. In system 200, physical qubit 210a is coupled to another physical qubit 210b. Energy source 230 may provide energy to physical qubit 210a, one or both of physical qubits 210b, and / or to the coupling mechanism to perform operations on the system, such as performing a gate operation on one or more of physical qubits 210a and / or 210b, applying other operations to one or more of physical qubits 210a and / or 210b (e.g., to correct detected errors, to construct cluster states using physical qubits 210a and / or 210b), or combinations thereof. Although FIG. 2 illustrates only two physical qubits 210a and 210b, it should be understood that in some embodiments there may be multiple physical qubits 210a / 210b, such that aspects of the technology described herein are not limited in this respect.

[0034] According to some embodiments, physical qubits 210a and / or 210b may include any bosonic system supporting multiple bosonic modes that may be implemented using any electromagnetic, mechanical, magnetic (e.g., quantized spin waves, also known as magnons) technique, and / or other techniques such as, but not limited to, any cavity resonator (e.g., microwave cavity), photonic qubits, trapped neutral atom qubits, and / or optomechanical qubits.

[0035] System 200 also includes system 201 in addition to energy source 230, controller 240, and storage medium 250. In some embodiments, a library of pre-computed drive waveforms may be stored on a computer-readable storage medium and accessed to apply the waveforms to the quantum system. For example, controller 240 may access drive waveforms 252 stored on storage medium 250 (e.g., in response to user input provided to the controller) and control energy source 230 to apply the drive waveforms to physical qubits 210a and / or 210b.

[0036] 3 illustrates another exemplary quantum system suitable for implementing aspects of the present application. System 300 includes a photon source 302 configured to generate single photons and / or photon pairs suitable for use as qubits.

[0037] In some embodiments, the output of the photon source 302 is coupled to an array of optical components 304. The array of optical components 304 may include any suitable optical components configured and aligned to perform one or more quantum operations on photonic qubits generated by the photon source 302. For example, the array of optical components 304 may include a beam splitter 304a, a phase shifter 304b, and / or a photodetector 304c.

[0038] In some embodiments, the array of optical components 304 may be configured to perform a series of operations on the photonic qubits generated by the photon source 302 to assemble a cluster state 306. The cluster state 306 may be assembled in any suitable manner, as described herein. Thus, after passing through the array of optical components 304, the photonic qubits may be output from the array 304 as a cluster state 306.

[0039] II. Building RHG cluster states using teleportation Foliation is a flexible approach to constructing fault-tolerant cluster states from stabilizer codes. As an example of foliation, each qubit in a stabilizer code can be replaced with a one-dimensional cluster state that can be used to teleport a single logical degree of freedom. These one-dimensional cluster states can be coupled such that the code's stabilizers are repeatedly measured during teleportation. These cluster states can then be used to fault-tolerantly store the original encoded state. Although the constructed cluster states do not apply logic gates during teleportation, there are methods to modify the basic fault-tolerant cluster states to enable universal fault-tolerant MBQC.

[0040] To explain the idea of ​​foliation, an overview of how to construct RHG cluster states by foliating a surface code is described herein. The process begins by constructing a one-dimensional cluster state that can teleport a single qubit. Each surface code qubit is then replaced with a one-dimensional teleportation cluster state. During teleportation, the one-dimensional cluster states are coupled such that the surface code stabilizer can be measured. The resulting RHG cluster state can then be used to detect errors.

[0041] 4A and 4B are schematic diagrams illustrating schematic representations of coupling between two qubits initialized in the |+> state and coupling between a qubit initialized in the |+> state and a qubit initialized in an arbitrary state |ψ>, where the coupling is generated by a CZ gate, according to some embodiments described herein. A solid black circle 400 indicates a qubit initialized in the |+> state, and a circle filled with diagonal lines 404 indicates a qubit initialized in an arbitrary state. The qubits connected by a line 402 have a CZ gate applied between them.

[0042] A standard one-dimensional teleportation cluster state is illustrated in Figures 4C and 4D. To prepare a cluster state, an arbitrary state |ψ> is initialized on the first qubit 410, and a |+> state is initialized on the remaining qubits 412. Adjacent qubits are then entangled by applying control phases or CZ gates to all pairs of neighbors. Importantly, the CZ gates can be interchanged and applied in any order.

[0043] The stabilizer formalism can be used to describe cluster states. Before the CZ gate, the logical Pauli operator on |Ψ> is L = X1 and Z L =Z1, and the state is stabilized by {X2,X3,...}. After applying the CZ gate, new logical operators and stabilizers are obtained from the old ones by conjugating with the CZ gate. This is

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[0044] The case of n=3 is illustrated in Figure 4C. This formalism makes it clear that if the first 2n qubits are measured in the X-based, the logical operators are teleported to qubits {2n+1, 2n+2}. For example, for n=3, qubits 1 through 6 are teleported to x with outcomes {x1, x2, x3, x4, x5, x6}. i If it is measured with = ±1, the logical operators are:

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[0045] To construct a fault-tolerant cluster state, individual one-dimensional teleportation cluster states can be combined, and the cluster can be modified to measure the check operator of the stabilizer code when the cluster is teleported. To generate a surface code, each qubit of the surface code shown in Figures 4A and 4B has a one-dimensional teleportation chain. As shown in Figure 4E, for each X stabilizer plaquette 106 of the surface code, an ancilla qubit can be attached to a respective pair of odd sites of the teleportation chain, and for each Z stabilizer plaquette 104, we attach an ancilla to a respective pair of even sites of the teleportation chain. These ancilla qubits are also initialized with |+> and entangled with their neighbors with CZ gates. Measuring an ancilla qubit in the X basis results in measuring the surface code stabilizer of the corresponding plaquette during teleportation.

[0046] The resulting three-dimensional cluster state, the RHG cluster state, is illustrated in FIG. 4F, with the unit cell 420 highlighted in dashed lines. To fault-tolerantly teleport information through this state, both the data qubits and the ancilla qubits are measured in the X-reference. For an error-free RHG cluster state, it can be checked that the product of X measurements around the face of the cell should be (+1), since the product of X operators on the face of the cell is a stabilizer of the cluster state.

[0047] As an example, consider applying a single Pauli error to an RHG cluster state. A Z or Y error 430 on a surface qubit will repel the syndromes of two neighboring cells, allowing for the detection of these errors as shown in FIG. 4G. Multi-qubit Pauli errors can be similarly detected by considering the syndromes they repel. Note that an X error 432 on the final cluster state has no effect, but an X error occurring between two CZ gates propagates to a Z error on the neighboring qubit. The error can be corrected by pairing the (-1) syndromes against each other using a minimum weight perfect matching (MWPM) decoder, under which the RHG cluster has a threshold for logical Pauli noise.

[0048] This method of generating fault-tolerant cluster states can also be used to generate cluster states that implement tuned or XZZX surface codes. However, these codes only offer improved thresholds over ordinary surface codes in that the effective probability of bit-flip errors is suppressed compared to phase-flip errors. Unfortunately, the one-dimensional teleportation procedure outlined above unbiasses the noise and converts physical Z errors into logical X errors. We can understand this phenomenon in two ways. First, Z L Since the operators involve physical X operators on qubits 2, 4, ..., 2n, the Z errors on these qubits are Z L and so X L Alternatively, if logical information from qubits 1 and 2 is teleported to, say, qubits 7 and 8, then accurate measurements x1,...,x6 are needed to recover the logical information about that state. A Z error on qubits 2, 4 or 6 will cause an erroneous measurement of the sign of x2, x4 or x6. This is equivalent to a Z L -Z L This results in a replacement with X L is equivalent to an error.

[0049] Thus, physical Z noise is transformed into logical X noise. As a result, the effective error channels of the one-dimensional teleportation chain, which are then combined to measure the check operators of the error-correcting code, are not biased. The resulting fault-tolerant cluster states therefore do not have enhanced thresholds even when the measured stabilizers correspond to codes that are specifically designed for biased noise, such as the XZZX surface code. In fact, the XZZX cluster states resulting from this approach do not perform better than standard RHGs at any bias.

[0050] Naively, one can deduce that since X-errors on the RHG cluster state have no effect on teleportation, the RHG lattice is not robust to Z-bias noise, but it should be robust to X-bias noise. However, even if the physical qubit experiences only X-errors, the CZ c,t , where c denotes the control and t denotes the target, is applied to c X for mistake c Z t Therefore, if X-errors occur during the construction of the RHG lattice, they propagate to Z-errors on the final cluster, and it cannot be assumed that there is X-biased noise after the construction of the cluster state. In contrast, the CZ gate commutes with Z-errors, and so Z-biased noise is compatible with constructing the cluster state.

[0051] III. Construction of XZZX cluster states using teleportation To construct a cluster state that realizes the XZZX code, the usual construction of the cluster state is Z Lcan be modified with the goal of generating one-dimensional teleportation cluster states that contain only physical Z operators. To achieve this, a generalized cluster state is constructed with two types of qubits, X-type and Z-type. The X-type qubits are initialized in the |+> state and measured in the X-reference, similar to the normal cluster state, and the Z-type qubits are initialized in the |0> state and measured in the Z-reference. As shown in Figures 5A-5C, the X-type qubits are indicated by small filled circles (e.g., qubits 500 and 506) and the Z-type qubits are indicated by large open circles (e.g., qubits 502 and 508). To entangle adjacent qubits, different gates are applied depending on the type of qubit to be entangled. To entangle two X-type qubits, a normal CZ gate is applied, and to entangle X and Z-type qubits, a CX gate 504 is applied, where the X-type qubit is the control qubit and the Z-type qubit is the target qubit. Importantly, entangling gates are still interchangeable and may be applied in any order. In this construction, both CX and CZ gates must be bias-preserving. Z-bias-preserving CX gates have already been proposed in multiple qubit platforms, and since Z errors are naturally commutative with CZ, CZ gates naturally preserve Z-bias.

[0052] This generalized construction allows the construction of two separate one-dimensional teleportation cluster states, shown in Figures 5D-5G. The first one, shown in Figures 5D and 5E, is an X-initial cluster state, which starts with state |Ψ> on the first qubit 506, followed by alternating Z-type qubits 502 and X-type qubits 500. The first qubit is not initialized in the |+> state, but it is treated as an X-type qubit during entanglement and measurement. The second one-dimensional teleportation cluster state, shown in Figures 5F and 5G, is a Z-initial cluster state, which starts with state |Ψ> on the first qubit 506, followed by alternating X-type qubits 500 and Z-type qubits 502. Here the first qubit 506 is treated as a Z-type qubit during entanglement and measurement.

[0053] After applying the entanglement gate, new logical operators and stabilizers are obtained from the previous ones by conjugation with the entanglement gate, which is:

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[0054] The case of n=3 for both clusters is illustrated in Figures 5D-5G. This formalism reveals that when the first n X-type qubits are measured in the X basis and the first n Z-type qubits are measured in the Z basis, the logical operators are teleported to qubits {2n+1,2n+2}. For example, when n=3 for the X starting cluster state, qubits 1-6 are teleported to x with outcomes {x1,z2,x3,z4,x5,z6}. i ,z i If the equations above are measured with the logical operators:

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[0055] Importantly, in this cluster state, Z L is the product of the physical Z operator, X L is the product of physical X operators, and physical Z errors are logical X L This means that it cannot introduce errors. Therefore, both teleportation clusters preserve the noise bias. These bias-preserving teleportation clusters can be used to foliate any biased noise stabilizer code to gain large threshold advantages that were not possible using conventional approaches.

[0056] To construct the bias-preserving cluster states that realize the XZZX code, it is most convenient to use an alternating grid of X-start and Z-start cluster states, as shown in FIG. 5H. For each plaquette 114 of the XZZX code, we add an X-type ancilla qubit. These ancilla qubits are also initialized in the |+> state and entangled with their neighbors. Measuring the ancilla qubit in the X basis results in measuring the XZZX code stabilizer of the corresponding plaquette during teleportation. The resulting XZZX cluster state is shown in FIG. 5I along with the unit cell 520.

[0057] Much like an XZZX surface code can be obtained from an ordinary surface code by conjugating stabilizers with H on alternating qubits, an XZZX cluster state can be obtained from an RHG cluster state by applying Hadamard (H) gates at the sites of Z-type qubits. However, since H operations do not preserve noise biases, it is important to physically construct the XZZX cluster state with CX and CZ gates rather than applying H gates to the RHG lattice.

[0058] Each cell of the XZZX cluster state has four X-type qubits and two Z-type qubits on its face; it is straightforward to show that for an error-free XZZX cluster state, the product of the X operator on the X-type qubit and the Z operator on the Z-type qubit is a stabilizer of the state, so the product of the corresponding X and Z measurements should be (+1). As shown in FIG. 5J, a Z or Y error on an X-type qubit 532 will shed syndromes of neighboring cells, as will an X or Y error on a Z-type qubit 530. Note that while an X error on the entire cluster state has no effect on the X-type qubit and a Z error on the entire cluster state has no effect on the Z-type qubit, a Z error on a Z-type qubit that occurs between two CX gates will propagate to the Z error on the neighboring X-type qubit and to the X error on the X-type qubit as well. The use of bias-preserving CX gates ensures that the Z error does not propagate to the X or Y error. Overall, the errors can be corrected again by pairing the (-1) syndromes against each other using the MWPM decoder. Importantly, we observe that Z-errors in the XZZX cluster state generate error chains that are confined to truncated 2D surfaces, while Z-errors in the RHG cluster state generate error chains that can meander in three dimensions. Intuitively, this makes it easier to decode the XZZX cluster state in the presence of biased noise. An effective reduction in the dimensionality of the adaptation graph in the biased noise case is the mechanism for the adjusted and increased threshold in the XZZX surface code.

[0059] To demonstrate the advantages of the XZZX cluster state in the presence of biased noise, we performed full circuit-level noise simulations for both the XZZX and RHG cluster states. The simulations use a physically well-inspired biased noise model. In this model, the CZ c,t The gate is a gate with probability p z Error with

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[0060] Also, to verify an earlier assertion that RHG cluster states should not have significantly higher thresholds under X-bias noise, RHG cluster states are simulated under X-bias noise. For X-bias noise in RHG cluster states, a physically inspired error model based on a specific implementation of the CZ gate is used. Note that even if the X-bias noise is estimated to be on the physical qubit, the noise of the CZ gate is not expected to be X-biased, since the CZ gate does not preserve the X-bias. In this model, the CZ c,t After applying the gate, error I c X t , X c I t , Z c X t and X c Z t The probability is 0.375p x occurs with error I c Y t , Y c It , Z c Y t and Y c Z t The probability is 0.125p x and all other errors occur with probability p x In addition to errors during the CZ gates, X errors occur with probability p x and Y and Z errors occur with probability p x It occurs with η.

[0061] In the Z-biased noise model, the total error probability of the CZ gate is

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[0062] FIG. 6 is a graph showing threshold error ratios as a function of bias for the XZZX and RHG cluster states, according to some embodiments described herein. In FIG. 6, results are plotted for thresholds of 1≦η≦10000. At η=1, the thresholds for all three cluster states are similar. As η is increased, the threshold for the RHG cluster state with X-biased noise outperforms the RHG cluster state with Z-biased noise; however, our error threshold for the XZZX cluster state strongly outperforms both versions of the RHG cluster state. For high bias, η≧1000, the threshold for the XZZX cluster state is p th >2.2%, and is greater than twice the threshold for RHG cluster states with Z-biased noise, and the p th <1.0%.

[0063] IV. Constructing XZZX cluster states using fusion In the most common measurement-based error correction (MBEC) framework, the cluster state is generated using a set of commuting two-qubit entanglement gates. Alternatively, one can start with several copies of a smaller few-field entangled state and then fuse them together with the two-qubit Pauli operator, also called a fusion or Bell measurement:

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[0064] Therefore, we developed a fusion-based architecture for error correction using XZZX cluster states. We developed two constructs, one based on the use of an ensemble of 4-qubit entangled resource states and the other based on the use of an ensemble of 6-qubit entangled resource states. Importantly, both constructs are biased in the noise in the fusion circuit,

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[0065] The fusion-based techniques described herein are inspired by dual-rail qubits in linear optical properties, the most widely tested platform in the FBEC framework. Linear optical property fusion on dual-rail qubits is inherently promising. The simplest fusion circuits fail with probability 1 / 2. Ancillary (2 n -2) Using photon entanglement, the failure rate is 1 / 2 n For the special case of n=2, four unentangled photons are sufficient to achieve a 1 / 4 failure probability. Notably, if a fusion attempt fails,

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[0066] Given the significant increase in the error threshold of the XZZX cluster state under bias noise, the next step is to consider which qubit platforms might naturally exploit the increased threshold. However, in this section, we make some preliminary proposals for quantum computing approaches that could benefit from realizing the XZZX cluster state, distinct from other forms of error correction. Each of the following proposals uses a two-step strategy to realizing the XZZX cluster state, where a small cluster state is first generated (the "initial resource state"), and then the XZZX cluster state is generated by fusing X-type qubits and Z-type qubits using Bell measurements. Such fusion is bias-preserving: Z errors on qubits involved in the fusion will produce Z errors on the final cluster state. Moreover, as shown for the specific platforms below, any further errors introduced during fusion are also Z-biased. Thus, all of these constructions produce a final cluster state with Z-biased noise.

[0067] The leading candidate for realizing the XZZX cluster state is the double-rail encoded photonic qubit, where the qubit is represented by a photon in one of two photonic modes.

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[0068] 7A-7J illustrate one way of constructing an XZZX cluster state by fusion. Figures 7A and 7B show two fusion processes for dangling X-type and Z-type qubits. Consider a cluster state defined on a graph G = (V, E). Let G have v i' A degree-1 vertex v with an edge to the qubit in i Z-qubits in and v j' ≠v i' A degree-1 vertex v with an edge to the qubit in j 7A and 7B, we have an X-shaped qubit. The qubit on the vertex of order 1 is referred to herein as a "dangling qubit." On a pair of dangling qubits,

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[0069] 7C and 7D are schematic diagrams of example three- and four-qubit cluster states according to some embodiments described herein. The three-qubit cluster state of FIG. 7C may be generated by initializing two X-type qubits and one Z-type qubit in the |+> state and performing a three-qubit Z measurement 700 to couple the three qubits. The four-qubit cluster state of FIG. 7D may be generated by initializing three Z-type qubits and one X-type qubit in the |+> state and performing a two-qubit Z measurement 702 to couple the qubits as shown.

[0070] 7E and 7F illustrate the creation of a larger five-qubit cluster state by fusing the elementary cluster states of FIGS. 7C and 7D, according to some embodiments described herein. The five-qubit cluster state of FIG. 7E is a non-destructive

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[0071] The five-qubit cluster state in FIG. 7E has a Z-type qubit at its center, and the five-qubit cluster state in FIG. 7F has an X-type qubit at its center. The center qubit is finally used to form the desired XZZX cluster state. The Z-centered and X-centered states are placed in the respective positions of the Z- and X-type qubits in the desired cluster state, as shown in FIG. 7H. This alignment of the five-body states ensures that adjacent suspended qubits are always of opposite types; adjacent suspended qubits can be fused according to FIGS. 7A and 7B. These initial resource states, including five-qubit cluster states in some embodiments, can then be fused to generate a larger XZZX cluster state. FIG. 7G illustrates the generation of a portion of the XZZX cluster state using the five-qubit cluster states of FIGS. 7E and 7F. To fuse these five-qubit cluster states, non-destructive transfer between the qubits of the two five-qubit cluster states is performed.

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[0072] The fused qubit is removed from the cluster and new coupling occurs between the central qubits to produce the desired XZZX cluster state. Finally, the cluster state qubits can be measured in the appropriate criteria described in the previous section on error correction. Note that each central qubit is entangled in the final cluster state after four fused measurements on its neighboring suspended qubits; consequently, four Pauli corrections are accounted for on this qubit. This can be done in software by simply reinterpreting the result of the final measurement of the cluster state qubit. This means that applying an X(Z) measurement of an X(Z) type qubit in a cluster state after Pauli Z(X) to it results in a classical flip of the X(Z) measurement, followed by the measurement result.

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[0073] It is possible to simplify the initial 5-qubit cluster state to a 4-qubit cluster state by noting that measuring the qubits that comprise the final cluster state is interchangeable with the fusion measurement, since these measurements are performed on different qubits and Pauli corrections, so that fusion can be simply accounted for in software. That is, the central qubit forming the XZZX cluster state can be measured before performing the fusion measurement. Once fusion is realized, the result of the previous central qubit measurement can be conditionally rejected on the fusion result. By measuring the central X and Z type qubits of the 5-qubit state in the X and Z references, respectively, a simpler 4-star resource state is generated, as shown in Figures 8A and 8B. Thus, the process of generating the XZZX cluster state can start directly with these 4-qubit resource states, presuming that the central qubit has been measured in the X / Z reference with the +1 measurement result. In this approach, the central qubit acts similarly to a virtual qubit. It is never physically realized, never physically measured, and its effective measurement result is entirely determined by the Pauli corrections tracked in software.

[0074] FIG. 8C is a schematic diagram of an example photonic circuit 800 for generating the four-qubit cluster state of FIG. 8A according to some embodiments described herein. Photonic circuit 800 includes several single photon sources 802, 50:50 beam splitters 804, and photodetectors 806. The four-qubit cluster state can be probabilistically generated from a single photon by a series of beam splitters and photon detectors that condition on specific detector outputs. Photonic circuit 800 proceeds when a single photon is output at each detector pair. Photonic circuit 800 can be modified to generate the four-qubit cluster state of FIG. 8B by further including a beam splitter applied to three of the four qubits that are output from photonic circuit 800.

[0075] Alternatively, an XZZX cluster state can be generated using a 6-qubit resource state, an example of which is shown in Figure 9A. The 6-qubit resource state can be generated based on the 3-qubit GHZ state illustrated in Figures 9B and 9C. The 6-ring resource state can then be constructed by performing fusion on a set of 3 GHZ states, as shown in Figure 6D.

[0076] In some embodiments, a three-qubit GHZ state can be generated using the circuit of FIG.

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[0077] To generate an XZZX cluster state using these 6-qubit resource states, qubits having 6-qubit resource states of the same type (e.g., X or Z) can be fused in some embodiments and as illustrated in Figures 9G and 9H.

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[0078] To generate an XZZX cluster state using the six-ring resource cluster state of FIG. 9A, two copies of the six-qubit cluster state can be placed at opposite corners of each unit cell of the XZZX cluster state, as shown in FIG. 9I. Two qubits of the same type share a face or edge.

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[0079] So far, imperfections in the actual hardware that can introduce photon losses have been ignored. Fortunately, an ideal fusion circuit would conserve the number of photons, i.e. all detector clicks should be equal to the number of input photons. Therefore, any photon loss in the fusion circuit would be preceded by observing fewer than expected clicks at the detectors, and both

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[0080] FIG. 10 is a threshold curve for XZZX cluster states generated using 4-qubit and 6-qubit resource states according to some embodiments described herein. Curves 1002 and 1006 show threshold curves as simulated for the techniques for constructing XZZX cluster states described herein. In contrast, curves 1004 and 1008 show threshold curves as simulated for previous techniques for constructing XZZX cluster states. Both curves 1002 and 1006 show higher error thresholds than curves 1004 and 1008. Furthermore, the threshold for the 6-ring construction is higher than the 4-star construction in having fewer fusions and therefore a lower probability of error per cluster state qubit. In the absence of photon loss, the numerically obtained thresholds for biased fusion failure using our scheme are about 30.5% for the 6-ring construction and about 19% for the 4-star construction. These numerically obtained thresholds are significantly higher than the thresholds for unbiased fusion failures obtained using previous proposals, which are correspondingly about 24% for the 6-ring construction and about 14.5% for the 4-star construction. As described earlier, this is because pure fusion failures result in two-dimensional system symmetry and therefore a two-dimensional syndrome graph that is easier to decode. In contrast, using previous strategies, the syndrome graph resulting from pure fusion failures is three-dimensional, which is more difficult to decode.

[0081] By utilizing the biased structure of fusion failure, we have introduced a novel resource state and fusion strategy for FBEC that allows more efficient error correction of biased fusion failure. This FBEC strategy is particularly relevant for linear optical quantum computers based on double-rail photonic qubits, where biased fusion failure is the dominant source of error. The resource state and fusion strategy described herein does not require additional overhead to implement, but produces a higher threshold for fusion failure for both 4-star and 6-ring resource states. In particular, in the 6-ring construction, the described technique has a threshold for fusion failure of more than 25%, which can be achieved using only 2-photon entangled ancillaries or 4-photon non-entangled ancillaries; therefore, the described construction overcomes a significant barrier for photonic quantum computer computation.

[0082] 11 is a flowchart describing a process 1100 for generating an XZZX cluster state using a fusion measurement according to some embodiments described herein. In some embodiments, the process 1100 may begin at act 1102, where a quantum state is initialized with a first qubit. The first qubit may include an X-type and a Z-type qubit.

[0083] In some embodiments, the first qubit may include a photonic qubit. In such embodiments, initializing a state in the first qubit may include generating the photonic qubit using one or more light sources. For example, the photonic qubit may be generated by one or more single photon sources.

[0084] In some embodiments, the first qubit may include a trapped neutral atom. In such embodiments, initializing the state in the first qubit may include exciting the state of the trapped neutral atom by driving the trapped neutral atom with a driving signal. For example, the driving signal may be an optical and / or microwave signal. Thus, initializing the state in the first qubit may include generating one or more optical or microwave signals using one or more optical or microwave sources; and transmitting the generated one or more optical or microwave signals to the first qubit to initialize the state.

[0085] In some embodiments, after act 1102, process 1100 may proceed to act 1104, where an initial resource state (e.g., an initial cluster state) is generated by performing a first Pauli product measurement on a set of X and / or Z qubits of the first qubit. The initial resource state may include a qubit cluster state including at least three qubits.

[0086] In some embodiments, generating the initial resource state includes generating a first three-qubit cluster state. The first three-qubit cluster state may be generated by first initializing a state in two X-type qubits and one Z-type qubit. A three-qubit Z measurement may then be performed on the initialized qubits to generate the first three-qubit cluster state by coupling the three qubits.

[0087] In some embodiments, generating the initial resource state includes generating a five-qubit cluster state using one or more of the first three-qubit cluster states. Generating the five-qubit cluster state may include fusing each of the two X-type qubits of the first three-qubit cluster state with a Z-type qubit of the second and third three-qubit cluster states. Fusing the two X-type qubits with the Z-type qubit may include making a fusion measurement between one of the X-type qubits and the respective Z-type qubit (e.g., performing a two-qubit Z measurement and a two-qubit X measurement, performing a Bell measurement).

[0088] In some embodiments, generating the initial resource state includes generating a four-qubit cluster state. The four-qubit cluster state may be generated by first initializing a state in three Z-type qubits and one X-type qubit. The four-qubit cluster state may then be generated by performing a two-qubit Z measurement between each of the three Z-type qubits and one X-type qubit. In some embodiments, a five-qubit cluster state may further be generated using the four-qubit cluster state. The five-qubit cluster state may be generated by first initializing a state in an additional X-type qubit and then performing a CZ gate between the additional X-type qubit and one X-type qubit of the four-qubit cluster state.

[0089] In some embodiments, generating the initial resource state includes separately generating a four-qubit cluster state. The four-qubit cluster state may be generated by first initializing a state with four X-type qubits. The four-qubit cluster state may then be generated by performing two-qubit X-measurements between pairs of qubits of the initialized four X-type qubits. In some embodiments, the four-qubit cluster state may be further generated by performing three Z-measurements of the initialized four X-type qubits.

[0090] In some embodiments, generating the initial resource state includes generating a six-qubit cluster state (e.g., a six-ring cluster state). The six-qubit cluster state may be generated by first generating three three-qubit cluster states. The three-qubit cluster state may be generated by initializing a state with six X-type qubits and three Z-type qubits, and performing at least four two-qubit X and / or Z measurements on each of the three three-qubit cluster states to generate three three-qubit cluster states. The qubits in the three-qubit cluster states may then be fused to generate a six-qubit cluster state. The six-qubit cluster state may include two Z-type qubits and four X-type qubits.

[0091] In some embodiments, after act 1104, process 1100 may proceed to act 1106, where the XZZX cluster state is generated by fusing two or more initial resource states, the fusing comprising performing a second Pauli product measurement between two or more qubits of the initial resource states. In some embodiments, the fusing of the two or more initial resource states comprises performing a Bell measurement between a first qubit of the first initial resource state and a second qubit of the second initial resource state.

[0092] An exemplary implementation of a classical computer system 1200 that may be used in connection with any of the embodiments of the disclosure provided herein is shown in FIG. 12. In some embodiments, any one of the processes described herein may be executed on and / or using the computer system 1200. The computer system 1200 may include one or more articles of manufacture including one or more processors 1210 and non-transitory computer-readable storage media (e.g., memory 1220 and one or more non-volatile storage media 1230). The processor 1210 may control the writing and reading of data to and from the memory 1220 and the non-volatile storage device 1230 in any suitable manner. To perform any of the functions described herein, the processor 1210 may execute one or more processor-executable instructions stored in one or more non-transitory computer-readable storage media (e.g., memory 1220) that may act as a non-transitory computer-readable storage medium that stores processor-executable instructions for execution by the processor 1210.

[0093] Although several aspects and embodiments of the technology described in this disclosure are thus described, it is understood that various changes, modifications, and improvements will be readily implemented by those skilled in the art. Such changes, modifications, and improvements are intended to be within the spirit and scope of the technology described herein. For example, one skilled in the art will readily envision various other means and / or structures for performing the functions and / or obtaining the results and / or one or more of the advantages described herein, and each of such changes and / or modifications is considered to be within the scope of the embodiments described herein. One skilled in the art will recognize or be able to ascertain using no more than routine experimentation many equivalents to the specific embodiments described herein. Thus, it is understood that the foregoing embodiments are presented by way of example only, and that within the scope of the appended claims and their equivalents, the embodiments of the invention may be practiced other than as specifically described. Also, any combination of two or more features, systems, articles, materials, kits, and / or methods described herein is included within the scope of the present disclosure, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent.

[0094] The above-described aspects may be implemented in any of many ways. One or more aspects and embodiments of the present disclosure, including the performance of a process or method, may utilize program instructions executable by a device (e.g., a computer, a processor, or other device) to perform or control the performance of the process or method. In this aspect, the various inventive concepts may be embodied as a computer-readable storage medium (or multiple computer-readable storage media) (e.g., a computer memory, one or more floppy disks, compact disks, optical disks, magnetic tapes, flash memory, circuitry within a field programmable gate array or other semiconductor device, or other tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform a method for performing one or more of the various aspects described above. The computer-readable medium(s) may be transportable such that the program(s) stored thereon may be loaded onto one or more different computers or other processors to perform various of the above-described aspects. In some embodiments, the computer-readable medium may be a tangible (e.g., non-transitory) computer-readable medium. In some embodiments, the computer-readable medium may include a persistent memory.

[0095] The term "program" or "software" is used herein in a general sense to refer to any type of computer code or set of computer executable instructions that can be used to program a computer or other processor to perform the various aspects described above.Furthermore, according to one aspect, it should be understood that one or more computer programs that, when executed, perform the methods of the present disclosure do not have to reside on a single computer or processor, but may be distributed in a modular manner among several different computers or processors to perform various aspects of the present disclosure.

[0096] Computer-executable instructions may exist in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0097] When implemented in software, the software code may be executed on any suitable processor or collection of processors, whether provided on a single computer or distributed among multiple computers.

[0098] Further, it should be understood that the computer may be embodied in any of a number of forms, such as, by way of non-limiting examples, a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, the computer may be embedded in a device not typically considered a computer, but having suitable processing capabilities, such as a personal digital assistant (PDA), a smartphone, or any other suitable portable or fixed electronic device.

[0099] A computer may also have one or more input and output devices. These devices may be used, among other things, to present a user interface. Examples of output devices that may be used to provide a user interface include a printer or display screen for visual presentation of output and a speaker or other sound generating device for audible presentation of output. Examples of input devices that may be used for a user interface include a keyboard and pointing devices, such as a mouse, a touchpad, and a digitizing tablet. As another example, a computer may receive input information via speech recognition or in other audible formats.

[0100] Such computers may be interconnected by one or more networks of any suitable form, such as local area networks or wide area networks, e.g., enterprise networks and intelligent networks (IN) or the Internet, etc. Such networks may be based on any suitable technology and may operate according to any suitable protocol, and may include wireless networks, wired networks or fiber optic networks.

[0101] Also, as described, some aspects may be embodied as one or more methods. The acts performed as part of a method may be ordered in any suitable manner. Thus, even if shown as sequential acts in an example embodiment, the acts may be performed in an order different from that shown, and embodiments may be constructed that may include performing some acts simultaneously.

[0102] All definitions defined and used herein should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0103] The indefinite articles "a" and "an," as used in the specification and claims, unless clearly indicated to the contrary, should be understood to mean "at least one."

[0104] The phrase "and / or" when used in the specification and claims should be understood to mean "either or both" of the elements so connected, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with "and / or" should be interpreted in the same manner, i.e., "one or more" of the elements so connected. Other elements, whether related or unrelated to those elements specifically identified, may optionally be present other than the elements specifically identified by the "and / or" clause. Thus, as a non-limiting example, a reference to "A and / or B", when used in conjunction with open-ended language such as "comprising", may refer in one embodiment to A only (optionally including elements other than B); in another embodiment to B only (optionally including elements other than A); in yet another embodiment to both A and B (optionally including other elements); and so forth.

[0105] As used in the specification and claims, the phrase "at least one" should be understood, in reference to a list of one or more elements, to mean at least one element selected from any one or more of the elements in the list of elements, without necessarily including at least one of each and every element specifically listed in the list of elements, but without excluding any combination of elements in the list of elements. This definition also allows for the optional presence of elements other than those specifically identified in the list of elements to which the phrase "at least one" refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, "at least one of A and B" (or, equivalently, "at least one of A or B) or, equivalently, "at least one of A and / or B") can refer, in one embodiment, to at least one that optionally includes more than one A in the absence of B (and optionally including elements other than B); in another embodiment, to at least one that optionally includes more than one B in the absence of A (and optionally including elements other than A); in yet another embodiment, to at least one that optionally includes more than one A, and at least one that optionally includes more than one B (and optionally including other elements); etc.

[0106] In the claims and the foregoing specification, all transitional phrases, such as "comprising," "including," "carrying," "having," "containing," "involving," "holding," "composed of," and the like, are understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases "consisting of" and "consisting essentially of" are intended to be closed or semi-closed transitional phrases, respectively.

[0107] The terms "approximately" and "about" can be used to mean, in some embodiments, within ±20% of a target value, in some embodiments, within ±10% of a target value, in some embodiments, within ±5% of a target value, and in some embodiments, within ±2% of a target value. The terms "approximately" and "about" can include the target value.

Claims

1. at least one controller; and 1. A quantum system comprising: at least one non-transitory computer-readable medium storing computer-readable instructions configured to cause at least one controller to generate an XZZX cluster state, wherein the generating comprises: initializing states in first qubits, where the first qubits include X-type and Z-type qubits; generating an initial resource state by performing a first Pauli product measurement on a set of X-type and / or Z-type qubits of the first qubit, where the initial resource state comprises a qubit cluster state including at least three qubits; and generating an XZZX cluster state by fusing two or more initial resource states, where fusing includes performing a second Pauli product measurement between qubits in the two or more initial resource states; Quantum systems, including:

2. 10. The quantum system of claim 1, further comprising a single photon source, wherein initializing the state in the first qubit comprises generating the photonic qubit using the single photon source.

3. To create the initial resource state: initializing states in two X-type qubits and one Z-type qubit; and generating a first three-qubit cluster state by performing a three-qubit Z measurement on the initialized qubits; 3. The quantum system of claim 1, further comprising generating the first three-qubit cluster state by:

4. 4. The quantum system of claim 3, wherein generating the initial resource state comprises generating a five-qubit cluster state by fusing each of two X-type qubits of the first three-qubit cluster state with a Z-type qubit of the second and third three-qubit cluster states.

5. Fusing each of the two X-type qubits of the first three-qubit cluster state with the Z-type qubits of the second and third three-qubit cluster states: performing a two-qubit Z measurement between each of the two X-type qubits and a Z-type qubit; and Performing two-qubit X measurements between each of the two X-type qubits and a Z-type qubit 5. The quantum system of claim 4, comprising:

6. To create the initial resource state: initializing states in three Z-type qubits and one X-type qubit; and Generating a four-qubit cluster state by performing two-qubit Z measurements between each of three Z-type qubits and one X-type qubit.

3. The quantum system of claim 1, further comprising generating a four-qubit cluster state by:

7. To create the initial resource state: initializing a state in a further X-type qubit; and Performing a CZ gate between an additional X-type qubit and one X-type qubit in a four-qubit cluster state 7. The quantum system of claim 6, comprising generating a five-qubit cluster state by:

8. To create the initial resource state: initializing states in four X-type qubits; and Generating a four-qubit cluster state by performing a two-qubit X measurement between pairs of qubits in an initialized four-X-type qubit cluster.

3. The quantum system of claim 1, further comprising generating a four-qubit cluster state by:

9. 10. The quantum system of claim 8, wherein generating the four-qubit cluster state further comprises performing three Z measurements of the initialized four X-type qubits.

10. To create the initial resource state: generating three three-qubit cluster states; and Fusing qubits from three three-qubit cluster states to generate a six-qubit cluster state containing two Z-type qubits and four X-type qubits 3. The quantum system of claim 1, further comprising generating a six-qubit cluster state by:

11. Three three-qubit cluster states can be generated: initializing states in six X-type qubits and three Z-type qubits; and performing at least four two-qubit X and / or Z measurements on each of the three three-qubit cluster states to generate three three-qubit cluster states; The quantum system of claim 10, comprising:

12. 3. The quantum system of claim 1, wherein fusing two or more initial resource states comprises performing a Bell measurement between a first qubit in a first initial resource state and a second qubit in a second initial resource state.

13. a plurality of physical qubits including neutral trapped atoms; and One or more optical or microwave sources coupled to multiple physical qubits The quantum system of claim 1 further comprising:

14. 1. A method of generating an XZZX cluster state for use in quantum information processing, comprising: initializing states in first qubits, where the first qubits include X-type and Z-type qubits; generating an initial resource state by performing a first Pauli product measurement on a set of X and / or Z qubits of the first qubit, where the initial resource state comprises a qubit cluster state including at least three qubits; and generating an XZZX cluster state by fusing two or more initial resource states, where fusing includes performing a second Pauli product measurement between qubits in the two or more initial resource states; A method comprising:

15. 15. The method of claim 14, wherein initializing a state in the first qubit comprises generating the photonic qubit using a single photon source.

16. To create the initial resource state: initializing states in two X-type qubits and one Z-type qubit; and generating a first three-qubit cluster state by performing a three-qubit Z measurement on the initialized qubits; 16. The method of claim 14 or 15, comprising generating the first three-qubit cluster state by:

17. 16. The method of claim 14 or 15, wherein generating the initial resource state comprises generating a five-qubit cluster state by fusing each of two X-type qubits of the first three-qubit cluster state with a Z-type qubit of the second and third three-qubit cluster states.

18. Fusing each of the two X-type qubits of the first three-qubit cluster state with the Z-type qubits of the second and third three-qubit cluster states: performing a two-qubit Z measurement between each of the two X-type qubits and a Z-type qubit; and Performing two-qubit X measurements between each of the two X-type qubits and a Z-type qubit 20. The method of claim 17, comprising:

19. To create the initial resource state: initializing states in three Z-type qubits and one X-type qubit; and Generating a four-qubit cluster state by performing two-qubit Z measurements between each of three Z-type qubits and one X-type qubit.

16. The method of claim 14 or 15, comprising generating a four-qubit cluster state by:

20. To create the initial resource state: initializing a state in a further X-type qubit; and Performing a CZ gate between an additional X-type qubit and one X-type qubit in a four-qubit cluster state 20. The method of claim 19, comprising generating a five-qubit cluster state by:

21. To create the initial resource state: initializing states in four X-type qubits; and Generating a four-qubit cluster state by performing a two-qubit X measurement between pairs of qubits in an initialized four-X-type qubit cluster.

16. The method of claim 14 or 15, comprising generating a four-qubit cluster state by:

22. 22. The method of claim 21, wherein generating the four-qubit cluster state further comprises performing three Z measurements of the initialized four X-type qubits.

23. To create the initial resource state: generating three three-qubit cluster states; and Fusing qubits from three three-qubit cluster states to generate a six-qubit cluster state containing two Z-type qubits and four X-type qubits 16. The method of claim 14 or 15, comprising generating a six-qubit cluster state by:

24. Three three-qubit cluster states can be generated: initializing states in six X-type qubits and three Z-type qubits; and performing at least four two-qubit X and / or Z measurements on each of the three three-qubit cluster states to generate three three-qubit cluster states; 24. The method of claim 23, comprising:

25. 16. The method of claim 14 or 15, wherein fusing two or more initial resource states comprises performing a Bell measurement between a first qubit in a first initial resource state and a second qubit in a second initial resource state.

26. The first qubit includes a plurality of neutral trapped atom qubits, and initializing a state in the first qubit includes: generating one or more optical or microwave signals using one or more optical or microwave sources; and communicating the generated one or more optical or microwave signals to the first qubit to initialize its state; 16. The method of claim 14 or 15, comprising:

27. 1. A method of constructing a fault-tolerant cluster state using a quantum system including a plurality of physical qubits, comprising: initializing an alternating grid of X-start and Z-start cluster states at physical qubits of the plurality of physical qubits; initializing at least one physical qubit of the plurality of physical qubits to be an X-type qubit; and measuring at least one physical qubit of the plurality of physical qubits in an X basis to measure an XZZX stabilizer of the corresponding cluster state; A method comprising:

28. 28. The method of claim 27, wherein initializing the alternating grid of X-start and Z-start cluster states comprises applying one or more CX and / or CZ gates to one or more of the plurality of physical qubits.

29. measuring an X-type physical qubit in an X-standard; and Measuring Z-type physics qubits in the Z standard 29. The method of claim 27 or 28, further comprising teleporting the logical information to another physical qubit of the plurality of physical qubits by

30. 29. The method of claim 27 or 28, further comprising detecting an error in one of the plurality of physical qubits by measuring one of a Z error on an X-type physical qubit or an X error on a Z-type physical qubit.

31. The step of detecting an error comprises:

31. The method of claim 30, comprising detecting the flipped stabilizer by measuring at least one X-type qubit.

32. Multiple physical qubits; at least one computer-readable medium storing a plurality of drive waveforms; and initializing an alternating grid of X-start and Z-start cluster states at physical qubits of the plurality of physical qubits; initializing at least one physical qubit of the plurality of physical qubits to be an X-type qubit; Measure at least one physical qubit of the multiple physical qubits in the X standard to measure the XZZX stabilizer of the corresponding cluster state At least one controller configured to Quantum systems, including:

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