QUANTUM CODE WITH SIMPLER PAIRWISE CHECKS

A method and apparatus for performing quantum error correction using an automorphism code related to the honeycomb code. The automorphism code is based on Kramers-Wannier (KW) duality. For embodiments on a hexagonal lattice using three repeated time steps, ⅓ of the pixels are active in a given time steps. In a given time step r, a KW circuit is applied to plaquettes labeled r mod 3, transferring quantum information from the active qubits at the beginning to a new set of active qubits at the end of the time step. Each of the three plaquette types is associated with one stabilizer either given by a product of six Z operators or three X operators. The KW circuit maps the product of X operators to the product of Z operators and vice-versa. The stabilizer group of the superlattice toric code changes every round.

FIELD

The field of the disclosure relates to quantum error correction and improvements thereof.

BACKGROUND

Quantum computing and information processing have great potential, but to achieve this potential several unique challenges must be overcome. Among these unique challenges is decoherence of quantum states arising from coupling between qubits and their environment. This decoherence challenge can be addressed is several ways, each of which have relative advantages and disadvantages. There are benefits to exploring multiple approaches to the decoherence challenge, including different types of qubit technologies, architectures, and codes.

One approach is quantum error correction, which is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction can be important to achieve fault-tolerant quantum computation that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum preparation, and faulty measurements.

Generally, quantum error correction includes a series of syndrome measurements resulting in an indication whether a fault (or error) has occurred. Decoding the results for the syndrome measurements may provide information about which fault occurred and regarding which unitary operator can be performed to correct the fault.

Different qubit configurations and codes operating thereon (e.g., syndrome measurements) have different advantages with respect to overhead, code distance, and complexity. For example, the honeycomb code has been shown to be a toric code that provides good code distance using two-qubit Pauli operators as check. The simplicity of the honeycomb code may make it useful for applications, especially in architectures where the basic operation is a pairwise measurement such as Majorana devices. Further, the honeycomb code has dynamically generated logical qubits, or “Floquet codes.”

The honeycomb code is a recently developed fault-tolerant quantum error correcting code. Beyond its possible practical application to Majorana hardware, this code has several interesting theoretical features. Although it is defined by a sequence of measurements of products of Paulis, it is not a stabilizer or subsystem code. Rather, the logical qubits are “dynamically generated”, being protected only because of the particular sequence of measurements chosen. Moreover, while at any instant the system is in a stabilizer state which is equivalent to the toric code (up to a local quantum circuit), the measurements implement an automorphism e↔m of the toric code. The checks are done in a repeating sequence, but after one period the electric and magnetic logical operators of the code are interchanged so that a state storing quantum information may be only invariant with twice the period.

Alternatives to the honeycomb code may prove desirable due to relative advantages with respect to complexity or practical implementation. Here, an alternative to the honeycomb code is disclosed using a measurement-based realization of Kramers-Wannier duality.

BRIEF SUMMARY

One embodiment illustrated herein includes a device that includes a quantum processor and a classical processor. The quantum processor includes a plurality of qubits arranged in a 3-colorable lattice of plaquettes. Each of the plaquettes is one of a first plaquette type, a second plaquette type, or third plaquette type, such that each plaquette is surrounded by plaquettes of different plaquette types than a plaquette type of the each plaquette. The classical processor controls quantum measurements on the plurality of qubits to perform a quantum error correction code based on Kramers-Wannier duality by performing a Kramers-Wannier circuit in a periodic sequence.

A further embodiment is the aforementioned embodiment of the device wherein each of said plaquettes consists of six qubits.

A further embodiment is any of the aforementioned embodiments of the device, wherein the classical processor performs the quantum error correction code as rounds of a repeating sequence of three steps.

A further embodiment is any of the aforementioned embodiments of the device, wherein at a beginning of each of the three steps one third of the qubits are active qubits, and which qubits are the active qubits is different for each of the three steps.

A further embodiment is any of the aforementioned embodiments of the device, wherein the quantum error correction code is an automorphism code. The automorphism code includes that at each time step, the Kramers-Wannier circuit maps a logical operator to another logical operator and vice versa. For each of the rounds of the three steps, the logical operator alternates with the another logical operator.

A further embodiment is any of the aforementioned embodiments of the device, wherein the classical processor controls the quantum measurements to perform the Kramers-Wannier circuit in a periodic sequence. The periodic sequence includes that a first step of the three steps is performed by performing the Kramers-Wannier circuit on qubits of the plaquettes of the first plaquette type. A second step of the three steps is performed by performing the Kramers-Wannier circuit on qubits of the plaquettes of the second plaquette type. A third step of the three steps is performed by performing the Kramers-Wannier circuit on qubits of the plaquettes of the third plaquette type.

A further embodiment is any of the aforementioned embodiments of the device, wherein the Kramers-Wannier circuit maps a product of Z operators to a product of X operators and maps products of X operators to respective products of Z operators.

A further embodiment is any of the aforementioned embodiments of the device, wherein the Kramers-Wannier circuit implements one of four types of Kramers-Wannier duality, and the classical processor uses results of the quantum measurements on the plurality of qubits to determine which of the four types of Kramers-Wannier duality was implemented.

A further embodiment is any of the aforementioned embodiments of the device, wherein the classical processor records measurement outcomes from each step of the periodic sequence of the quantum error correction code. The classical processor determines how plaquette stabilizers evolve based on the record measurement outcomes. The classical processor checks whether the record measurement outcomes agree with expected outcomes. When the record measurement outcomes do not agree with the expected outcomes, (i) the classical processor signals a fault occurred and (ii) the classical processor determines which fault occurred and applies one or more corrective unitary operators to the plurality of qubits.

A further embodiment is any of the aforementioned embodiments of the device, wherein the Kramers-Wannier circuit maps a product of X operators to a product of Z operators and vice-versa.

A further embodiment is any of the aforementioned embodiments of the device, wherein the Kramers-Wannier circuit comprises a set of Hadamard gates, a set of single-qubit Z measurements, a set of two-qubit XX measurements, a set of two-qubit ZZ measurements, and a set of single-qubit X measurements.

A further embodiment is any of the aforementioned embodiments of the device, wherein the Kramers-Wannier circuit comprises, a set of single-qubit X measurements, a set of two-qubit XZ measurements, a set of two-qubit ZX measurements, and another set of single-qubit X measurements.

A further embodiment is any of the aforementioned embodiments of the device, wherein the Kramers-Wannier circuit comprises a first set of single-qubit X measurements, a first set of controlled Z gates, a second set of controlled Z gates, and a second set of single-qubit X measurements.

Another embodiment illustrated herein is a method of performing an error correction code. The method includes using a classical processor to control quantum measurements on a plurality of qubits to perform a quantum error correction code, the quantum error correction code being based on Kramers-Wannier duality, and the quantum error correction code includes performing a Kramers-Wannier circuit in a periodic sequence. The plurality of qubits is arranged in a 3-colorable lattice of plaquettes, each of the plaquettes being one of a first plaquette type, a second plaquette type, or third plaquette type, such that each plaquette is surrounded by plaquettes of different plaquette types than a plaquette type of the each plaquette.

A further embodiment is the aforementioned embodiments of the method, wherein the classical processor controls the quantum measurements on the plurality of qubits such that the quantum error correction code is performed as rounds of a repeating sequence of three steps.

A further embodiment is any of the aforementioned embodiments of the method, wherein the classical processor controls the quantum measurements on the plurality of qubits to perform the periodic sequence that includes three steps. In a first step of the three steps, the Kramers-Wannier circuit is performed on qubits of the plaquettes of the first plaquette type. In a second step of the three steps, the Kramers-Wannier circuit is performed on qubits of the plaquettes of the second plaquette type. In a third step of the three steps, the Kramers-Wannier circuit is performed on qubits of the plaquettes of the third plaquette type.

A further embodiment is any of the aforementioned embodiments of the method, wherein the classical processor controls the quantum measurements to perform the quantum error correction code that is an automorphism code such that, at each step, the Kramers-Wannier circuit maps a logical operator to another logical operator and vice versa, and, for each of the rounds of the three steps, the logical operator alternates with the another logical operator.

A further embodiment is any of the aforementioned embodiments of the method, wherein the classical processor controls the quantum measurements to perform the Kramers-Wannier circuit, and Kramers-Wannier circuit maps a product of Z operators to a product of X operators and maps products of X operators to respective products of X operators.

A further embodiment is any of the aforementioned embodiments of the method, wherein the classical processor controls the quantum measurements on the plurality of qubits to perform the Kramers-Wannier circuit, wherein the Kramers-Wannier circuit implements one of four types of Kramers-Wannier duality, and the classical processor uses results of the quantum measurements on the plurality of qubits to determine which of the four types of Kramers-Wannier duality was implemented.

A further embodiment is any of the aforementioned embodiments of the method, wherein the classical processor performs the following steps of: (1) recording measurement outcomes from each step of the periodic sequence of the quantum error correction code; (2) determining how plaquette stabilizers evolve based on the record measurement outcomes; (3) checking whether the record measurement outcomes agree with expected outcomes; and (4), when the record measurement outcomes do not agree with expected outcomes, the classical processor performs steps of (i) signaling a fault occurred, (ii) determining which fault occurred, and (iii) applying one or more corrective unitary operators to the plurality of qubits

DETAILED DESCRIPTION

As discussed above, the honeycomb code is a fault-tolerant quantum memory defined by a sequence of checks which implements a nontrivial automorphism of the toric code. Here, a toric code is presented. The toric code is an alternative to the honeycomb code and is related to Kramers-Wannier duality, as discussed in D. Aasen, Z. Wang, and M. Hastings, “Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes,” Phys. Rev. B, vol. 106, p. 085122 (Aug. 15, 2022), and discussed in https://arxiv.org/abs/22033.11137, both of which are incorporated herein by reference in their entirety. The code is similar to the honeycomb code in that it implements a non-trivial automorphism of the toric code after three rounds of measurement. In contrast to the honeycomb code, the code presented herein advantageously involves only two qubit XX and ZZ measurements in addition to single-qubit X and Z measurements.

The code disclosed herein is a measurement-based quantum code that is often referred to as the “e↔m automorphism code.” According to certain embodiments, the e↔m automorphism code can be defined by a periodic sequence of measurements on a hexagonal lattice with qubits on the edges. The instantaneous stabilizer group of the e↔m automorphism code is equivalent to the stabilizer group of a triangular super-lattice toric code with additional decoupled degrees of freedom. The triangular super-lattice varies from round to round, similar to the honeycomb code. The instantaneous stabilizer group can be defined to be the stabilizer group after some given number of rounds of the circuit. Like the honeycomb code, the e↔m automorphism code implements a non-trivial automorphism of the super-lattice toric code once per period. Up to measurement-dependent signs, the e↔m automorphism code implements the adiabatic path discussed in the previous section. According to certain embodiments, the primary tool used for defining this code is a measurement-based realization of Kramers-Wannier duality, which ultimately comes from the non-trivial invertible bimodule over VecZ2. The Kramers-Wannier duality can also be implemented using the method disclosed in N. Tantivasadakarn et al., “Long-range entanglement from measuring symmetry-protected topological phases,” arXiv:2112.01519, which is incorporated herein in its entirety. The trivial invertible domain wall will also result in a Floquet code with instantaneous stabilizer code given by the usual toric code stabilizers. The resulting Floquet code simply measures different subsets of the usual toric code stabilizers at different times. A person of ordinary skill in the art will recognize that a straightforward generalization of the e↔m automorphism code can be implemented on any 3-colorable graph, as illustrated inFIG.1A.

The e↔m automorphism code has one qubit per edge of the hexagonal lattice and period three.FIGS.1A and2show the geometry and a nonlimiting example of the code. The thick edges correspond to dead/inactive qubits, while the thinner edges correspond to active qubits (e.g., qubits encoding relevant quantum information). At any given time-step, ⅓ of the qubits are decoupled from the system (thick edges). At time step r, j=(r mod 3) and the Kramers-Wannier circuit KW(j)(e.g., any one of those shown inFIGS.3A-C) is performed on the type j plaquettes. During the Kramers-Wannier measurement sequence, the dead qubits are transferred from the boundary of the type r+1 mod 3 plaquettes to the boundary of the type r+2 mod 3 plaquettes. The measurement outcomes of the Kramers-Wannier circuit at time step r determine the vertex stabilizers of a toric code living on a triangular super-lattice whose vertices are identified with the r+2 mod 3 plaquettes. After implementing the first three rounds of the Kramers-Wannier circuit, all super-lattice toric code stabilizers are measured.

Now, additional description of the e↔m automorphism code is provided. First, the Kramers-Wannier circuit itself is described. Then, the computation of the instantaneous stabilizer group (ISG) of the e↔m automorphism code are described. It is shown that at any fixed time, the ISG is generated by the stabilizers of the toric code on a triangular super-lattice.

Referring now toFIG.1A, a hexagonal lattice100is shown. The coding method can be described with reference to a mapping of qubits to a 3-colorable hexagonal lattice in which each edge is assigned a qubit from a set of qubits. The hexagons are 3-colorable meaning that they can be labeled such that any two neighboring hexagons have different labels. Cell labels are also referred to herein as cell types. As shown inFIG.1A, a hexagonal lattice100is defined in which each hexagon is assigned one of a first, second, or third hexagon label such that each hexagon is surrounded by hexagons having different hexagon labels. In this example, “0,” “1,” and “2,” are used as hexagon labels but any other convenient labels can be used such as red, green, blue or A, B, C. For purposes of illustration, cells associated with each label have a common shading. Hexagons110,111,112are representative type 0, 1, and 2 hexagons, respectively. In the coding methods disclosed herein, certain qubits are denoted as active or dead during some steps. These designations are shown inFIG.1Ain which a representative edge120is associated with an active qubit and a representative edge121is associated with a dead qubit. Dead qubits are associated with edges shown with thicker lines.

In the explanations, methods and associated apparatus refer in some instances to operations or configurations of edges for purposes of illustration. It should be understood that such references pertain to qubits associated with particular edges. The hexagonal lattice, lattice labeling, and designation of edges as active or dead are illustrative and do not require a particular physical arrangement of qubits. Hexagonal cells are also referred to herein as plaquettes. In the following description, cell edges are referred to as even or odd based on a labeling of a representative cell150shown inFIG.1B.

Representative Processing

Referring toFIG.2, a portion200of a hexagonal lattice used in the e↔m automorphism code. Each edge has one qubit. During a given time step of a toric code (note, as used herein toric code refers to the application of the e↔m automorphism code on a torus, which has the benefit of periodic boundary conditions), the 0-labeled hexagonal cells are designated as inactive as noted with thick lines; active qubits are associated with thin lines. In this example, odd edges such as representative odd edges211,213, and215of 2-labeled hexagon cell202are active. As shown inFIG.2, the active qubits form a triangular-super lattice with vertices identified with the 0-type plaquettes. For the given time step, all edges of the 0-labeled cells are marked as “dead”; the 2-labeled cells are marked similarly to the 2-labeled hexagon cell202.

InFIG.2, a cell220has odd-labeled qubits (at odd edges) denoted a1, a3, and a5, and has even-labeled qubits (at even edges) denoted b2, b4, and b6. From the given time step to a next time step, a Kramers-Wannier (KW) circuit, such as KW circuit300illustrated inFIG.3A, transforms the quantum information in these odd qubits a1, a3, and a5, which are active in the given time step, to the even qubits a2, a4, and a6, which are active in the next time step.

Referring toFIG.3A, a representative Kramers-Wannier (KW) circuit300is illustrated. KW circuit300includes: (i) a set of Hadamard gates302; (ii) a set of single-qubit Z measurements304with measurement results r2, r4, r6; (iii) a set of two-qubit XX-measurements306with measurement results m1, m3, m5; (iv) a set of two-qubit ZZ-measurements308with measurement results m2, m4, m6; and (v) a set of X-measurements310with measurement results r1, r3, r5. A second KW circuit300′ is illustrated inFIG.3B, and a third KW circuit300″ is illustrated inFIG.3C. The methods described herein can be performed with any one of these three alternatives (i.e., with either KW circuit300, KW circuit300′, or KW circuit300″).

Referring toFIG.3B, a representative Kramers-Wannier (KW) circuit300′ is illustrated, which is an alternative circuit configuration to KW circuit300. KW circuit300′ is a Kramers-Wannier circuit without the Hadamard gates302, which are in KW circuit300. The Hadamard transformation on a qubit can be discarded if one interchanges all subsequent X and Z measurements on that qubit. Thus, the KW circuit300is equivalent to KW circuit300′ with only X and Z measurements. KW circuit300′ includes: (i) a set of single-qubit Z-measurements312with measurement results r2, r4, r6; (ii) a set of two-qubit XZ-measurements314with measurement results m1, m3, m5; (iii) a set of two-qubit XZ-measurements316with measurement results m2, m4, m6; and (iv) a set of X-measurements318with measurement results r1, r3, r5.

Referring toFIG.3C, a representative Kramers-Wannier (KW) circuit300″ is illustrated, which is another alternative circuit configuration to KW circuit300. KW circuit300″ includes: (i) a set of single-qubit X-measurements322with measurement results r2, r4, r6; (ii) a first set of two-qubit controlled-Z (CZ) operations324; (iv) a second set of two-qubit controlled-Z operations326; and (iv) a set of single-qubit X-measurements328with measurement results r1, r3, r5. The matrix elements of a control-Z operation for two qubits are given by

which can be written in matrix form as

Returning toFIGS.2and3A, the coding method is described for the cell220, which is a highlighted plaquette. At time step j=2, the KW circuit300is coupled to the all the, including the active qubits a1, a3, and a5and the inactive qubits b2, b4, and b6. The same KW circuit300is applied to all 2-labeled plaquettes.

The sequence of time steps is performed modulo 3, such that, at the subsequent time step j=3 mod 3 which is equal to 0, the KW circuit300is applied to all 0-labeled plaquettes. Then, at time step j=4, the KW circuit300is applied to all 1-labeled plaquettes, and so forth. For example, at the end of time step j=2, the qubits labeled a2, a4, and a6at the output of the KW circuit300will the active qubits on the 0-labeled plaquettes at the beginning of time step j=3. Thus, from time step to time step, the KW circuit300changes which qubits are active. The KW circuit applied at time step j is denoted as KW(j mod 3).

Generally, at time step j, a KW circuit KW(j)is applied to edges (qubits) of j-labeled cells and produces measurement outcomes r, m wherein r=r1, . . . , r6and m2, . . . , m6. The qubits b1, b3, and b5are decoupled from the system into the X-basis due to a prior round of measurements. Depending on the measurement outcomes {rj} and {mj} the circuit will implement one of four types of Kramers-Wannier duality, as described below.

The Kramers-Wannier circuit plays a critical role in the e↔m automorphism code. Now, the Kramers-Wannier on 2N qubits is discussed for the e↔m automorphism code for the nonlimiting case N=3. More generally, the code may be applied to any 3-colorable lattice and N is not limited to the value 3.

The initial measurements in the Kramers-Wannier circuit serve to disentangle the odd qubits from the even qubits through the onsite Z measurements. Hence, without loss of generality it may be assumed that the incoming wavefunction is not entangled with the odd qubits. Similarly, the single-qubit X measurements at the end of the circuit disentangle the even qubits out of the wavefunction. As the name suggests, the circuit takes a generic state on the incoming odd qubits and outputs the Kramers-Wannier dual on the outgoing even qubits. There is one caveat, that the Kramers-Wannier dual will depend on the measurement outcomes. For example, if all measurement outcomes are +1, then the standard Kramers-Wannier duality is obtained.

The evolution of the quantum state of the qubits due to the Kramers-Wannier circuit can be straightforwardly computed. Here, only the matrix elements between the even and odd qubits are discussed, as the remaining degrees of freedom are decoupled and determined by the single-qubit X and Z measurements on the odd and even qubits respectively. Up to normalization, the matrix elements are given by,

Wherein r=(r1, . . . , r2N) and m=(m1, . . . , m2N) are lists of measurement outcomes and take values in d. The r measurements determine the values of the incoming and outgoing ancillas. We mention that KW0,0is the standard Kramers-Wannier transformation.

It is helpful to analyze the commutation relations of KWr,mwith the operators ZZ and X, resulting in

In particular,

Therefore, four kinds of Kramers-Wannier duality are manifest, determined by the mod 2 value of Σjm2j+r2jand Σjm2j-1+r2j-1. Equivalently, the above two equations show that the Kramers-Wannier circuit measures X1X3. . . X2N-11 on the initial state, with measurement outcome (−1)Σjm2j+r2jand prepares a given X2X4. . . X2Nin the final state, with eigenvalue (−1)Σjm2j-1+r2j-1.

The Toric Floquet Code and its Instantaneous Stabilizer Group

Now, a description is provided a description of the toric Floquet code (also called e↔m automorphism code) and its instantaneous stabilizer group (ISG). More particularly, the toric Floquet code and compute its instantaneous stabilizer group are presented.

A nonlimiting example of the code is given by the following measurement schedule. First, initialize the code by measuring all type 0 and type 2 edges in the X basis. Then, at time r run the Kramers-Wannier circuit on all type r mod 3 plaquettes of the hexagon lattice shown inFIG.2, starting with r=0.

Now, the instantaneous stabilizer group (ISG) is calculated, according to one embodiment. The ISG is defined to be the stabilizer group after a given number of rounds of the circuit. The following disclosure relies on the following additional notation. Let P(r)be the set of plaquettes of type r, and E(r)be the set of edges of type r. An edge e∈E(r)terminates on two type r plaquettes. The edges at the boundary of a plaquette p is referred to by ∂p. Similarly, the notation e∈pris used to label all edges terminating on a plaquette pr∈P(r).

Starting with the maximally mixed state, at time step r the Kramers-Wannier circuit is implemented on all type r mod 3 plaquettes starting with r=0. The Kramers-Wannier circuit is run on all type 0 plaquettes. The type 0 plaquettes have type 1 and 2 edges at there boundaries. After running the Kramers-Wannier circuit on the type 0 plaquettes, the type 1 edges will be dead, and the type 2 edges will be active. The type 0 edges will be unmodified. The equation

indicates that the value of Πe∈∂p0∩E(2)Xefor each p0∈ P(0)is determined by the measurements done on the boundary of plaquette p0. Thus, the instantaneous stabilizer group is given by

Next, the Kramers-Wannier circuit is measured on the plaquettes of type 1. Initially, all type 2 qubits are active and all type 1 qubits are dead. After the Kramers-Wannier circuit on the type 1 plaquettes, the type 1 and 2 edges will be dead, and the type 0 edges will be active. The circuit will measure a new set of stabilizers associated with the type 1 plaquettes given by Πe∈∂p1∩E(0)Xefor p1∈ P(1). As above, the stabilizer eigenvalue can be inferred by the measurement outcomes via the equation

The circuit will also transfer the stabilizer associated with the type 0 plaquettes in ISG0, from the product of three X operators, to a product of six Z operators due to the relation given in the equations

Specifically, the stabilizer Πe∈∂p0∩E(2)Xe∈ ISG0becomes ±Zewhere denotes a type 0 edge e that terminates on a plaquette p0. Again, the overall\pmsign of the stabilizer can be inferred from the measurement outcomes. Thus,

Finally, the Kramers-Wannier circuit is run on the type 2 plaquettes. The circuit measures a new X type stabilizer associated with the type 2 plaquettes given by ±Xefor p2∈P(2). The sign of the stabilizer can be inferred from the measurement outcomes. The circuit also transforms the X stabilizer associated with the type 1 plaquettes to a Z type stabilizer associated with the 6 edges terminating on the type 1 plaquette. Thus,

It can be seen that ISG2is simply the stabilizers of a triangular lattice toric code whose vertices are identified with the type 2 plaquettes. All subsequent steps can be analyzed very similarly.

In summary, the instantaneous stabilizer groups for the e↔m after implementing KW(r mod 3)are given by,

FIG.4Aillustrates a diagrammatic description of the instantaneous stabilizer groups (ISGs)400for the e↔m automorphism code. Again, thick edges denote “dead” qubits, decoupled from the system in the X basis. Thinner edges denote “active” qubits participating in the toric code existing on the triangular super-lattice with vertices given by the r=1; 2; 0 plaquettes going from left to right. Associated with each plaquette that is surrounded by a thick line is a stabilizer given by the product of six Z operators on the edges terminating on the plaquette. Associated to all other plaquettes are stabilizers given by a product of three X operators on the thinner edges surrounding the plaquette.

FIG.4Billustrates a diagrammatic description of ISGr≥2. Each plaquette shown inFIG.4Bis associated with one stabilizer, either given by a product of six Z operators or three X operators. Each thick blue denotes a dead qubit, and correspondingly contributes one single site X stabilize per thick edge, shown on the right ofFIG.4B. When r≥2 the stabilizers are exactly those of a triangular super-lattice toric code with vertices identified with r+2 mod 3 plaquettes.

Now, one detail is elucidated. For a superlattice toric code on a closed manifold, the product of all plaquette stabilizers is equal to +1 and the product of all vertex stabilizers is also equal to +1. How does this constraint get passed between rounds of measurement? The new plaquette stabilizers inferred after each round of measurement can be ±1, and so it is not obvious why the product of the superlattice vertex or plaquette stabilizers should be fixed. Suppose a state is stabilized by ISGr-1, with each stabilzer's expectation value given by sp(r−1). Where p runs over every plaquette of the honeycomb lattice. Note that (r−1) mod 3 does not necessarily coincide with plaquette type in the quantity sp(r−1). The Kramers-Wannier circuit KW(r mod 3)is performed on every pr∈P(r mod 3). Denote the measurement outcomes of the Kramers-Wannier circuit on plaquettes pr∈P(r mod 3)as rp=(rp,1, . . . rp,6) and mp=(mp,1, . . . , mp,6). Based on the equations

it can be seen that

where p∈P(r). That is, for each plaquette p∈P(r mod 3), sp(r−1)is measured and fixed sp(r)is prepared. It is also seen how the stabilizers on plaquettes p∈P(r−1 mod 3)P(r+1 mod 3)are updated, resulting in

Wherein p′p∩P(r mod 3)denotes a plaquette p′∈P(r mod 3), which is neighboring p. The function ƒp′, and gp′, can be determined using equations

Thus, it can be inferred that the evolution of the plaquette stabilizers from ISGr-1to ISGrunder the Kramers-Wannier circuit KW(r). Note, that only the r mod 3 plaquette stabilizers are measured at this step.

Now it can be computed how the product of the superlattice toric code stabilizers evolve under KW(r). The updated products of plaquette stabilizers are given by,

To derive the first equation above, the following expression has been used

To derive the second equation above, a very similar expression has been used. It can be further determined that

The left side of the equation is the product over all vertex stabilizers of the triangular superlattice toric code at time step r, while the right side is the product over all plaquette stabilizers of the triangular superlattice time step r−1. Similarly, it can be determined that

The left side is the product of all plaquette stabilizers of the triangular superlattice toric code at time step r, which is equal to the product of all vertex stabilizers of the triangular superlattice toric code at time step r−1.

Logical Operators

The superlattice toric code has well-known logical operators: a product of Pauli X operators on a homologically nontrivial loop on the lattice or a product of Pauli Z operators on a homologically nontrivial loop on the dual lattice. Arbitrarily, one of these may be called electric and the other may be called magnetic.

If some logical operator of the superlattice toric code is measured, the measured logical operator maps to some other logical operator of the toric code after applying the KW circuit.

To see this, note that it has been verified that the ISG is that of the superlattice toric code. Measuring some logical operator of the superlattice toric code increases the rank of the ISG by one, and the increase in rank must be maintained from one round to the next since the rank of the ISG cannot reduce under measurement. So, the result after any number of further rounds must also be a superlattice toric code with one additional stabilizer, and that additional stabilizer must be a logical operator.

Without doing any calculation, it can be inferred that the electric and magnetic operators interchange every round. Indeed, the KW circuit maps a product of X operators to a product of Z operators and vice-versa. Of course, the stabilizer group of the superlattice toric code changes every round, but it is periodic mod3; since 3 is odd, the electric and magnetic operator interchange every period.

Error Detection

Now error detection and correction are discussed. At time step r, ISGris given by

Further, ISGris diagrammatically represented inFIG.4B. Suppose a state is stabilized by ISGr, with each stabilizers expectation value given by sp(r), where p runs over every plaquette of the honeycomb lattice. Note that r mod 3 does not necessarily coincide with plaquette type in the quantity sp(r). Now, the Kramers-Wannier circuit KW(r+1 mod 3)is run on every p∈P(r+1 mod 3). The measurement outcomes of the Kramers-Wannier circuit on plaquettes p′∈P(r+1 mod 3)are denoted as rp′, and mp′. Based on the equations

it can be seen that

Where p′∈P(r+1 mod 3)denotes a plaquette p′∈P(r+1 mod 3)which is neighboring p, and ƒp′is linear in rp′and mp′. The functions ƒ and g are determined by

Thus, the evolution of the plaquette stabilizers between round r and round r+1 can be inferred. However, only the r+1 mod 3 plaquette stabilizers are measured at this step (the reason ultimately boils down to equations

By recording the measurement outcomes at each step, it can be tracked how the plaquette stabilizers evolve. After three rounds of measurement, the plaquette stabilizers can be compared, and it can be checked whether the results of the measurements agree with the expected outcome. When they do not agree an occurrence of a fault can be detected.

Representative Classical Computing Environment

FIG.5illustrates a method500of performing the automorphism e↔m code. Method500works for any three-colorable lattice, including the hexagonal lattice discussed above. In step505of method500, the qubits are initialized. In step510of method500, the KW(0)circuit is performed on the type 0 plaquettes, and the values resulting from the quantum measurements are recorded. In step520of method500, the KW(1)circuit is performed on the type 1 plaquettes, and the values resulting from the quantum measurements are recorded. In step530of method500, the KW(2)circuit is performed on the type 2 plaquettes, and the values resulting from the quantum measurements are recorded. In step540of method500, the measurement results from the previous three steps are analyzed to determine if there were any errors. If one or more errors occurred, the measurement results can then be decoded to determine which errors occurred, and what unitary operations are to be performed on the qubits to correct the errors. As indicated by step550, this process can be repeated until error correction is no longer needed, and the process is finished.

With reference toFIG.6, an exemplary system for implementing the disclosed technology includes computing environment600. In computing environment600, a compiled quantum computer circuit description (including quantum circuits for performing any of the disclosed techniques as disclosed herein) can be used to program (or configure) one or more quantum processors such that the quantum processor(s) implement the circuit described by the quantum computer circuit description.

The environment computing600includes one or more quantum processors602and one or more readout device(s)608. The quantum processor(s) execute quantum circuits, such as the KW circuits disclosed herein, that are precompiled and described by the quantum computer circuit description. The quantum processor(s) can be one or more of the following nonlimiting examples: (a) a superconducting quantum computer; (b) an ion trap quantum computer; (c) a fault-tolerant architecture for quantum computing; and/or (d) a topological quantum architecture (e.g., a topological quantum computing device using Majorana zero modes). The precompiled quantum circuits, including any of the disclosed circuits, can be sent into (or otherwise applied to) the quantum processor(s) via control lines606at the control of quantum processor controller620. The quantum processor controller (QP controller)620can operate in conjunction with a classical processor610to implement the desired quantum computing process. In the illustrated example, the QP controller620further implements the desired quantum computing process via one or more QP subcontrollers604that are specially adapted to control a corresponding one of the quantum processor(s)602. For instance, in one example, the quantum controller620facilitates implementation of the compiled quantum circuit by sending instructions to one or more memories (e.g., lower-temperature memories), which then pass the instructions to low-temperature controller(s) (e.g., QP subcontroller(s)604) that transmit, for instance, pulse sequences representing the gates to the quantum processor(s)602for implementation. In other examples, the QP controller(s)620and QP subcontroller(s)604operate to provide appropriate magnetic fields, encoded operations, or other such control signals to the quantum processor(s) to implement the operations of the compiled quantum computer circuit description. The quantum controller(s) can further interact with readout devices608to help control and implement the desired quantum computing process (e.g., by reading or measuring out data results from the quantum processors once available, etc.).

With reference toFIG.6, compilation is the process of translating a high-level description of a quantum algorithm into a quantum computer circuit description comprising a sequence of quantum operations or gates, which can include the circuits as disclosed herein (e.g., the circuits configured to perform one or more of the procedures as disclosed herein or resulting from any of the disclosed techniques). The compilation can be performed by a compiler622using a classical processor610of the computing environment600which loads the high-level description from memory or storage devices612and stores the resulting quantum computer circuit description in the memory or storage devices612.

In other embodiments, compilation and/or verification can be performed remotely by a remote computer660(e.g., a computer having a computing environment) which stores the resulting quantum computer circuit description in one or more memory or storage devices662and transmits the quantum computer circuit description to the computing environment600for implementation in the quantum processor(s)602. Still further, the remote computer660can store the high-level description in the memory or storage devices662and transmit the high-level description to the computing environment600for compilation and use with the quantum processor(s). In any of these scenarios, results from the computation performed by the quantum processor(s) can be communicated to the remote computer after and/or during the computation process. Still further, the remote computer can communicate with the QP controller(s)620such that the quantum computing process (including any compilation, verification, and QP control procedures) can be remotely controlled by the remote computer660. In general, the remote computer660communicates with the QP controller(s)620, classical processor610via communication interface650.

In particular embodiments, the computing environment600can be a cloud computing environment, which provides the quantum processing resources of the computing environment600to one or more remote computers (such as remote computer660) over a suitable network (which can include the internet).