Patent ID: 12260157

DETAILED DESCRIPTION

Quantum computing entails coherently processing quantum information stored in the quantum bits (qubits) of a quantum computer. Superconducting quantum computing is a promising implementation of solid-state quantum computing technology in which quantum information processing systems are formed, in part, from superconducting materials. To operate quantum information processing systems that employ solid-state quantum computing technology, such as superconducting qubits, the systems are maintained at extremely low temperatures, e.g., in the 10s of mK. The extreme cooling of the systems keeps superconducting materials below their critical temperature and helps avoid unwanted state transitions. To maintain such low temperatures, the quantum information processing systems may be operated within a cryostat, such as a dilution refrigerator.

Even at the extremely low qubit operating temperatures, qubits may still suffer from decoherence and gate errors. As such, large-scale quantum error correction algorithms can be deployed to compensate for the gate errors and qubit decoherence. An error-corrected quantum processor leverages redundancy to synthesize protected logical qubits from ensembles of error-prone qubits. Implementations of current superconducting quantum systems therefore may use a large number of qubits to implement error correction algorithms.

As processors scale to more qubits, high-fidelity operation of superconducting qubits requires fine-tuning numerous experimental parameters and this becomes increasingly challenging. In particular, in a square array of qubits, each pair of neighboring qubits can be connected by an adjustable coupler and these couplers can complicate system calibration. The resonance frequencies of the two qubits and the coupler are tunable with biases respectively applied to the qubits and a bias applied to the coupler. For example, it may be difficult to determine coupler bias settings that turn off qubit-qubit interactions due to unwanted qubit-qubit interactions and due to the coupler bias affecting the qubit frequencies.

Conventionally, qubit-coupler systems were calibrated by human experts often iterating back and forth between qubit calibration and changing the coupler bias to try to find parameters where the coupling is off. Therefore, a procedure to reliably find a set of control parameters to turn off the coupling between neighboring qubits may be necessary.

This specification provides such procedure considering a pair of qubits and a coupler that mediates their interactions.

FIG.1is a schematic that illustrates a qubit-coupler-qubit system

The qubit-coupler-qubit system100includes a first qubit110, a second qubit120and a coupler130. The first qubit110is labelled as ‘Qubit A’ and the second qubit120is labelled as ‘Qubit B’ inFIG.1.

The first qubit110and the second qubit120may be any type of qubits, or quantum systems including energy levels which can be approximated as a quantum mechanical 2-level system. The examples of the first qubit110and the second qubit120include quantum LC oscillators or superconducting qubits such as transmon qubits, fluxmon qubits, charge qubits, or gmon qubits. The examples of the first qubit110and the second qubit120may also include other solid state systems such as semiconductor quantum dots and diamond NV centers and atomic systems such as trapped ions. However, the examples of the first qubit110and the second qubit120are not limited to these examples.

In some implementations, the first qubit110and the second qubit120may be the same type of qubits.

In some implementations, the first qubit110and the second qubit120may be different types of qubits. For example, when the first qubit110and the second qubit120are superconducting qubits, but the first qubit110may be a transmon qubit and the second qubit120may be a flux qubit or a phase qubit.

In some implementations, the qubit-coupler-qubit system100may be a part of a larger system containing additional qubits and couplers or a plurality of the qubit-coupler-qubit system100. For example, the qubit-coupler-qubit system100may be a part of a 2-dimensional arrangement of qubits, such as a square grid, where each adjacent pair of qubits is interposed and mediated by the coupler130. As part of the larger system of qubits and couplers, the first qubit110and the second qubit120of the qubit-coupler-qubit system100may further be connected to other qubits. For example, the first qubit110may be coupled to three more qubits in addition to the second qubit120via three respective couplers.

The first qubit110and the second qubit120exhibit at least one resonance frequency arising from the 2-level quantum mechanical transition. The corresponding resonance frequencies will be referred to as a first resonance frequency111and a second resonance frequency121, respectively.

The first qubit110is configured such that the first resonance frequency111is controlled by providing a first bias112, labelled as ‘A’ inFIG.1. Therefore, the first resonance frequency111is modelled to be dependent on the first bias112or as a function of the first bias112labelled as ‘f(A)’ inFIG.1.

The second qubit120is configured such that the second resonance frequency121is controlled by providing a second bias122, labelled as ‘B’ inFIG.1. Therefore, the second resonance frequency121is modelled to be dependent on the second bias122or as a function of the second bias122labelled as ‘f(B)’ inFIG.1.

In some implementations, the first bias112and the second bias122may be a magnetic flux applied to the first qubit110and the second qubit120, respectively. For example, when the first qubit110and the second qubit120are superconducting qubits, the first bias112and the second bias122may be provided by the magnetic flux threaded through the respective SQUID loops of the first qubit110and the second qubit120.

The examples of the first bias112and the second bias122are not limited to magnetic flux or magnetic field and depend on the specific design or inherent property of the first qubit110and the second qubit120. For example, when the first qubit110and the second qubit120are superconducting qubits, the first bias112and the second bias122may be provided by AC signals inductively or capacitively coupled to the first qubit110and the second qubit120. When the first qubit110and the second qubit120are an atomic system or other solid state systems, an electric field may be applied to induce a Stark shift of the resonance frequencies111,121or a magnetic field may be applied to induce a Zeeman shift of the resonance frequencies111,121.

Therefore, the first bias112and the second bias122will be used in this specification to refer to any control means or control variables of the first qubit110and the second qubit120facilitating the control of the first resonance frequency111and the second resonance frequency121.

The coupler130is configured to mediate the coupling between the first qubit110and the second qubit120.

The coupler130exhibits at least one resonance frequency. The corresponding resonance frequency will be referred to as a third resonance frequency131. The coupler130is configured such that the third resonance frequency131is controlled by providing a third bias132, labelled as ‘C’ inFIG.1. Therefore, the third resonance frequency131is modelled to be dependent on the third bias132or as a function of the third bias132labelled as ‘f(C)’ inFIG.1.

In some implementations, the coupler130may be a qubit.

In some implementations, the coupler130may be an LC resonator with a Josephson junction as the inductance.

In some implementations, the coupler130may be the same kind qubit as the first qubit110and/or the second qubit120.

In some implementations, the coupler130may be a different type of qubit than the first qubit110and the second qubit120. For example, when the first qubit110and the second qubit120are transmon qubits, the coupler130may be a gmon qubit.

In some implementations, the coupler130may be a resonator in which the resonance frequency, the third resonance frequency131, is controlled by providing the third bias132. For example, the coupler130may be a high quality factor resonator which couples to both the first qubit110and the second qubit120.

The degree of coupling mediated by the coupler130between the first qubit110and the second qubit120will be referred to as a first degree of coupling140-1,140-2in this specification.

The first degree of coupling140-1,140-2is dependent on the first resonance frequency111, the second resonance frequency121and the third resonance frequency131, as will be explained in more detail later. Therefore, for given initial frequencies of the first qubit110and the second qubit120, the first resonance frequency111and the second resonance frequency121, respectively, the third bias132can be adjusted to control the first degree of coupling140-1,140-2.

In some implementations, for given initial frequencies of the first qubit110and the second qubit120, the first resonance frequency111and the second resonance frequency121, respectively, the third bias132can be adjusted to control the first degree of coupling140-1,140-2such that the first degree of coupling140-1,140-2is minimized or turned off. For example, the third bias132may be applied to the coupler130such that the third transition frequency131is shifted far from the first transition frequency111and the second transition frequency121.

In some implementations, the first degree of coupling140-1,140-2may be dependent on the third resonance frequency131in view of the first resonance frequency111and the second resonance frequency121. For example, when the third resonance frequency131and the second resonance frequency121are close to each other, the second part of the first degree of coupling140-1may be larger than when they are further apart.

Therefore, the first degree of coupling140-1,140-2in this specification represents a part of the coupling between the first qubit110and the second qubit120mediated by the coupler130, which can be controlled by the third bias132applied to the coupler130. In other words, the first degree of coupling140-1,140-2corresponds to an indirect coupling between the first qubit110and the second qubit120, facilitated by the coupler130.

A first part140-1of the first degree of coupling represents the coupling between the first qubit110and the coupler130. A second part140-2of the first degree of coupling represents the coupling between the second qubit120and the coupler130.

The extent to which the first qubit110and the second qubit120are coupled not mediated by the coupler130will be referred to as a second degree of coupling150in this specification.

In some implementations, the first degree of coupling140-1,140-2may arise from the spatial proximity between the coupler130and the first qubit110or the second qubit120and therefore depend on the distance between the coupler130and the first qubit110or the second qubit120, respectively. For example, the mechanism of the first degree of coupling140-1,140-2may include capacitive coupling, inductive coupling, dipolar coupling although possible mechanisms of coupling are not limited to these examples.

In some implementations, the first degree of coupling140-1,140-2may arise from a common channel coupled to both the coupler130and first qubit110or the second qubit120. For example, even if the coupler130is spatially far apart from the first qubit110and the second qubit120such that inductive coupling, capacitive coupling or dipolar coupling is negligible, the coupler130and the first qubit110or the second qubit120may be both coupled to a common bus such as a waveguide or a resonator, which can efficiently mediate the coupling between the coupler130and the first qubit110or the second qubit120.

The second degree of coupling150represents the direct coupling between the first qubit110and the second qubit120which is present regardless of whether the first degree of coupling140-1,140-2is present or not including when the first degree of coupling140-1,140-2is made negligible. For example, when the third resonance frequency131is detuned far from the first resonance frequency111and the second resonance frequency121, the first degree of coupling140-1,140-2can be rendered negligible. The first qubit110and the second qubit120may still be coupled without the mediation of the coupler130.

In the qubit-coupler-qubit system100described inFIG.1, due to the direct coupling, or the second degree of coupling150, even when the coupler130is far detuned from the first resonance frequency111and the second resonance frequency121such that the direct coupling or the first degree of coupling140-1,140-2is negligible, the first resonance frequency111may shift in response to the change of the second bias122changes, and vice versa.

In some implementations, the second degree of coupling150may arise from the spatial proximity between the first qubit110and the second qubit120and therefore depend on the distance between the first qubit110and the second qubit120. For example, the mechanism of the second degree of coupling150may include capacitive coupling, inductive coupling, dipolar coupling although the mechanism of the second degree of coupling are not limited to these examples.

In some implementations, the second degree of coupling150may arise from a common channel coupled to both the first qubit110and the second qubit120. For example, even if the first qubit110and the second qubit120are spatially far apart such that inductive coupling, capacitive coupling or dipolar coupling is negligible, they may be both coupled to a common bus such as a waveguide or a resonator, which can efficiently mediate the coupling between the first qubit110and the second qubit120. In this case, the first qubit110and the second qubit120can be coupled to each other without the mediation of the coupler130.

In some implementations, the second degree of coupling150may be dependent on the first resonance frequency111and the second resonance frequency121. For example, when the first resonance frequency111and the second resonance frequency121are close to each other, the second degree of coupling150may be larger than when they are further apart.

Therefore, the qubit-coupler-qubit system100described inFIG.1depends on at least three control variables: the first bias112, the second bias122and the third bias132. The first resonance frequency111and the second resonance frequency121can be modelled as a function of the first bias112, the second bias122and the third bias132. For example, as discussed above, the first resonance frequency111can be controlled by providing the first bias112to the first qubit110, but the first resonance frequency111is also dependent on the second bias122and the third bias132due to the indirect coupling140-1,140-2and the direct coupling150present in the qubit-coupler-qubit system100.

This specification relates to a procedure to calibrate the qubit-coupler-qubit system100. This specification also relates to a procedure to initialize the qubit-coupler-qubit system100using the model obtained from the calibration procedure, where the coupling between the first qubit110and the second qubit120is suppressed or turned off. Given the idle frequencies or the initial resonance frequencies111,121of the first qubit110and the second qubit120, the obtained model may be used to predictively set the third bias132of the coupler130to counteract the second degree of coupling150such that the first degree of coupling140-1,140-2cancels out the second degree of coupling150.

FIG.2is a flowchart that illustrates a method of calibrating a qubit-coupler-qubit system.

The method200of calibrating the qubit-coupler-qubit system100as described inFIG.1includes the following steps.

In step210, a first model, a qubit model, is provided which describes the behavior of the first qubit110and the second qubit120. In particular, the first model may be a function which outputs the first resonance frequency111and the second resonance frequency121when the first bias112and the second bias122are provided, respectively, as the variable. The first model corresponds to the functions labelled as ‘f(A)’ and ‘f(B)’ inFIG.1.

In case the first qubit110and the second qubit120are the same type of qubits, one type of function may be used as the first model. In other words, ‘f(A)’ and ‘f(B)’ may be the same type of function with different set of parameters or coefficients.

In case the first qubit110and the second qubit120are different types of qubits, the first model may comprise two separate functions which respectively describe the behavior of the first resonance frequency111and the second resonance frequency121as a function of the first bias112and the second bias122, respectively.

In some implementations, the first model, providing the first resonance frequency111and the second resonance frequency121, may be a function of the respective bias, a first bias112and/or a second bias122independent of the first degree of coupling140-1,140-2and the second degree of coupling150.

In some implementations, the first model may describe the first resonance frequency111as a function of the first bias112and the second resonance frequency121as a function of the second bias122when the first qubit110and the second qubit120are isolated from other qubits or couplers.

In some implementations, if some conditions for the variables of the first model are known a priori, these may be incorporated into the first model.

The parameters of the first model will be referred to as a first set of parameters. The first set of parameters may be specific to experimental settings and/or the characteristics of the specific type of qubits110,120.

In step220, the first set of parameters is determined such that the first model can be used to estimate the first resonance frequency111at a given value of the first bias112and the second resonance frequency121at a given value of the second bias122.

In some implementations, the first set of parameters may be obtained by generating a first data set by measuring the first resonance frequency111at a plurality of values of the first bias112and by measuring the second resonance frequency121at a plurality of values of the second bias122and subsequently by fitting the first model to the first data set.

In some implementations, when the first data set is obtained at one of the qubits110,120, the resonance frequency of the other qubit110,120and the coupler130may be shifted such that both the first degree of coupling140-1,140-2and the second degree of coupling150are minimized or suppressed such that the influence of the indirect coupling140-1,140-2and the direct coupling150is negligible. For example, when the first set of data for the first qubit110is obtained, the second resonance frequency121may be shifted far enough such that the direct coupling, the second degree of coupling150is negligible. Also, the third resonance frequency131may be shifted such that the coupler130does not mediate any coupling between the first qubit110and the second qubit120.

In some implementations, in case the first qubit110and the second qubit120are the same type of qubits, as discussed in step210, one type of function may be used as the first model. In this case, the first set of data may be obtained from both the first qubit110and the second qubit120to determine the first set of parameters. In this case, in some implementations, the first set of parameters may include two separate sets of parameters for the first qubit110and the second qubit120although one function is used as the first model.

For example, in case the first qubit110and the second qubit120are the same type of superconducting qubits, the first data set may include the first resonance frequencies111of the first qubit110at each value of the first bias112and the second resonance frequencies121of the second qubit120at each value of the second bias122. The first data set including the data from both the first qubit110and the second qubit120may be used to fit the first model to determine the first set of parameters.

In some implementations, in case the first qubit110and the second qubit120are different types of qubits, as discussed in step210, the first model may comprise two separate functions which respectively describe the behavior of the first resonance frequency111and the second resonance frequency121as a function of the first bias112and the second bias122, respectively. In this case, the data obtained with the first qubit110, namely the first resonance frequencies111at each value of the first bias112is used to fit the part of the first model for the first qubit110and the data obtained with the second qubit120, namely the second resonance frequencies121at each value of the second bias122is used to fit the part of the first model for the second qubit120.

For example, when the first qubit110is a transmon qubit and the second qubit120is a flux qubit, the first model includes a model for the transmon qubit with undetermined parameters and a model for the flux qubit with undetermined parameters. These parameters may be evaluated separately for the first qubit110and the second qubit120.

As a result of steps210and220, the first model is obtained which evaluates the first resonance frequency111and the second resonance frequency121for a given value of or in response to the first bias112and the second bias122, respectively when they are isolated from the other qubits and couplers.

In step230, a second model, a qubit-coupler model, is provided which describes the behavior of the qubit-coupler system, namely a coupled system comprising the first qubit110and the coupler130or comprising the second qubit120and the coupler130. In particular, the second model may be a function which outputs the first resonance frequency111or the second resonance frequency121with the third bias131as a variable. The second model contains the first part of the first degree of coupling140-1and the second part of the first degree of coupling140-2as parameters.

The second model is a function of the third bias131. In particular, the second model describes a system containing the coupler130and only one of the qubits110,120. For example, the second model may describe the behavior of the system containing the first qubit110and the coupler130when the second resonance frequency121of the second qubit120is shifted such that neither the direct coupling140-1,140-2and the direct coupling150between the first qubit110and the second qubit120are suppressed. In this case, the second model contains the first part of the first degree of coupling140-1as a parameter to be determined.

If some conditions for the variables of the second model are known a priori, these may be incorporated into the second model.

In case the first qubit110and the second qubit120are the same type of qubits, one type of function with undetermined parameters may be used as the second model, which applies both to the coupling between the first qubit110and the coupler130, the first part of the first degree of coupling140-1, and to the coupling between the second qubit120and the coupler130, the second part of the first degree of coupling140-2.

In case the first qubit110and the second qubit120are different types of qubits, the first model may comprise two separate functions which respectively describe the behavior of the first resonance frequency111as a function of the third bias131and the behavior of the second resonance frequency121as a function of the third bias131.

The parameters of the second model will be referred to as a second set of parameters. The second set of parameters may be specific to experimental settings and/or the characteristics of the specific type of qubits110,120and the coupler130.

In some implementations, the second set of parameters may comprise two separate sets of parameters, respectively for the first system including the first qubit110and the coupler130and the second system including the second qubit120and the coupler. The former includes the first part of the first degree of coupling140-1and the latter includes the second part of the first degree of coupling140-2.

In step240, the second set of parameters of the qubit-coupler model or the second model is determined such that the second model can be used to estimate the first resonance frequency111at a given value of the third bias132and the second resonance frequency121at a given value of the third bias132.

In some implementations, the second set of parameters may be obtained by generating a second data set by measuring the first resonance frequency111at a plurality of values of the third bias132and by measuring the second resonance frequency121at a plurality of values of the third bias132and subsequently by fitting the second model to the second data set.

In some implementations, when the second data set is obtained at one of the qubits110,120, the resonance frequency of the other qubit110,120may be shifted such that the second degree of coupling150are minimized. For example, in order to determine the second set of parameters of the second model which relates to a system containing the first qubit110and the coupler130, the second resonance frequency121may be shifted far enough such that the direct coupling, the second degree of coupling150is negligible or suppressed.

In some implementations, the second data set may be obtained at a plurality of values of the first bias112or at a plurality of values of the second bias122. For example, when the part of the second data set is obtained from a system containing the first qubit110and the coupler130, the data set may be obtained at a plurality of values of the first bias112such that the accuracy of determining the second set of parameters can be improved.

In case the first qubit110and the second qubit120are the same type of qubits, one type of function may be used as the second model. In this case, the first set of data may be obtained between two sets of experiments, namely between the first qubit110and the coupler130and the second qubit120and the coupler130, to determine the second set of parameters.

In some implementations, the second set of parameters may include separate sets of parameters for a first pair including the first qubit110and the coupler130and a second pair including the second qubit120and the coupler130. Each of the two sets of experiments may be used to fit the separate sets of the parameters. The set of parameters for the first pair including the first qubit110and the coupler130includes the first part of the first degree of coupling140-1. The set of parameters for the second pair including the second qubit120and the coupler130includes the second part of the first degree of coupling140-2.

For example, in case the first qubit110and the second qubit120are the same type of superconducting qubits, the resonance frequency of each qubit110,120can be recorded while sweeping the magnetic flux bias applied to the coupler130. Then the second data set includes the first resonance frequencies111of the first qubit110at a plurality of values of the third bias132and the second resonance frequencies121of the second qubit120at a plurality of values of the third bias132. The second data set including the data from both the first pair including the first qubit110and the coupler130and the second pair including the second qubit120and the coupler130may be used to fit the second model to determine the second set of parameters.

In case the first qubit110and the second qubit120are different types of qubits, as discussed in step230, the second model may comprise two separate functions which respectively describe the first resonance frequency111as a function of the third bias132and the second resonance frequency121as a function of the third bias132. In this case, the data obtained with the first pair including the first qubit110and the coupler130is used to fit the corresponding part of the second model, which describes the coupling between the first qubit110and the coupler130and the data obtained with the second pair including the first qubit110and the coupler130is used to fit the corresponding part of the second model, which describes the coupling between the second qubit120and the coupler130.

For example, when the first qubit110is a transmon qubit and the second qubit120is a flux qubit, the second model includes a model for the coupling between the coupler130and the transmon qubit with undetermined parameters and a model for the coupling between the coupler130and the flux qubit with undetermined parameters. These parameters may be evaluated separately.

In some implementations, the second model may comprise a coupled two-level-system model when the coupler130comprises a system which can be described as a two-level-system such as a transmon qubit.

As a result of steps230and240, the second model, qubit-coupler model, is obtained which evaluates the first resonance frequency111at a given value of the third bias132when the second qubit120is decoupled from the first qubit110and evaluates the second resonance frequency121at a given value of the third bias132when the first qubit110is decoupled from the second qubit120.

Also, the coupling efficiencies between the first qubit110and the coupler130and between the second qubit120and the coupler130are evaluated, corresponding to the first part of the first degree of coupling140-1and the second part of the first degree of coupling140-2, respectively.

In step250, a third model, a qubit-qubit model, is provided which describes the behavior of the qubit-qubit system, namely a coupled system comprising the first qubit110and the second qubit120. In particular, the third model may be a function which outputs the first resonance frequency111with the second bias122as a variable or outputs the second resonance frequency121with the first bias112as a variable. The third model contains the second degree of coupling150as a parameter.

In some implementations, the third model may include two functions, one of which is formulated as a function which outputs the first resonance frequency111with the second bias122as a variable, and the other of which is formulated as a function which outputs the second resonance frequency121with the first bias112as a variable.

Alternatively, in some implementations, the third model may be formulated as a function of both the first bias112and the second bias122as variables.

In some implementations, the third model may be formulated as a function of the first bias112, the second bias122and the third bias132as variables.

If some conditions for the variables of the third model are known a priori, these may be incorporated into the third model. The parameters of the third model will be referred to as a third set of parameters. The third set of parameters may be specific to experimental settings and/or the characteristics of the specific type of qubits110,120.

In some implementations, the third set of parameters may include the values for the variables of the third model other than the first bias112or the second bias122which can be controlled and set as desired. For example, when the third model is formulated as a function of both the first bias112and the second bias122as variables, either the value of the first bias112or the second bias122may be included in the third model. For another example, the coupling between the first qubit110and the second qubit120may be dependent on the third bias132and a specific value of the third bias132may be incorporated into the third model. Alternatively, the value of the third bias132may be a part of the third set of parameters.

In step260, the third set of parameters of the qubit-qubit model or the third model is determined such that the third model can be used to estimate the first resonance frequency111at a given value of the second bias122or to estimate the second resonance frequency121at a given value of the first bias112.

In some implementations, the third set of parameters may be obtained by generating a third data set by measuring the first resonance frequency111at a plurality of values of the second bias122and by measuring the second resonance frequency121at a plurality of values of the first bias112and subsequently by fitting the third model to the third data set.

In some implementations, when the third data set is obtained, the resonance frequency of the coupler130may be set at a predetermined position. The corresponding value of the third bias132may be incorporated into the third model or the third bias132may be part of the third set of parameters such that it can be evaluated by fitting the third data set to the third model. For example, when the coupler130is a transmon qubit, the third bias132may be set such that the third transition frequency131is at the maximum frequency of the transmon qubit.

Alternatively, in some implementations, when the third data set is obtained, the resonance frequency of the coupler130may be shifted far from the first resonance frequency111and the second resonance frequency121such that the first degree of coupling140-1,140-2is negligible or suppressed.

As discussed in step230, in some implementations, the third model may be formulated as two functions. One function may output the first resonance frequency111with the second bias122as a variable, and the other function may output the second resonance frequency121with the first bias112as a variable. Alternatively, in some implementations, the third model may be formulated as a function of both the first bias112and the second bias122as variables. In both cases, the third data set may be generated by measuring the first resonance frequency111at a plurality of values of the second bias122and by measuring the second resonance frequency121at a plurality of values of the first bias112.

In some implementations, the third data set may be generated by measuring only the first resonance frequency111at a plurality of values of the second bias122or by measuring only the second resonance frequency121at a plurality of values of the first bias112. This is in case the third data set obtained in only one of the measurements provides a necessary level of accuracy in determining the third set of parameters.

In some implementations, the third model may comprise a coupled two-level-system model.

As a result of steps250and260, the third model, qubit-qubit model, is obtained which evaluates the first resonance frequency111in response to the second bias122and/or the second resonance frequency121in response to the first bias112. Also, the coupling efficiency between the first qubit110and the second qubit120is evaluated, corresponding to the second degree of coupling150.\

In step270, a fourth model is provided which describes the behavior of the complete qubit-coupler-qubit system100, namely a coupled system comprising the first qubit110, the second qubit120and the coupler130as described inFIG.1.

The fourth model may include the first part of the first degree of coupling140-1, the second part of the first degree of coupling140-2and the second degree of coupling150as parameters or coefficients.

In some implementations, the fourth model may be a three coupled two-level-system model. For example, the fourth model may be formulated as a 3×3 Hamiltonian in the single-excitation subspace in the rotating frame, as will be discussed in more detail later. In particular, the fourth model may describe the system dynamics of the qubit-coupler-qubit system100and may be formulated as a function of the first resonance frequency111, the second resonance frequency121and the third resonance frequency131. As a result of steps210and220, the first resonance frequency111and the second resonance frequency121may be converted to the first bias112and the second bias122, respectively and vice versa.

In some implementations, the fourth model may describe the system dynamics of the qubit-coupler-qubit system100and may be formulated as a function of the first bias112, the second bias122, and the third bias132.

FIG.3is a flowchart that illustrates a method of initializing a qubit-coupler-qubit system.

The method300of initializing the qubit-coupler-qubit system100as described inFIG.1includes the following steps.

In step310, the qubit-coupler-qubit system100is calibrated and a model describing the qubit-coupler-qubit system100is provided. Step310is described in detail inFIG.2. As a result of the calibration process of the qubit-coupler-qubit system100, as described inFIG.2, the respective relations between the first resonance frequency111and the second resonance frequency121and the first bias112and the second bias122are identified (steps210and220). In addition, the first degree of coupling140-1,140-2is evaluated (steps230and240). The second degree of coupling150is evaluated (steps250and260). A model is provided describing the system dynamics of the qubit-coupler-qubit system100and may be formulated as a function of the first resonance frequency111, the second resonance frequency121and the third resonance frequency131(step270).

In step320, a first idle resonance frequency111′ of the first qubit110and a second idle resonance frequency121′ of the second qubit120are provided.

The first idle resonance frequency111′ and the second idle resonance frequency121′ are the desired initial qubit frequencies. These may be determined by the requirement of the computation algorithm to be performed. As discussed above, the qubit-coupler-qubit system100may be a part of a larger system containing additional qubits and couplers or a plurality of the qubit-coupler-qubit system100. Therefore, the first idle frequency111′ and the second idle frequency121′ may be determined in view of the initialization of the larger system.

In step330, a value of the third resonance frequency131is determined, at which a difference between the first degree of coupling140-1,140-2and the second degree of coupling150is minimized or suppressed.

The value of the third resonance frequency131or the third bias132may be evaluated using the fourth model. When the first resonance frequency111and the second resonance frequency121are substituted by the first idle resonance frequency111′ and the second idle frequency121′, respectively, the only remaining variable of the fourth model is the third resonance frequency131or equivalently, the third bias132. The third resonance frequency131can be determined such that the coupling between the first qubit110and the second qubit120, including the indirect coupling140-1,140-2and direct coupling150, is minimized. At the value of the third resonance frequency131which minimizes the coupling between the first qubit110and the second qubit120, the indirect coupling mediated by the coupler130, which includes the first part of the first degree of coupling140-1and the second part of the first degree of coupling140-2may match the second degree of coupling150in magnitude but acts in an opposite direction such that the coupling mediated by the coupler130and the direct coupling between the first qubit110and the second qubit120cancel each other. In this case, the difference between the coupling mediated by the coupler130and the direct coupling between the first qubit110and the second qubit120may be approximately or effectively minimized. The term “approximately” here may refer to a tolerance below, for example, 5% of the stated value.

For example, in case the fourth model is formulated as may be as a 3×3 Hamiltonian in the single-excitation subspace in the rotating frame representing a three coupled two-level-system model, the odd mode qubit eigenfrequency and the even mode qubit eigenfrequency may be evaluated from the Hamiltonian. The third resonance frequency131at which the coupling between the first qubit110and the second qubit120is minimized may be found by equating the even mode qubit eigenfrequency and the odd mode qubit eigenfrequency. This will be discussed in more detail later.

In step340, the third bias132is provided to the coupler130according to the determined value of the third resonance frequency131of step330.

The concepts described inFIGS.1to3apply to any types of qubits as the first qubit110and the second qubit120as long as the transition frequencies111,121are controlled by providing appropriate biases112,122.

As an example, the procedures described inFIGS.2and3will be exemplified with frequency tunable transmon (FT-XMON) qubits as the first qubit110and the second qubit120in this specification. The transmon qubits are superconducting qubits where a specific ratio of junction critical current of the SQUID and capacitance is met to be in the transmon regime. However, other designs are possible for the first qubit110and the second qubit120. For example, a ratio of junction critical current of the SQUID and capacitance not in the transmon regime may be chosen, such as fluxmon qubits or gmon qubits, among others.

Similarly, the concepts and the procedures described inFIGS.1to3apply to any types of resonator as the coupler130in which the resonance frequency131can be controlled by providing appropriate bias132. However, in the examples below, the coupler130will be assumed to be another superconducting qubit mentioned above or a quantum LC resonator containing a Josephson junction as a nonlinear element such as a DC SQUID or an RF SQUID.

FIG.4shows a schematic of a frequency tunable transmon qubit and an energy diagram.

A frequency tunable transmon (FT-XMON) qubit, or a transmon qubit400is a parallel LC circuit. The LC circuit comprises a capacitor401and a nonlinear inductor402. The nonlinear inductor402is configured such that its inductance varies according to the level of the applied current.

The transmon qubit400may be constructed with a superconducting material, a material which becomes superconducting below a transition temperature. For example, the LC circuit may be constructed as a coplanar waveguide within an aluminum layer deposited on a dielectric substrate such as Sapphire or Silicon. The superconducting transition temperature of the aluminum is around 1K, and the transmon qubit400may be operated at around 10 mK temperature.

In some implementations, the capacitance, which is represented as the capacitor401inFIG.4a, may be determined by the spatial extent within the plane of the substrate or the geometry of the coplanar waveguide forming the transmon qubit400.

In some implementations, the nonlinear inductor402may include a Josephson junction403,404. The Josephson junctions403,404may be formed by a thin insulating barrier interrupting two superconducting electrodes. For example, a Josephson junction may form an aluminum-aluminum oxide-aluminum structure.

In some implementations, the nonlinear inductor402may be a superconducting quantum interference device, or a SQUID402. As in the example ofFIG.4a, the SQUID402may be formed as a superconducting loop including two parallel Josephson junctions, a first Josephson junction403and a second Josephson junction404. The transition frequency or the resonance frequency of the transmon qubit400can be dynamically changed or tuned by providing a magnetic flux threaded through the loop of the SQUID402because the magnetic flux threading the SQUID loop402changes the effective inductance of the SQUID loop402.

In some implementations, in place of the SQUID402with two Josephson junctions403,404, a single Josephson junction can be used as the nonlinear inductor402.

In the examples below, the first qubit110and the second qubit120will be taken to be the transmon qubit400. The transition frequency of the transmon qubit400corresponds to the first resonance frequency111or the second resonance frequency112in the diagram ofFIG.1and the magnetic field or the magnetic flux corresponds to the first bias112or the second bias122in the diagram ofFIG.1.

The magnetic flux is provided with an external magnetic flux drive (not shown inFIG.4), which will be referred to as a Z control line in this specification. For example, the Z control line may include a current source and a coil positioned in the vicinity of the transmon qubit400such that a magnetic flux is generated when the current flows through the coil. Since the mutual inductance between the external magnetic flux drive and the SQUID loop402depends on the specific geometry of the transmon qubit400and the Z control line and the experimental setting such as the distance between the transmon qubit400and the Z control line, the mutual inductance may be difficult to design or predict a priori. Therefore, the mutual inductance between the Z control line and the transmon qubit400may be set as a free parameter of the first model and calibrated once the transmon qubit400is operational with the Z control line at a cryogenic condition, as part of steps210and220discussed above.

The inductance of the Josephson junction403,404, so-called Josephson inductance, is known to be dependent on the phase difference across the junction and be proportional to the critical current of the Josephson junction. Assuming a fixed critical current, Josephson inductance is proportional to inverse cosine of the phase across the Josephson junction.

L~1cos⁢(π⁢φφ0)

φ represents the magnetic flux through the SQUID402, which is here assumed to be a DC SQUID. φ0represents the magnetic flux quantum. Therefore, the Josephson inductance varies as a function of the magnetic flux provided through the loop of the DC SQUID402.

Since the transmon qubit400is constructed as an LC circuit, where L corresponds to the Josephson inductance of the Josephson junction402,403. It is well known that the resonance frequency of an LC circuit is given by

f=12⁢π⁢LC

Therefore, the resonance frequency, or the transition frequency of the transmon qubit400may be expressed as a function of the magnetic flux through the DC SQUID as

fQubit(φ)=fmax⁢cos⁡(π⁢φφ0)

fmaxrepresents the maximum resonance frequency of the qubit. This description applies in general to any superconducting qubits constructed as an LC circuit with the L as a nonlinear inductance. The maximum resonance frequency of the qubit, fmaxalso represents a flux insensitive point. Around the flux insensitive point, the resonance frequency of the qubit400will be more robust to the fluctuation of the magnetic flux than other values of the magnetic flux. This is because the cosine squared curve is symmetric around the flux insensitive point and the rate of change of the resonance frequency with respect to the change of the magnetic flux is smallest. Therefore, the maximum resonance frequency or the flux insensitive point may be relatively robust to the noise arising from spurious fluctuation of the magnetic flux from the environment. The measurement data of the cosine squared behavior will be shown inFIG.5b.

When the first qubit110and the second qubit120are transmon qubits, the first resonance frequency111and the second resonance frequency121are proportional to the square root of cosine of the magnetic flux applied to the first qubit110and the second qubit120, respectively, namely the first bias112and the second bias122. Therefore, the first transition frequency111and the second transition frequency121are a periodic function of the first bias112and a second bias122, respectively.

The maximum resonance frequency of the qubit, fmaxdepends on the specific geometry of the transmon qubit400, for example, the capacitance value of the transmon qubit400and the dimensions of the Josephson junctions403,404. Therefore, the maximum resonance frequency may be difficult to determine or predict a priori. Therefore, the maximum frequency may be set as a free parameter of the first model and calibrated once the transmon qubit400is operational with the Z control line at a cryogenic condition, as part of steps210and220discussed above. As discussed above, since the transition frequency of the transmon qubit400is a periodic function of the magnetic flux, fmaxmay occur for a plurality of values of the magnetic flux through the SQUID loop402. For a transmon qubit, typical fmaxmay be between about 3 GHz and about 10 GHz.

Since the LC circuit forming the transmon qubit400includes a nonlinear inductor402, the transmon qubit400is a nonlinear resonator with a resonance frequency in the microwave regime. The anharmonicity of the nonlinear resonator leads to unequally spaced energy levels as shown in a potential energy diagram430of the transmon qubit400. The potential energy diagram430shows the potential energy of the transmon qubit400in a vertical axis432as a function of the magnetic flux traversing the SQUID loop402of the transmon qubit400in a horizontal axis431. The potential energy diagram430shows that the potential energy shows a periodic behavior as the magnetic flux increases. As discussed above, since the magnetic flux changes the transition frequency of the transmon qubit400, one of the potential wells may be chosen for a desired operation of the transition frequency. The lowest two levels of one of these potential wells, are used as a qubit, namely ‘0’ and ‘1’ states. The quantum mechanical two-level-system defined by these two levels will be referred to as a qubit transition, a 0-1 transition or a single excitation subspace in this specification.

The transmon qubit400is coupled to a readout resonator410. The transmon qubit400is connected to an interface circuitry420via the readout resonator410.

In some implementations, the transmon qubit400may be coupled to the readout resonator410via a capacitor, as shown inFIG.4a.

In some implementations, the readout resonator410is coupled inductively to the interface circuitry420.

In some implementations, the interface circuitry420may include a transmission line.

In some implementations, the interface circuitry420may include driving electronics. The driving electronics may include an arbitrary waveform generator (AWG). The driving electronics may be configured to generate a probing signal, an RF or microwave signal, to drive the readout resonator410for the state detection of the transmon qubit400.

In some implementations, the interface circuitry420may include detection electronics. The detection electronics may include an amplifier network comprising one or more of an impedance matched parametric amplifier (IMPA) and a high electron mobility transistor (HEMT). When the RF or microwave signal generated at the driving electronics as part of the interface circuitry420is sent into the readout resonator410, the reflected signal is sent into the amplifier network for the state detection of the transmon qubit400.

The readout resonator410is introduced to enhance the coherence of the qubit400by protecting the transmon qubit400from the dissipation caused by the interface circuitry420. In order to act as a short at the qubit transition frequency, the readout resonator410is detuned from the transition frequency of the transmon qubit400. Since the transmon qubit400and the readout resonator410are strongly coupled, the shift of the transition frequency of the qubit400causes the resonance frequency of the readout resonator410to shift. The resonance frequency of the readout resonator410changes depending on the state of the qubit400.

Since the coupling efficiency between the transmon qubit400and the readout resonator410depends on the specific geometry such as the physical distance and the relative arrangement and the experimental setting, it may be difficult to determine or predict a priori. Therefore, the coupling efficiency between the transmon qubit400and the readout resonator410may be calibrated once the transmon qubit400is operational at a cryogenic condition, in steps210and220discussed above.

In some implementations, the state detection of the transmon qubit400is achieved using a dispersive detection scheme, or equivalently, dispersive measurement or dispersive readout scheme. In order to read out or detect the state of any qubit, a probing signal, e.g., a travelling microwave, may be sent along a readout transmission line of the interface circuitry420into the readout resonator410. The frequency of the probing signal generated by the driving electronics may be in the vicinity of the resonance frequency of the readout resonator410. Depending on the internal quantum mechanical state of the qubit400, the intensity or phase of the probing signal transmitted along the readout transmission line may be altered because the reflectivity of the readout resonator coupled to the qubit changes depending on the state of the qubit. This allows for the state detection of the qubits.

In some implementations, the interface circuitry420may include a Purcell filter. A high degree of coupling between the driving electronics of the interface circuitry420and the readout resonator410may improve the detection speed but may also affect the coherence time of the qubit400by providing further decay channels. In order to mitigate this effect, the Purcell filter, configured as a relatively low quality factor resonator, around Q of 30, may be introduced between the readout resonator410and the driving electronics. In some implementations, the readout resonator410is coupled inductively to the Purcell filter.

In the rest of the specification, the method200of calibrating the qubit-coupler-qubit system100and the method300of initializing the qubit-coupler-qubit system100will be described with two transmon qubits400as the first qubit110and the second qubit120. However, as stated above the methods200,300described herein are not limited to the transmon qubits400for the first qubit110and the second qubit120.

The coupler130of the qubit-coupler-qubit system100in the following example will be taken to be another transmon qubit400. However, as stated above the methods200,300described herein are not limited to the transmon qubits400as the coupler130. The coupler130may be a qubit or a resonator whose resonance frequency131is controllable with the third bias132.

FIG.5ashows a measurement data of the resonance frequency of the readout resonator of a transmon qubit as a function of the flux bias applied to the transmon qubit.

A first graph500shows a measurement of the resonance frequency of the readout resonator410coupled to the transmon qubit400, which acts as, for example, the first qubit110of the qubit-coupler-qubit system100. Similar measurement can be repeated for the readout resonator410of the transmon qubit400which acts as the second qubit120of the qubit-coupler-qubit system100.

As part of steps210and220, the resonance frequency of the readout resonator410is measured as the first bias112, the magnetic flux applied by the Z control line into the SQUID loop402is changed. The first bias112is represented in a vertical axis510, labelled as ‘Qubit bias (dacamp).’ ‘dacamp’ represents a DAC voltage controlling the Z control line.

The frequency of the probing signal, labelled as ‘Resonator drive frequency (GHz)’, is represented in a horizontal axis520.

As discussed above, the probing signal, generated by the driving electronics of the interface circuitry420is reflected from the readout resonator410and couples back into the interface circuitry420to be detected by the detection electronics. The lines on the first graph500represents the frequency position of the readout resonator410and the width of the line in the horizontal direction represents the linewidth of the readout resonator410.

As the first bias112is changed, the first transition frequency111correspondingly changes. Since the transmon qubit400and the readout resonator410are strongly coupled, the resonance frequency of the readout resonator410shifts accordingly.

Although the resonance frequencies of the qubits110,120,400and the resonance frequency of the corresponding readout resonators410are designed to be at predetermined frequencies, these can be confirmed with the measurement shown in the first graph500, thereby providing a calibration of the isolated qubits110,120. Also the observation that the resonance frequency of the readout resonator410shifts as a function of the first bias112confirms that the readout resonator410and the transmon qubit400are coupled to each other and that the Z control line is connected and is operational.

The first graph500shows three segments of measured traces, which are separated by a first line501and a second line502, namely one above the first line501, one between the first line501and the second line502and one below the second line502. Two of the segments are tangential to the first line501which is parallel to the horizontal axis, representing a value of the first bias112, or the flux threaded through the SQUID loop402, around 0.6 dacamp. Two of the segments are tangential to the second line502which is parallel to the vertical axis, representing a value of the first bias112, or the flux threaded through the SQUID loop402, around −0.75 dacamp. The segment shown in the middle is tangential to both the first line501and the second line502. For example, around the first line501, the middle segment of the trace diverges towards the negative direction in the horizontal axis, and the top segment of the trace diverges towards the positive direction in the horizontal axis. This so-called “avoided crossing” behavior of the resonance frequencies of the coupled resonator system is a signature that two resonators, in this case the transmon qubit400and the readout resonator410, are strongly coupled. A degree of separation between the two segments of the measurement curves on either side of the first line501and the second line502reflects the coupling efficiency between the readout resonator410and the transmon qubit400. Two coupled resonators or oscillators can be described by the coupled two-level-systems model. Therefore, in some implementations, the first model, or the qubit model for steps210and220may be formulated based on the coupled two-level systems model.

A third line503, which is parallel to the horizontal axis, represents a value of the first bias112, or the flux threaded through the SQUID loop402, around 0.00 dacamp, which represents DAC amplitude. The third line503points to a point of the measurement trace where the resonance frequency curve is flat with respect to the change of the first bias112or the magnetic flux. This point corresponds to the maximum frequency of the qubit400or the flux insensitive point, as discussed above. As explained above, this point occurs theoretically when the magnetic flux is zero, but due to the stray magnetic fields from the experimental setting near the transmon qubit400and the readout resonator410, the flux insensitive point may be offset from zero. Therefore, in some implementations, the first model, or the qubit model for steps210,220may include the flux offset as a free parameter.

When the flat point occurs near the third line503, the line tangential to the measurement trace points to, corresponds to the maximum frequency of the transmon qubit400in the horizontal axis520. InFIG.5a, the maximum frequency is approximately 4.520 GHz. Although the maximum frequency is predetermined in the design stage, the actual maximum frequency may turn out to be different from the design value. Therefore, in some implementations, the first model, or the qubit model for steps210and220may include the maximum frequency of the qubit400as a free parameter.

The vertical axis510inFIG.5ais in ‘dacamp’ which represents DAC amplitude. InFIG.5a, the DAC amplitude ranges from −1V to 1V. The Z control line configured to generate magnetic flux for the first bias112, may include a current source, controlled by the DAC voltage and a transducer to generate magnetic flux from the current generated by the current source. Therefore, the DAC amplitude in the vertical axis510is in general proportional to the magnetic flux threaded through the SQUID loop420. However, the conversion of the DAC amplitude and the physical units of the actual magnetic flux threaded through the SQUID loop420may depend on various factors, such as the mutual inductance between the Z control line and the SQUID loop420of the qubit400. Therefore, in some implementations, the first model, or the qubit model for steps210and220may include the mutual inductance between the qubit and the Z control line as a free parameter.

Steps210and220are performed by taking measurements of the resonance frequency of the readout resonator410as a function of the first bias112or the flux bias applied to the qubit400, and fitting the data to the first model.

In some implementations, the first model, or the qubit model is formulated based on the coupled two-level-system model and may include the flux offset of the flux insensitive point, the mutual inductance between the qubit400and the Z control line, the maximum frequency of the qubit400, and the coupling efficiency between the qubit400and the readout resonator410. The first model may include more free parameters depending on the design of the qubit400or the experimental settings, such as the interface circuitry420. Also, some of the parameters above may be removed from the first model if the value of that parameter can be identified a priori with sufficient accuracy.

As mentioned above for steps210and220, when the measurements shown inFIG.5ais performed on the first qubit110, the second qubit120and the coupler130may be shifted away from the relevant frequency range for the first qubit110, namely the resonance frequency of the transmon qubit400and the readout resonator410such that they are not coupled to the first qubit110. In some implementations, the resonance frequencies of the second qubit120and the coupler130, when the coupler130is another qubit, may be shifted towards their minimum frequencies.

The measurements in the first graph500shows the resonance frequency of the readout resonator410coupled to the qubit400and the measured frequency is not the bare resonance frequency of the readout resonator410. The resonance frequency in the first graph500corresponds to an eigenmode of the coupled two-level-system model, equivalently, a hybridized mode or a mixed state. Therefore, as a result of performing steps210and220, including the measurements shown inFIGS.5aand5b, the bare frequency of the qubit400, isolated from the readout resonator400, and the bare resonance frequency of the readout resonator410isolated from the qubit400may be inferred. These may be used for the three coupled two-level-system model describing the qubit-coupler-qubit system100which is on the bare transition frequency of the qubits111,121and the bare transition frequency of the coupler131.

The calibration of a single transmon qubit400may include the calibration of three inputs. The first is the Z control line, as discussed above. The second input is an XY control line for microwave control of the state of the qubit400, for example, to achieve π and π/2 pulses which lead to the controlled rotation of the qubit state and to perform Rabi oscillations of the qubit400. Gate operations on qubits110,120,400may involve a combination of one or more of Z control signals and one or more of XY control signals. The third input is the readout line through the interface circuitry420, mentioned above. However, this specification will focus on the aspect of resonance frequency control and the initialization of the qubits110,120,400and the second input, the XY control line and the third input, the readout line, will be assumed to have been calibrated before or be calibrated concurrently with the procedure described inFIGS.2and3.

FIG.5bshows a measurement data of the resonance frequency of a transmon qubit as a function of the flux bias applied to the transmon qubit.

A second graph550shows a measurement of the resonance frequency of the transmon qubit400coupled to the readout resonator410, which acts as, for example, the first qubit110of the qubit-coupler-qubit system100. Similar measurement can be repeated for the transmon qubit400which acts as the second qubit120of the qubit-coupler-qubit system100.

The resonance frequency of the transmon qubit400, represented in a vertical axis560, is measured as the first bias112, the magnetic flux applied by the Z control line into the SQUID loop402, represented in a horizontal axis570, labelled as ‘DAC Z Bias’, is changed.

Also as part of steps210and220, this measurement maps the DAC voltage in the horizontal axis570to the frequency of the qubit400. At each value of the first bias112, qubit spectroscopy is performed via the XY control line to identify the resonance frequency of the qubit400. The first bias112, the flux bias to the transmon qubit400applied by the Z control line, is represented in the horizontal axis570. The resonance frequency of the transmon qubit400is represented in the vertical axis560, labelled as ‘Peak Spectroscopy Freq. (GHz).’

In some implementations, at each value of the first bias112, the flux bias to the transmon qubit400applied by the Z control line, represented in the horizontal axis570, the XY control line connected to the transmon qubit400is driven with a microwave pulse which is fixed at the desired or the target first transition frequency111known from the design, or inferred from the measurements shown inFIG.5a. Multiple pulses are sent to the qubit400via the XY control line with variable amplitude. When the first bias112places the qubit400on resonance, the qubit400undergoes Rabi oscillations, cycling between 0 and 1 states as the amplitude of the XY control pulse increases. From these oscillations at each amplitude, the resonance frequency of the qubit400can be identified. This can be detected with the readout line of the interface circuitry420. This measurement can be repeated by varying the frequency of the XY control pulses.

In some implementations, when the first transition frequency111is known for at least one value of the first bias112, the measurement may be made adaptively, namely by changing the first bias112and scanning the frequency range of the XY control pulses within a reasonable range such that the full possible frequency range does not have to be scanned with the XY control pulse and the Z control signal, or the first bias112. For example, the measurements in the second graph550may be performed near the idle bias, which corresponds to a desired initialized frequency of the first qubit110. The idle bias of the transmon qubit400may be determined to be near the flux insensitive point.

The measurements shown in the second graph550provides the measurement of the maximum frequency of the qubit400, which may be more accurate from the maximum frequency obtained from the measurements inFIG.5a. Therefore, the first model, or single qubit model for the measurements in the second graph550may include the formula

fQubit(φ)=fmax⁢cos⁡(π⁢φφ0)
which was presented above and represented as a curve551inFIG.5b. In some implementations, the flux offset and the mutual inductance, explained inFIG.5a, may be included in this formula as free parameters.

In the second graph550, the measurement data points, represented as dots, are fitted to the curve551to evaluate the maximum frequency of the qubit400.

FIG.6shows the resonance frequency of a transmon qubit as a function of the flux bias applied to a coupler.

A first graph600shows the resonance frequency of the transmon qubit400, which acts as, for example, the first qubit110of the qubit-coupler-qubit system100. Similar measurement can be repeated for the transmon qubit400which acts as the second qubit120of the qubit-coupler-qubit system100.

As mentioned above for steps230and240, when the measurements shown inFIG.6is performed on the first qubit110and the coupler130, the second qubit120may be shifted away from the relevant frequency range for the first qubit110, namely the resonance frequency of the transmon qubit400and the readout resonator410such that the second qubit120is not coupled to either the first qubit110or the coupler130.

As part of steps230and240, the resonance frequency of the transmon qubit400is measured as the third bias132applied to the coupler130is changed. For example, the third bias132can be a magnetic flux applied by the Z control line into the SQUID loop of the coupler130when the coupler130is a superconducting qubit. For example, the coupler130may be a transmon qubit400.

The third bias132is represented in a horizontal axis620, labelled as ‘Bias [V].’ As the third bias132is varied from about −0.65V to 0.35V, the frequency of the probing signal, labelled as ‘Frequency [arb.]’, represented in a vertical axis610in an arbitrary unit, is also varied.

As discussed above, the probing signal, generated by the driving electronics of the interface circuitry420is reflected from the readout resonator410and couples back into the interface circuitry420to be detected by the detection electronics. The lines in the first graph600corresponds to a resonance frequency or an eigenmode of the qubit-coupler system, which corresponds to a hybridized mode or a mixed state of a coupled two-level system model of the qubit-coupler system.

As the third bias132is changed, the first transition frequency111changes because the first qubit110and the coupler130are coupled to each other via the first part of the first degree of coupling140-1. As shown inFIG.5a, since the transmon qubit400and the readout resonator410are strongly coupled, the resonance frequency of the readout resonator410of the first qubit110shifts accordingly.

The first graph600shows three segments of measured traces, which are separated by a first line601and a second line602, namely one on the left hand side of the first line601, one between the first line601and one on the right hand side of the second line602. Two of the segments are tangential to the first line601representing a value of the third bias132, or the flux applied to the coupler130, around −0.6V. Two of the segments are tangential to the second line602representing a value of the third bias132around 0.3V. The segment shown in the middle is tangential to both the first line601and the second line602. The “avoided crossing” behavior, discussed above inFIG.5ais shown around the first line601and the second line602, which suggests that the first qubit110or the transmon qubit400and the coupler130are strongly coupled. A degree of the separation between the two segments of the measurement curves on either side of the first line501and the second line502relate to the coupling efficiency between the first qubit110and the coupler130.

Since this behavior can be described by the coupled two-level-systems model, in some implementations, the second model, or the qubit-coupler model for steps230and240may be formulated based on the coupled two-level systems model.

A third line603, which is parallel to the vertical axis, represents a value of the third bias132around −0.15V. The third line603points to a point of the measurement trace where the resonance frequency curve is flat with respect to the change of the third bias132. This point, corresponding to the maximum frequency of the qubit400or the flux insensitive point, may be offset from zero. Therefore, in some implementations, the second model, or the qubit-coupler model for steps230,240may include the flux offset as a free parameter.

When the flat point occurs near the third line603, the line tangential to the measurement trace points to corresponds to the maximum frequency of the transmon qubit400in the vertical axis610. Therefore, in some implementations, the second model, or the qubit-coupler model for steps230and240may include the maximum frequency of the qubit400as a free parameter.

The horizontal axis620inFIG.6may represent DAC amplitude for the third bias132or the coupler bias. When the coupler130is a transmon qubit, the Z control line configured to generate magnetic flux for the third bias132, may include a current source, controlled by the DAC voltage and a transducer to generate magnetic flux from the current generated by the current source. Therefore, the DAC amplitude in the horizontal axis620is in general proportional to the magnetic flux threaded through the SQUID loop of the coupler130or the coupler qubit. However, the conversion of the DAC amplitude and the physical units of the actual magnetic flux may depend on various factors, such as the mutual inductance between the Z control line and the SQUID loop420of the coupler qubit130,400. Therefore, in some implementations, the second model, or the qubit-coupler model for steps230and240may include the mutual inductance between the coupler qubit130and the Z control line as a free parameter.

Steps230and240are performed by taking measurements of the resonance frequency of the first qubit110,400as a function of the third bias132applied to the coupler130and fitting the data to the second model.

In some implementations, the second model, or the qubit-coupler model is formulated based on the coupled two-level-system model and may include the flux offset or the flux insensitive point, the mutual inductance between the coupler and the Z control line coupled to the coupler130, the maximum frequency of the first qubit110,400, and the coupling efficiency between the first qubit110and the coupler130. In particular the coupling efficiency between the first qubit110and the coupler130corresponds to the first part of the first degree of coupling140-1. If the same measurements are repeated between the second qubit110and the coupler130, the coupling efficiency included in the second model corresponds to the second part of the first degree of coupling140-2.

The second model may include more free parameters depending on the design of the qubit400or the experimental settings, such as the interface circuitry420. Also, some of the parameters above may be removed from the second model if the value of that parameter can be identified a priori with sufficient accuracy.

Steps230and240, the measurements and the fitting in the first graph600may be repeated for the second qubit120and the coupler130. Such measurements may provide, among others, the second part of the first degree of coupling140-2.

The first graph600also shows a result of fitting the data to the second model. The second model fitted to the data is shown as a solid line.

In some implementations, when the qubit-coupler-qubit system100is symmetrical in the sense that the first qubit110and the second qubit120are identical or the first part of the first degree of coupling140-1and the second part of the second degree of coupling140-2is identical, the steps230and240may be performed only on either with the first qubit110and the coupler130or with the second qubit120and the coupler130.

In some implementations, the steps230and240may be performed at multiple frequencies of the first qubit110or the second qubit120to improve accuracy in evaluation of the fitted parameters.

FIG.7shows the resonance frequency of a first transmon qubit as a function of the flux bias applied to a second transmon qubit.

A first graph700shows the resonance frequency of the transmon qubit400, which acts as, for example, the first qubit110of the qubit-coupler-qubit system100. In particular, the resonance frequency corresponds to an eigenmode of the qubit-qubit system, or a hybridized mode or a mixed state of a coupled two-level system model of the qubit-qubit system. Similar measurement can be repeated for the readout resonator410of the transmon qubit400which acts as the second qubit120of the qubit-coupler-qubit system100.

As part of steps250and260, the resonance frequency of the transmon qubit400is measured as the second bias122applied to the second qubit120is changed. For example, the second bias122can be a magnetic flux applied by the Z control line into the SQUID loop of the second qubit120.

In the first graph700, for example, the coupler130may be a transmon qubit400and set at the maximum frequency. However, in order to measure the coupling between the first qubit110and the second qubit120, the coupler qubit130does not necessarily have to be set at the maximum frequency. For example, if the minimum frequency of the coupler qubit130is known, the third bias132may be set such that the coupler qubit130is sent near the minimum frequency.

The second bias122is represented in a horizontal axis720, labelled as ‘bias (dacamp)’ As the third bias132is varied from about −0.65V to 0.65V, the delay in the Ramsey fringe measurement related to ‘Ramsey qubit frequency [arb.]’ in a vertical axis710, is also varied. In order to measure the first transition frequency111, Ramsey fringe measurements may be made.

In Ramsey fringe measurements, the first qubit110or the transmon qubit400is prepared along the x-axis of the Block sphere by applying a π/2 pulse using the X control line. After a time delay, a second π/2 pulse is applied to the transmon qubit400using the X control line. If the frequency of the π/2 pulse, known and controllable a priori, coincides with the transition frequency of the transmon qubit400, or the first transition frequency111, the measured qubit state after the second π/2 pulse is ‘1’ state or the excited state regardless of the delay, on the assumption that the coherence time of the transmon qubit400is much longer than the longest delay used for the Ramsey measurement. However, when the frequency of the π/2 pulse does not match the first transition frequency111, the measured qubit state after the second π/2 pulse appears to precess at a frequency which corresponds to the difference between the frequency of the π/2 pulse and the first transition frequency111.

From the precession frequency of along the vertical line of measured data at each value of the second bias122and the known frequency of the π/2 pulse used for the Ramsey measurement, the first transition frequency111of the first qubit110or the transmon qubit400can be evaluated by taking the difference.

An alternative method of measuring the qubit transition frequency111,121was presented inFIG.5b, namely the qubit spectroscopy. Ramsey measurements may facilitate the measurements of the qubit transition frequency111,121with a higher accuracy than the qubit spectroscopy. However, in some implementations, the first transition frequency111may be obtained with the qubit spectroscopy as discussed inFIG.5b.

The first graph700shows the first transition frequency111evaluated using the Ramsey measurements as dots. A vertical axis710of the first graph700, labelled as ‘Ramsey qubit frequency [arb.]’ represents the first transition frequency111in arbitrary unit.

As the second bias122is changed, the first transition frequency111changes because the first qubit110and the second qubit120are coupled to each other via the second degree of coupling150. The first graph700shows that the coupling between the first qubit110and the second qubit120show “avoided crossing” behavior around a first line701and a second line702. This suggests that the first qubit110and the second qubit120are strongly coupled.

Since this behavior can be described by the coupled two-level-systems model, in some implementations, the third model, or the qubit-qubit model for steps250and260may be formulated based on the coupled two-level systems model.

In some implementations, the third model, or the qubit-qubit model for steps250,260may include the coupling efficiency between the first qubit110and the second qubit120as a free parameter.

In some implementations, the third model, or the qubit-qubit model for steps250,260may include the flux offset as a free parameter.

In some implementations, the third model, or the qubit-qubit model for steps250,260may include the maximum frequency of the first qubit110as a free parameter.

The horizontal axis720of the first graph700inFIG.7may represent DAC amplitude for the second bias122. The Z control line configured to generate magnetic flux for the second bias122may include a current source, controlled by the DAC voltage and a transducer to generate magnetic flux from the current generated by the current source. Therefore, the DAC amplitude in the horizontal axis720of the first graph700is in general proportional to the magnetic flux threaded through the SQUID loop of the second qubit120. However, the conversion of the DAC amplitude and the physical units of the actual magnetic flux may depend on various factors, such as the mutual inductance between the Z control line and the SQUID loop420of the second qubit120. Therefore, in some implementations, the third model, or the qubit-qubit model for steps250and260may include the mutual inductance between the second qubit120and the Z control line as a free parameter.

Steps250and260are performed by taking measurements of the resonance frequency of the first qubit110as a function of the second bias122and fitting the data to the third model, or the qubit-qubit model.

In some implementations, the third model, or the qubit-qubit model is formulated based on the coupled two-level-system model and may include the flux offset or the flux insensitive point, the mutual inductance between the second qubit120and the Z control line coupled to the second qubit120, the maximum frequency of the first qubit110,400, and the coupling efficiency between the first qubit110and the second qubit120. In particular the coupling efficiency between the first qubit110and the second qubit120corresponds to the second degree of coupling150.

The second model may include more free parameters depending on the design of the qubit400or the experimental settings, such as the interface circuitry420. Also, some of the parameters above may be removed from the second model if the value of that parameter can be identified a priori with sufficient accuracy or evaluated in previous steps210to240.

Steps250and260or the measurements and the fitting in the first graph700may be repeated by varying the first bias112and measuring the second transition frequency121of the second qubit120. This may lead to more accurate measurement of the second degree of coupling.

The first graph700shows a result of fitting the data shown as dots to the third model. The second model fitted to the data is shown as a solid line on the first graph700.

In some implementations, the steps250and260may be performed at multiple frequencies of the first qubit110to improve accuracy in evaluation of the fitted parameters.

As discussed inFIG.3, once the qubit-coupler-qubit system100is calibrated as discussed above (step310), the qubit-coupler-qubit system100may be initialized.

For gate operations or measurements required for performing algorithms or computations, the transition frequencies111,121of the first qubit110and the second qubit120may be dynamically shifted by adjusting the first bias112and the second bias122. However, initially, the qubits110,120may be set at their respective initial frequencies111,121or idle frequencies111′,121′ and such that the coupling between the qubits140-1,140-2,150is minimized.

For given desired initial transition frequencies111,121or for a first idle frequency111′ for the first qubit110and the second idle frequency121′ for the second qubit120, the third bias132at which the coupling between the first qubit110and the second qubit120is minimized or suppressed.

In steps230and240andFIG.6, the first part of the first degree of coupling140-1and the second part of the first degree of coupling140-2was evaluated. This corresponds to an indirect coupling mediated by the coupler130.

In steps250and260andFIG.7, the second degree of coupling150was evaluated. This corresponds to a direct coupling between the first qubit110and the second qubit120. In some implementations, when the first qubit110and the second qubit120are transmon qubits400, the direct coupling or the second degree of coupling150may be capacitive coupling between the LC oscillators which form the transmon qubits400.

The overall coupling or the net coupling between the first qubit110and the second qubit120may be decomposed into an even (symmetric) mode and an odd (antisymmetric) mode. When the first idle frequency111′ and the second idle frequency121′ are equal, the first degree of coupling140-1,140-2mediated by the coupler130affects the even mode and the second degree of coupling150, direct coupling between the first qubit110and the second qubit120affects the odd mode. Since the direct coupling the net coupling corresponds to half the difference between the even mode and the odd mode, in some implementations, the third bias132may be set such that the direct coupling150and the indirect coupling140-1,140-2cancel each other. For example, the third transition frequency131of the coupler may be set to be higher than the first transition frequency111and the second transition frequency121.

In some implementations, the first degree of coupling140can be evaluated based on the second degree of coupling150and by measuring the net coupling between the first qubit110and the second qubit120.

In order to evaluate the third bias132at which the net coupling is minimized or suppressed such that the first qubit110and the second qubit120are decoupled (step330), a three coupled two-level-systems model can be made for the qubit-coupler-qubit system100. In formulating a Hamiltonian for the three coupled two-level-systems model, the first qubit110, the second qubit120and the coupler130will be labelled as a, b, c, respectively for notation. For example, the bare transition frequencies111,121,131will be noted as fa, fb, fc, respectively, and the coupling rate between the second qubit120and the coupler130will be noted as gbc. The Hamiltonian in the single-excitation subspace in the rotating frame is given by

ℋ=(fagabgacgabfbgbcgacgbcfc)

faand fbare bare transition frequencies111,121of the first qubit110and the second qubit120, respectively evaluated in steps210and220or inFIGS.4aand4b. As discussed above, a bare transition frequency of a transmon qubit correspond to the transition frequency when it is decoupled from its readout resonator. This is evaluated in steps210and220or inFIGS.5aand5b.

fcis a third transition frequency131of the coupler130.

gac, the coupling rate between the first qubit110and the coupler130, corresponds to the first part of the first degree of coupling140-1, evaluated in steps230and240or inFIG.6.

gbc, the coupling rate between the second qubit120and the coupler130, corresponds to the second part of the first degree of coupling140-2, evaluated in steps230and240or inFIG.6.

gab, the coupling rate between the first qubit110and the second qubit120, corresponds to the second degree of coupling150, evaluated in steps250and260or inFIG.7.

In some implementations, the third bias132or the third transition frequency131at which the net coupling is minimized can be found by diagonalizing the matrix and equating the odd mode eigenvalue or eigenfrequency and the even mode eigenvalue or eigenfrequency.

For example, when the qubit bare frequencies are equal such that fa=fb=f and the first part of the first degree of coupling140-1and the second part of the first degree of coupling140-2are equal such that gac=gbc=gcthe Hamiltonian can be diagonalized analytically without use of numerical evaluations as follows.

Since the coupling rate is proportional to the geometric mean of the resonance frequencies, the Hamiltonian can be rewritten as

ℋ=(fkq⁢f/2kc⁢f⁢fc/2kq⁢f/2fkc⁢f⁢fc/2kc⁢f⁢fc/2kc⁢f⁢fc/2fc)

The odd mode eigenfrequency and the even mode eigenfrequency are equated when
fc=f(2−kq)/(2−kc2/kq).

Therefore, as a result of step S330, the third resonance frequency131, fc, can be obtained based on the first part of the first degree of coupling140-1, the second part of the first degree of coupling140-2and the second degree of coupling150.

In step340, the third bias132can be set according to the third resonance frequency131, fc, obtained in step330to turn off the net coupling between the first qubit110and the second qubit120.

Implementations of the subject matter and operations described in this specification can be implemented in suitable circuitry where input powers are low enough, operating temperatures are below the superconducting temperature of the device, and low loss and low insertion loss are required. Examples of such circuitry may include quantum computational systems, also referred to as quantum information processing systems, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The terms “quantum computational systems” and “quantum information processing systems” may include, but are not limited to, quantum computers, quantum cryptography systems, topological quantum computers, or quantum simulators.

The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, e.g., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In some implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

Quantum circuit elements (also referred to as quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, the quantum circuit elements are configured to make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent and operate on information in more than one state simultaneously. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUID or DC-SQUID), among others.

In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and/or input/output operations on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and/or receive data from the quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors.

Fabrication of the quantum circuit elements and classical circuit elements described herein can entail the deposition of one or more materials, such as superconductors, dielectrics and/or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. Processes for fabricating circuit elements described herein can entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process can include, e.g., wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography).

During operation of a quantum computational system that uses superconducting quantum circuit elements and/or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (alternatively superconducting) material can be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of about 1.2 kelvin), indium (superconducting critical temperature of about 3.4 kelvin), NbTi (superconducting critical temperature of about 10 kelvin) and niobium (superconducting critical temperature of about 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature.

While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various components in the implementations described above should not be understood as requiring such separation in all implementations.

A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims. Quantum computing is usually formulated in terms of ideal two-level systems, called qubits. The two levels used in quantum computations are canonically denoted |0> and |1>, and are together called the computational states and said to form a computational subspace.

However, physical realizations of qubits often have additional states, known as non-computational states, which are said to form a non-computational subspace. These are often higher energy levels of the physical system being used as a qubit, such as the |2>, |3> etc. states. During the implementation of a quantum algorithm, leakage into these states from the computational states can be problematic and result in errors when executing the quantum algorithm. This leakage is, however, difficult to avoid, especially in weakly non-linear qubit systems, such as transmons. Moreover, non-computational states may be used in some implementations of quantum gates to execute a particular operation of the computational subspace. Leakage may occur during the execution of such quantum gates.

Minimizing leakage is an important design consideration when making an accurate quantum computer. However, tuning parameters of the quantum computer to minimize leakage requires that the population of non-computational states after executing an algorithm be known in order to determine the leakage. In practice, the populations of non-computational states are difficult to determine using readout apparatus that is used for measuring states in the computational subspace; often the readout apparatus, while capable of distinguishing between states in the computational subspace, cannot easily discriminate between a state in the non-computational subspace and a state in the computational subspace.

Instead of measuring the populations of the non-computational states directly, the systems and methods disclosed herein use readout apparatus that can only distinguish between a subset of the available states of the physical qubit, and apply shuffling sequences of control pulses to qubits in order to exchange the populations of states in the qubits before measurement. The readouts from the readout apparatus from these shuffled states can collectively be used to determine the populations of the states of the qubits.