Patent Publication Number: US-2022215283-A1

Title: Global flux bias

Description:
TECHNICAL FIELD 
     This present subject matter relates to control of qubits. 
     BACKGROUND 
     Large-scale quantum computers have the potential to provide fast solutions to certain classes of difficult problems. Multiple challenges in the design and implementation of quantum architecture to control, program and maintain quantum hardware impede the realization of large-scale quantum computing. 
     SUMMARY 
     The present disclosure describes technologies for implementing a qubit control cable and an attenuator as part of the cable. 
     In general, one innovative aspect of the subject matter of the present disclosure may be embodied in methods that include: providing an offset magnetic flux bias to a plurality of superconducting qubits; and providing respective control magnetic flux biases, for performing a computation, to the plurality of qubits using a plurality of control lines coupled respectively to each qubit. The qubits are configured such that respective resonance frequencies of the qubits are controlled by the offset magnetic flux bias and the respective control magnetic flux biases. The qubits are arranged to perform the computation when the respective resonance frequencies of the qubits are within an operational dynamic range. 
     The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. 
     In some implementations, providing the offset magnetic flux bias comprises setting the resonance frequencies of all of the qubits at a frequency within the operational dynamic range. 
     In some implementations, the method further includes: identifying a first group of qubits, in which the resonance frequency of each of the qubits is not controllable; and identifying a second group of qubits, in which the resonance frequency of each of the qubits is controllable. Providing the offset magnetic flux bias includes: setting the offset magnetic flux bias such that the resonance frequencies of the qubits of the first group of qubits and the second group of qubits are outside the operational dynamic range when the control magnetic flux biases are not provided. Providing the control magnetic flux biases further includes: setting the control magnetic flux biases for the second group of qubits such that the resonance frequencies of the second group of qubits are within the operational dynamic range when the offset magnetic flux bias is provided. 
     In some implementations, providing the offset magnetic flux bias is by providing a global magnetic field. 
     In some implementations, the global magnetic field is generated by driving a current through a coil. The coil is arranged such that the magnitude of the global magnetic field is substantially uniform to the plurality of qubits. 
     In some implementations, the coil is wound around the plurality of qubits such that the plurality of qubits are exposed to the global magnetic field through an axis of the coil. 
     In some implementations, the coil is disposed on a substrate on which the plurality of qubits are disposed. 
     In some implementations, providing the offset magnetic flux bias includes, in the following sequence: arranging a temperature around the plurality of qubits to be above the superconducting transition temperature; providing the global magnetic field; lowering the temperature below the superconducting transition temperature; and turning off the global magnetic field, such that the offset magnetic flux bias is conserved within all of the plurality of qubits after turning off the global magnetic field. 
     In some implementations, the operational dynamic range comprises one or more frequency ranges which fall between 4 GHz and 6 GHz. 
     In some implementations, each of the plurality of qubits comprises a DC SQUID. 
     In some implementations, each of the plurality of qubits further comprises a parallel LC circuit. An inductor of the parallel LC circuit comprises the DC SQUID. An inductance of the DC SQUID is determined by the offset flux bias and the control flux bias provided to the qubit. 
     In some implementations, an operation bandwidth of the offset magnetic flux is below 10 Hz, an operation bandwidth of a control magnetic flux bias is 300 to 700 MHz. 
     Another innovative aspect of the subject matter of the present disclosure may be embodied in an apparatus for providing an offset magnetic flux bias for a plurality of superconducting qubits that includes: an offset magnetic flux bias generator arranged to generate an offset magnetic flux bias to the plurality of qubits. The plurality of qubits are configured such that respective resonance frequencies of the qubits are controlled by the offset magnetic flux bias. 
     The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. 
     In some implementations, the offset magnetic flux bias generator further includes: a driving circuit; and a transducer. 
     In some implementations, the transducer includes: a coil; and a plurality of control lines coupled respective to each qubit. 
     In some implementations, the driving circuit is arranged to drive the coil to provide the offset magnetic flux bias. 
     In some implementations, the coil is wound around the plurality of qubits, the coil arranged to generate a global magnetic field which is substantially uniform for the plurality of qubits. 
     In some implementations, the coil is wound around the plurality of qubits such that the plurality of qubits are exposed to the global magnetic field through an axis of the coil. 
     In some implementations, the coil is disposed on a substrate on which the plurality of qubits are disposed. 
     In some implementations, the driving circuit is arranged to drive the plurality of control lines to provide the offset magnetic flux bias. 
     By providing an offset magnetic flux bias, or a global magnetic flux bias, to all of the qubits using a single transducer, the level of noise transmitted through the respective Z control lines and the heat generated by the Z control lines within a cryostat may be reduced. 
     The magnetic flux bias may be provided in the form of persistent currents within the qubits, which removes the necessity of constantly providing current. This may further reduce the heat load and suppress decoherence of qubits arising from the fluctuation of the magnetic flux bias. 
     The offset magnetic flux bias may be used to selectively decouple any faulty qubits from an array of qubits. Therefore, generating the offset magnetic flux bias may minimize the number of qubits excluded from the computation. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic that illustrates an exemplary quantum computing system. 
         FIG. 2  is a plot that illustrates a transition frequency of a superconducting as a function of magnetic flux applied to the qubit. 
         FIG. 3  is a flowchart of controlling transition frequencies of qubits using a global magnetic flux bias. 
         FIG. 4 a    is a schematic that illustrates an exemplary embodiment of a global magnetic flux generator. 
         FIG. 4 b    is a schematic that illustrates an exemplary embodiment of a global magnetic flux generator on a qubit chip. 
         FIG. 5  is a flowchart of generating the global magnetic flux bias using a global magnetic flux generator. 
         FIG. 6 a    is a schematic that illustrates a 2-dimensional array of qubits. 
         FIG. 6 b    is a schematic diagram that illustrates a procedure of selectively decoupling a faulty qubit from a 2-dimensional array of qubits. 
         FIG. 7  is a flowchart of isolating faulty qubits within a network of coupled qubits. 
     
    
    
     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. 
     In some implementations, control signals are generated in higher-temperature environments, and are transmitted to the quantum information processing system using shielded impedance-controlled GHz capable transmission lines, such as coaxial cables. The cryostat may step down from room-temperature (e.g., about 300 K) to the operating temperature of the qubits in one or more intermediate cooling stages. For instance, the cryostat may employ a stage maintained at a temperature range that is colder than room temperature stage by one or two orders of magnitude, e.g., about 30-40 K or about 3-4 K, and warmer than the operating temperature for the qubits (e.g., about 10 mK or less). 
     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. At least one room-temperature co-axial cable per qubit is used to provide the qubit control signal. As the number of qubits increases, the thermal dissipation arising from the current level of the control signals also increases, which may raise an issue in view of a limited cooling capability of a cryostat. Furthermore, if any of the cables fail, the corresponding qubit can no longer be controlled. Furthermore, since a qubit is often coupled to one or more neighboring qubits, those neighboring qubits may have to be excluded from the computational system. For example, in a 2-dimensional grid of qubits, as many as 5 qubits may have to be excluded if one of the cables is found to be faulty. 
     This application relates to addressing these problems by providing a global flux bias to a group of qubits simultaneously. 
       FIG. 1  illustrates a schematic of an exemplary quantum computing system. The quantum computing system includes a qubit chip  100  coupled to qubit control electronics  20 . The qubit chip  100  includes one or more qubits  102 , such as superconducting qubits, and may be operated using a final stage  11  of a cryostat  10  at extremely low temperatures (e.g., at around 10 mK or less, subject to the minimum possible temperature achievable by the cryostat, generally below 100 mK for a dilution refrigerator as the cryostat  10 ). The qubit control electronics  20  may be placed outside the cryostat  10 , which may be at ambient condition. Alternatively, all or part of the qubit control electronics  20  may be placed in one of the stages of the cryostat  10 , which will be discussed below to mitigate thermal dissipation of the qubit control electronics  20 . 
     For the purposes of this disclosure, the qubits operated by the qubit control electronics  20  are assumed to be frequency tunable transmon (FT-XMON) qubits. A specific ratio of junction critical current of the SQUID and capacitance may be met to be in the transmon regime. Other designs are possible. For example, a ratio of junction critical current of the SQUID and capacitance not in the transmon regime may be chosen. However, the concept of this application applies to any design of qubits in which the transition frequencies are controlled by providing magnetic flux. However, the qubit control electronics  20  described herein are not limited to working with transmon qubits and may also be used with other qubit configurations, such as fluxmon qubits or gmon qubits, among others. 
     The cryostat  10  may be a dilution refrigerator. However, as long as the cryostat  10  can provide a sufficiently low temperature for the coherence of the qubits  102 , the exact type of the cryostat  10  is not limited to a dilution refrigerator. The final stage  11  of the cryostat  10  provides a lowest possible temperature the cryostat  10  is capable of providing. For example, in case the cryostat  10  is a dilution refrigerator, the final stage  11  may be a part of the cryostat that is in thermal equilibrium with the mixing chamber of the dilution refrigerator, which usually provides a temperature around 10 mK. Also in case the cryostat  10  is a dilution refrigerator, an intermediate stage  12  of the cryostat  10  may be a part of the cryostat that is in thermal equilibrium with a chamber including liquid Helium (He4) cooled to below room temperature but above the qubit operating temperature, e.g., at around 3-4K or in thermal equilibrium with a pulse-tube cooler. The final stage  11  and the intermediate stage  12  of the cryostat  10  may be thermally enclosed within an initial stage  13  of the cryostat  10 . The initial stage  13  may include parts of the cryostat  10  which provides an initial shielding from the room temperature condition. The initial shielding may include a vacuum shield or a liquid nitrogen shield and the rest of the components of the cryostat  10  such as vacuum systems and thermal shielding layers. This description of the cryostat  10  refers mainly to the common design of a dilution refrigerator. However, as discussed, the requirement from the cryostat is that it provides a sufficiently low temperature for the qubits to maintain a coherence time required for the operation. Therefore, the details of the design of the cryostat  10  may differ, such as the exact number of the stages, depending on the type of the cryostat  10  used. For example, the intermediate stage  12  temperature may be achieved with a pulse tube cooler. 
     Regardless of the type of the cryostat  10 , the temperature gradient from the room temperature to the temperature at which the qubits  102  operate often raises considerable challenge in connecting the qubit control electronics  20  and the qubit chip  100 . For brevity of the description, in the rest of the specification the cryostat  10  will be assumed to be a dilution refrigerator with the first stage  13 , the intermediate stage  12 , and the final stage  11 . 
     Each qubit  102  of the qubit chip  100  may be coupled to a Z drive qubit circuit element  106  (e.g., a resonator or an inductor), an XY drive qubit circuit element  110  (e.g., a capacitor), and a qubit readout resonator  112 . The qubits  102  and associated circuit elements formed on the qubit chip  100  can be formed from patterned superconductor materials on a dielectric substrate (e.g., aluminum on a silicon or sapphire substrate). 
     The qubit chip  100  is coupled to the qubit control electronics  20 , which are operated outside the cryostat  10 , for example at room temperature (e.g., about 300 K). Data lines  22 ,  24 ,  26  that connect the control electronics  20  to the qubit chip  100  may pass through one or more low temperature stages of the cryostat  10 , namely through the initial stage  13 , the intermediate stage  12  and the final stage  11 . 
     The data lines  22 ,  24 ,  26  may comprise at least one qubit Z control line  22 , at least one qubit XY control line  24 , and at least one qubit readout line  26 . 
     Microwave gate operations on qubits  102  can be carried out by generating an XY control signal at the control electronics  20  and then applying the XY control signal, when the qubit is operating at its resonant frequency, to the XY drive qubit circuit element  110 , resulting in a deterministic rotation of the qubit state about an axis in the XY plane of the Bloch sphere, where the axis and angle of rotation are determined by the carrier phase and integrated envelope amplitude of the microwave signal, respectively. Exemplary pulse durations and envelope amplitudes, referenced to the XY-drive qubit circuit element  110 , are 10-30 ns and 10-100 μV, respectively. 
     The control of the non-linearity of the qubit  102 , therefore the tuning of the transition frequency of the qubit, can be carried out by generating a Z control signal at the control electronics  20  and then applying the Z control signal to the Z drive qubit circuit element  106 . 
     Gate operations on qubits  102  may involve a combination of one or more of Z control signals and one or more of XY control signals. 
     In case there are a plurality of qubits  102 , a plurality of qubit Z control lines  22  may be provided to control the transition frequencies of the respective qubits  102  individually. 
     The qubit control electronics  20  may include standard control circuits operating at room temperature use high-speed (˜1 GSPS or higher) and high-resolution (˜14-bit) digital to analog converter (DAC) waveform generators to generate each qubit XY control signal and Z control signal. 
       FIG. 1  also shows an exemplary arrangement of electronics components disposed within the cryostat  10  necessary for transmitting signals to control the qubits  102 . 
     The data lines  22 ,  24 ,  26  may include a first attenuator  31 , a second attenuator  32 , and a third attenuator  33  and an amplifier  34 . The first attenuator  31  may be disposed on the qubit Z-control line  22  in the intermediate stage  12 . The third attenuator  33  may be disposed on the qubit XY-control line  24  in the intermediate stage  12 . These attenuators are to suppress the noise from the qubit control electronics  20  disposed at room temperature (around 300K). In particular, 300K thermal noise (Johnson-Nyquist noise), generated from using resistance at room temperature and transmitted through the qubit Z-control line  22  and the qubit XY-control line  24 , is attenuated. 
     The attenuators  31 ,  32 ,  33  may provide 20 dB attenuation or more. However, the exact degree of attenuation may depend on the exact parameters of the hardware. 
     The amplifier  34  may be disposed on the qubit readout line  26  in the intermediate stage  12 , to amplify the readout signal from the qubit chip  100 . 
     For the qubit Z control line  22 , placing another attenuator in the final stage  11  may often not be feasible because the power of the signal required for Z control is typically mW level. Placing an attenuator to produce this level of power at the final stage  11  may generate heat comparable to or over the cooling capacity of the final stage  11  of a dilution fridge. Therefore, further attenuation may be provided for the qubit Z control line  22  near the qubit control electronics  20  outside the cryostat  10 . The qubit  102  is a non-linear resonator with a resonance frequency in the microwave regime. For the case of a frequency tunable transmon (FT-XMON) qubit, the qubit  102  includes a capacitor in parallel with a pair of Josephson Junctions wired in a loop to form a SQUID whose effective inductance can be tuned by threading the loop with an external magnetic flux drive (e.g., provided by the qubit Z control line  22 ). 
       FIG. 2  shows a plot  200  that illustrates a resonance frequency of a qubit  102  as a function of magnetic flux applied to the qubit  102  with references to  FIG. 1 . 
     A vertical axis  210  of the plot  200  represents a normalized transition frequency, or a normalized resonance frequency of a qubit  102 . The scale of the vertical axis  210 , 0 to 1, will be discussed in more detail later. A horizontal axis  220  of the plot  200  represents the magnetic flux through a SQUID loop of the qubit  102  normalized to the magnetic flux quantum, as will be explained in more detail below. 
     Each qubit  102  includes at least one Josephson junction. Each Josephson junction may be arranged to act as a variable inductance controlled with magnetic field. The inductance of the Josephson junction, 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 
             ∼ 
             
               1 
               
                 cos 
                 ⁡ 
                 
                   ( 
                   
                     πφ 
                     
                       φ 
                       0 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     φ represents the magnetic flux through the DC SQUID of each qubit  102 . φ 0  represents the magnetic flux quantum. Therefore, the Josephson inductance varies as a function of the magnetic flux provided through the loop of the DC SQUID within each qubit  102 . 
     A qubit  102  may be constructed as an LC circuit, where L corresponds to the Josephson inductance of the Josephson junction of the qubit  102 . It is well known that the resonance frequency of an LC circuit is given by 
     
       
         
           
             f 
             = 
             
               1 
               
                 2 
                 ⁢ 
                 π 
                 ⁢ 
                 
                   
                     L 
                     ⁢ 
                     C 
                   
                 
               
             
           
         
       
     
     Therefore, the resonance frequency, or the transition frequency of a qubit may be expressed as a function of the magnetic flux through the DC SQUID as 
     
       
         
           
             
               
                 f 
                 Qubit 
               
               ⁡ 
               
                 ( 
                 φ 
                 ) 
               
             
             = 
             
               
                 f 
                 
                   m 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ax 
                 
               
               ⁢ 
               
                 
                   cos 
                   ⁡ 
                   
                     ( 
                     
                       πφ 
                       
                         φ 
                         0 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     f max  represents the maximum resonance frequency of the qubit. The vertical axis  210  of the plot  200  represents a transition frequency of a qubit  102  normalized to f max , therefore ranges from 0 to 1. For a typical transmon qubit, the qubit  102  may be designed such that f max  may be a frequency from about 3 GHz to about 10 GHz. However, the qubit  102  may be designed such that f max  is any other frequency. The horizontal axis  220  of the plot  200  represents 
     
       
         
           
             
               φ 
               
                 φ 
                 0 
               
             
             , 
           
         
       
     
     the magnetic flux φ through a SQUID loop of the qubit  102  normalized to the magnetic flux quantum ω 0 . The horizontal axis  220  may contain a negative number, which represents the opposite direction of the magnetic flux with respect to the SQUID loop of the qubit  102 . For brevity, in the rest of the application, only the positive numbers in the horizontal axis  220  will be considered, which means that only the magnitude of the magnetic flux varies, but the direction of the magnetic flux will not reverse. However, it is understood that the concept presented in this application encompasses varying the direction of the magnetic flux to achieve the desired effect. 
     A first point  201  represents a flux insensitive point. When magnetic flux φ is not actively provided to the qubit  102 , the resonance frequency of the qubit  102  will remain around the first point  201 . Around the first point  201 , the resonance frequency of the qubit  102  around the first point  201  will be more robust to the fluctuation of the magnetic flux than other points in the curve  200 . This is because the curve  200  is symmetric around the first point  201  and the rate of change of the resonance frequency with respect to the change of the magnetic flux is smallest. Therefore, the first point  201  may be relatively robust to the noise arising from spurious fluctuation of the magnetic flux from the environment. 
     The resonance frequencies of the qubits  102  may be shifted away from the flux insensitive point  201  for operation such as performing an algorithm or a calibration. This is partly because the qubits  102  are affected the most by the two-level system (TLS) defects which are present in amorphous dielectric oxide layers. The noise spectral density of the two-level system (TLS) may be centered at a frequency which is higher than those of the qubits  102  and may gradually decrease with frequency on both sides of the center frequency. Therefore, the effect of the two-level system (TLS) defects decreases as the resonance frequencies of the qubits  102  are lowered. In addition to the two-level system (TLS), there can be other undesired parasitic resonances. Therefore, for performing an algorithm or a calibration, magnetic flux φ may be provided such that the resonance frequencies of the qubits  102  are shifted away from the flux insensitive point  201 . Also, for a given design of an array of qubits, there may be a range of the flux insensitive points of the qubits due to distribution in fabrication tolerances. Therefore, when all of the resonance frequencies of an array of qubits are not identical, shifting the resonance frequencies of the qubits as described in  FIG. 2  may provide uniformity in frequency control. 
     The operations may take place when the resonance frequencies of the qubits  102  are within a range defined by a second point  202  and a third point  203 . The second point  202  and the third point  203  may be reached by increasing the magnetic flux φ from the first point  201 ′ on the horizontal axis  220  to the second point  202 ′ and the third point  203 ′ on the horizontal axis  220 , respectively. The range between the second point  202  and the third point  203  may be referred to as an operation dynamic range  206 . 
     During an algorithm or a calibration, often the ability to move between at least two frequencies may be required. Also, the resonance frequencies of the qubits  102  may be adjusted to control the degree of interaction between neighboring qubits. Therefore, the magnitude of the operation dynamic range  206  may be determined such that during performing an algorithm or a calibration, at least two different bands of resonance frequencies of the qubits  102  can be defined. For example, the operation dynamic range  206  of a transmon qubit may be from 4 to 7 GHz or 5 to 8 GHz, although an exact range may depend on the design of the qubits  102 . 
     A fourth point  204  represents a point substantially far from the first point  201  and from the operation dynamic range  206  in frequency, substantially towards the half flux quantum point, corresponding to the resonance frequency near DC, namely 0 in the vertical axis  210 . The fourth point  204  may be determined such that the qubit whose resonance frequency is at the fourth point  204  is substantially decoupled from the qubits whose resonance frequencies are within the operation dynamic range  206 , between the second point  202  and the third point  203 , even if the qubit at the fourth point  204  and the qubit within the operation dynamic range  206  are spatially in close proximity. 
     In some implementations, the exact position of the fourth point  204  in terms of the magnetic flux may depend on the degree of capacitive coupling between the qubits  102 , namely the geometry of each qubit  102  and the arrangement of the qubits  102 , in particular, the distance between two neighboring qubits  102 . Any two qubits  102  may be regarded as substantially decoupled when the degree of coupling between the two qubits  102  are sufficiently small such that it does not significantly affect performing algorithm or calibration. 
     There can be a trade-off relationship between the extent of the operation dynamic range and the suppression of noise. The main source of noise may be the qubit control electronics  20 , often at room temperature. A large fraction of this noise is attenuated at the first attenuator  31 , in the intermediate stage  12  of the cryostat  10 . The first attenuator  31  may be placed in the intermediate stage  12  because the power requirement of the Z control signals is such that an appropriate cooling power is provided only at the intermediate stage  12 , but not at the final stage  11 . The noise from the room temperature qubit control electronics  20  may be therefore dissipated at the first attenuator  31  and be thermalized. Then the spectrum of the noise may be assumed to be roughly white when it reaches the qubit chip  100 . 
     Since the noise near the qubit chip  100  may be largely white in spectrum, the total amount of noise is directly related to the bandwidth of the Z control signals. In other words, the qubit control electronics  20  outputs a constant white noise level, and there is a constant noise level per unit flux to the qubit chip  100 . A large operation dynamic range between the second point  202  and the third point  203  may allow a proportionally larger amount of noise. When a qubit is cooled through the critical temperature Tc under zero magnetic field, the resonance frequency is set at the first point  201 , which is a flux insensitive point. Then in order to operate up to the third point  203 , the end of the operating range  206 , a current is required with a magnitude corresponding to an interval between the first point  201 ′ and the third point  203 ′ on the horizontal axis. If instead the qubit is cooled under a magnetic field such that the resonance frequency is set at begin at a point  205  within the operating range  206 , less current is required to reach the frequency at the third point  203 . This allows more attenuation of the fixed white noise level from the control electronics, which provides an enhanced signal-to-noise ratio. 
     If the qubit is cooled under a magnetic field such that the frequency is set at the point  205  within the operating range  206 , there is a trade-off between the signal to noise ratio and the width of the operation dynamic range between the second point  202  and the third point  203 . Therefore, to minimise the noise leading to decoherence of the qubits  102 , the computation may be performed with the minimum possible operation dynamic range. 
     Since this noise is transmitted via the Z control line  22 , the noise level may be mitigated if at least part of the flux bias to the qubits  102  can be provided independent of the Z control line  22 . In other words, if the qubits  102  receive a global flux bias such that a fixed amount of magnetic flux can be applied through the SQUID loop of the qubits  102  without using the Z control line  22 . 
     For example, a magnetic field may be generated around the qubit chip  100  such that all of the qubits  102  experience substantially uniform magnetic field. If this magnetic field is such that all of the qubits  102  are brought to the second point  202 , or the third point  203 , or any one point  205  within the operation dynamic range  206 , the magnetic flux which should be applied via the Z control line for performing an algorithm or a calibration is reduced to the range between  202 ′ to  203 ′. This is smaller than the magnetic flux from the first point  201 ′ to the third point  203 ′. Therefore, if offset magnetic flux bias can be generated for all of the qubits  102  without using the Z control lines  22 , the noise level may be reduced, which may lead to suppression of decoherence of the qubits  102  and improving the signal-to-noise ratio of the measurements of the state of the qubit. 
       FIG. 3  shows a flowchart of controlling the transition frequencies of one or more of qubits  102  using a global magnetic flux bias, with reference  FIGS. 1 and 2 . 
     In step S 310 , a global magnetic flux bias, or an offset global magnetic flux bias, is provided to a plurality of qubits  102 . The global magnetic flux bias or the offset magnetic flux bias correspond to the magnetic flux bias through the SQUID coil of the qubits  102  and shifts the resonance frequencies of the plurality of qubits  102 . Once the offset magnetic flux bias is provided to the plurality of qubits  102 , further adjustment of the transition frequencies of the individual qubits  102  may start from this offset value for performing an algorithm or an operation. For brevity of explaining the concept, it will be assumed in this application that the offset magnetic flux bias or the global magnetic flux bias is assumed to be substantially uniform for all of the qubits  102 . However, depending on the method of generating the offset magnetic flux bias, any two of the qubits  102  may experience slightly different magnetic flux. How the offset magnetic flux bias is generated will be discussed in more detail later. As discussed in  FIG. 2 , the offset magnetic flux bias or the global magnetic flux, may be such that the qubits  102  are brought to the beginning of the operation dynamic range, the second point  202 , or the end of the operation dynamic range, the third point  203 , or any point  205  within the operation dynamic range. 
     In step  320 , control magnetic flux biases are provided to the plurality of qubits  102  using individual Z control lines  22  connected to each qubit  102 , to perform a computation, which may include performing a calibration or an algorithm. To perform any desired calibration or algorithm for computation, the movement of the transition frequencies will be performed with individual Z control lines  22  connected respectively to the qubits  102  while the offset magnetic flux bias is kept substantially at DC or within a small bandwidth, covering a speed relatively slower than the XY control signals. The XY control signals may be provided via the XY control lines  24 . For the rest of the application, the magnetic flux bias applied to the qubits  102  via the Z control lines  22  will be referred to as control magnetic flux bias CZ and the offset magnetic flux bias affecting all of the qubits  102  will be referred to as the global magnetic flux bias GZ, to distinguish two different magnetic flux biases. 
       FIG. 4 a    shows a schematic that illustrates an exemplary embodiment of an apparatus for providing an offset magnetic flux bias, or a global magnetic flux bias with references to  FIG. 1 .  FIG. 4  shows the qubit chip  400  disposed in the final stage  13  of the cryostat. A global flux bias generator  440  may comprise a driving circuit  441 , a global Z drive qubit circuit element  442  and a transducer  443 . The driving circuit  441  may be included within the qubit control electronics  20 . Alternatively, the driving circuit  441  may be a separate unit from the qubit control electronics  20 . The driving circuit  441  may be disposed outside the cryostat  10  or in one of the stages, the initial stage  13 , the intermediate stage  12 , or the final stage  11 . The driving circuit  441  may be arranged to provide a current or a voltage required for generating the global magnetic flux bias. The global Z drive qubit circuit element  442  may be included in the qubit chip  400 , to the transducer  443  via the global Z drive qubit circuit element  442 . The global Z drive qubit circuit element  442  may serve as an interface between the driving circuit  441  and the transducer  443 . The driving circuit  441  and the global Z drive qubit circuit element  442  may be connected to each other via a driving line  444 . The driving line  444 , in the similar fashion as the data lines  22 ,  24 ,  26 , may be disposed traversing one or more of the stages  11 ,  12 ,  13  of the cryostat  10 . The global Z drive qubit circuit element  442  may be a connector which interfaces the driving line  444  and the qubit chip  400 . The global Z drive circuit element  442  may include any other circuit elements such as a filter or a resonator which facilitates the operation of the transducer  443 . The transducer  443  may convert the current or the voltage received from the driving circuit  441  into a magnetic field. The transducer  443  may be disposed on the qubit chip  400 . Alternatively, the transducer  443  may be placed in the vicinity of the qubit chip  400  such that the magnetic field generated at the transducer  443  can be efficiently provided to the qubits  102 . The example of the transducer  443  may include a coil or a loop into which the current generated by the driving circuit  441  is sent. 
       FIG. 4 b    shows a schematic that illustrates an exemplary embodiment of the global flux bias generator  440  on the qubit chip  400  with references to  FIG. 1 . The qubit chip  400  comprises a plurality of qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . In the example of  FIG. 4 , the qubit chip  400  includes six qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . However, this number of qubits is only exemplary and the number of qubits is not limited to six. The qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  may be connected to respective Z drive qubit circuit elements  421 ,  422 ,  423 ,  424 ,  425 ,  426  via respective inductive elements  411 ,  412 ,  413 ,  414 ,  415 ,  416 . As explained above in  FIG. 1 , each qubit  401 ,  402 ,  403 ,  404 ,  405 ,  406  may also be connected to respective XY drive qubit circuit element  110  and respective qubit readout resonators  112  within the qubit chip  400 . However, since this application mainly relates to providing the magnetic flux bias for Z control,  FIG. 4 b    only depicts only the element directly relevant to Z control. The control magnetic flux bias CZ for each qubit  401 ,  402 ,  403 ,  404 ,  405 ,  406  may be provided by the qubit control electronics  20 , which may be disposed outside the cryostat  10  or in the initial stage  13  or in the intermediate stage  12  of the cryostat  10 . 
     In the example of  FIG. 4 b   , the transducer  443  of the global magnetic flux bias generator  440  is disposed on the qubit chip  400  and in the form of a loop of conductor around the qubit chip  400 , which encloses the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . In the example of  FIG. 4 , the transducer  443  is formed as a part of the superconducting layer from which the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  are formed. However, the arrangement of the transducer  443  is not limited to a loop on the surface of the qubit chip  400 . The transducer  443  may assume any shape which is for efficient coupling of the magnetic flux to the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . For example, the transducer  443  may be in the form of multiple loops. In some implementations, the transducer  443  may be fabricated within the same layer of the superconducting material as the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  and the inductive elements  411 ,  412 ,  413 ,  414 ,  415 ,  416 . Alternatively, the transducer  443  may be freestanding conducting wires wound around the qubit chip  400 . Alternatively, the transducer  443  may be fabricated on the qubit chip  400  but from a different layer from the superconducting layer which includes the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . Alternatively, the transducer  443  may be disposed on a front surface of a separate chip which may be brought into the proximity of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . For example, the separate chip may have a loop of conducting material as the transducer  443  and may be placed vertically over the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . 
     The transducer  443  may be arranged to generate global magnetic flux bias GZ which reaches all of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  within the qubit chip  400 . This may be achieved, for example, by running a current through the loop of conducting material of the transducer  443 . However, any other suitable means to generate magnetic flux may be employed and the transducer  443  is not limited to a conducting loop. In some implementations, the distribution of the global magnetic flux bias GZ generated by the transducer  443  is such that all of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  experience substantially the same amount of the magnetic flux. In some implementations, the transducer  443  may generate global magnetic flux bias GZ whose direction is substantially perpendicular to the surface of the qubit chip  400  such that it couples efficiently to the SQUID loop of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . However, this may depend on the design of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . Therefore, as long as the global magnetic flux bias GZ generated by the global magnetic flux generator  430  is coupled efficiently to the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 , the direction of the global magnetic flux bias GZ may deviate from perpendicular to the qubit chip  400 . 
     In some implementations, there may be more than one transducer  443  such that the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  are grouped into more than one group of qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  and each group experiences a distinct magnetic flux. 
     In order to reduce thermal noise, the currents to generate the control magnetic flux bias CZ may be attenuated at the intermediate stage  12  with attenuators. This may place a significant heat load on the cryostat  10  as the number of qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  increases because the number of Z drive qubit circuit elements  421 ,  422 ,  423 ,  424 ,  425 ,  426  also increases. If the global magnetic flux generator  440  is employed, the current level to generate the control magnetic flux bias CZ for each qubit  401 ,  402 ,  403 ,  404 ,  405 ,  406  may be reduced. 
     In case the global magnetic flux bias GZ is generated by running a current through a conducting part of the transducer  443 , the global Z drive qubit circuit element  442  or the driving circuit  441  may include a low pass filter with a narrow bandwidth such that the global magnetic flux bias GZ is substantially at DC and does not cause any fluctuation of the resonance frequencies of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  which may lead to decoherence. The bandwidth of the low pass filter may be determined based on the required noise level of the global magnetic flux bias GZ. Alternatively, the bandwidth of the low pass filter may be determined based on the frequency at which the resonance frequency of each qubit is shifted for performing a calculation or a calibration. 
     In case the global magnetic flux bias GZ is generated by running a current through a conducting part of the transducer  443 , the degree of thermal dissipation may be kept within the cooling power of the final stage  13  of the cryostat  10 . 
     As the number of qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  increases, the area to be enclosed by the transducer  443  may increase, which may require a higher level of currents to be constantly supplied throughout the operation. This may cause heating of the final stage  13  of the cryostat  10 . This issue of thermal dissipation may be addressed if an alternative method of generating the global magnetic flux bias GZ is followed as described below and in  FIG. 5 . 
       FIG. 5  shows a flowchart of generating the global magnetic flux bias GZ using the global magnetic flux generator  440 , with references to  FIGS. 2, 4   a  and  4   b.    
     In step S 510 , the temperature of the final stage  13  of the cryostat  10  may be arranged such that the temperature is above the superconducting transition temperature of the material used in fabricating the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . Above the superconducting transition temperature, the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  lose coherence and do not function as a quantum mechanical 2-level system. All of the circuit elements fabricated within the same plane as the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  such as the inductive elements  411 ,  412 ,  413 ,  414 ,  415 ,  416  become lossy above the superconducting transition temperature. 
     In step S 520 , the global magnetic flux bias GZ may be provided. As discussed above in  FIGS. 4 a  and 4 b   , this may be achieved by transmitting signal from the driving circuit  441  to the global Z drive qubit circuit element  442 , then to the transducer  443 . The driving circuit  441  may provide necessary level of current to the conducting part of the transducer  443 , for example, a coil around the qubit chip  400 . The driving circuit  441  may provide necessary level of voltage to the conducting part of the transducer  443 , via the global Z drive qubit circuit element  442 . In case the transducer  443  is a coil, the global Z drive qubit circuit element  442  may be arranged to generate the necessary level of current. The necessary level of current, therefore the global magnetic flux bias GZ, may be determined such that once the temperature of the final stage  13  is cooled down below the transition temperature, the resonance frequencies of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  are brought to the point  205  between the second point  202  and the third point  203  of  FIG. 2 , namely within the operation dynamic range  206 . 
     In some implementations, the transducer  443  may be fabricated within the superconducting layer on which the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  are fabricated, the transducer  443  above the superconducting transition temperature will exhibit a finite level of dissipation because the temperature has been elevated above the transition temperature in step S 510 . The current level may be determined taking into consideration of the finite dissipation. In some implementations, the transducer  443  may be a separate element from the superconducting layer of the qubits,  401 ,  402 ,  403 ,  404 ,  405 ,  406 , for example, an external coil. The current level may be determined taking into consideration of the finite dissipation of the material of the transducer  443  at the elevated temperature. The material of the transducer  443  separate from the qubit chip  100  may be chosen to be a material which becomes superconducting at the elevated temperature of step S 510 . 
     In step S 530 , the temperature of the final stage  13  may be lowered below the superconducting transition temperature, while the global magnetic flux bias GZ provided in step S 520  is maintained. As well-known as the Meissner effect, a superconductor expels magnetic field during the transition, thereby cancelling all of the magnetic field within the body of the superconductor. Therefore, during the superconducting transition as the temperature of the final stage  13  lowers, each qubit  401 ,  402 ,  403 ,  404 ,  405 ,  406  comprising a superconductor material, will generate a so-called persistent current internally to cancel the magnetic field distribution within itself formed by the global magnetic flux bias GZ provided in step S 520 . It is well known that the decay of the persistent current in time is negligible as long as the temperature is kept below the superconducting transition temperature. 
     In step S 540 , the global magnetic flux bias GZ may be turned off after the temperature of the final stage  13  is lowered below the superconducting transition temperature of the superconducting material of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 . Also as discussed above in  FIGS. 4 a  and 4 b   , this may be achieved by interrupting the current or the voltage signal from the driving circuit  441  to the transducer  443 . Due to the persistent current, even after the global magnetic flux bias GZ is turned off, the global magnetic flux bias GZ is provided as long as the temperature of the final state  13  is kept lower than the superconducting transition temperature. Since it is equal in magnitude of the global magnetic flux bias GZ, as soon as the global magnetic flux bias GZ is turned off, the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  are brought to the point  205  between the second point  202  and the third point  203  of  FIG. 2  within the operation dynamic range. 
     Generating the global magnetic flux bias GZ as described above and in  FIG. 5  does not require the current through the transducer  443  constantly on. Therefore, the issue of heating of the intermediate stage  12  or the final stage  13  of the cryostat  10  may be mitigated. Also, since the global magnetic flux bias GZ is conserved within the superconducting material of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406 , the magnitude of the global magnetic flux bias GZ stays inherently stable. Therefore, a low pass filter with small bandwidth may not be necessary to minimize the noise level of the global magnetic flux bias GZ. 
     In some implementations, the global magnetic flux bias GZ may also be used to isolate one or more faulty qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  from the others. The failure of the qubit may originate from the qubit itself, or one or more of the circuit elements,  106 ,  110 ,  112 ,  421 ,  422 ,  423 ,  424 ,  425 ,  426 ,  440  or any point within the control lines  22 ,  24 ,  26  connected to the qubit such that the qubit  401 ,  402 ,  403 ,  404 ,  405 ,  406 . 
     Each Z control line  22  may be faulty at various failure points such as faulty cabling, loosened connectors, errors in PCB packaging, wirebonds and lithography such as the qubit Z control line  22 , the first attenuator  31 , the Z drive qubit circuit element  106 ,  421 ,  422 ,  423 ,  424 ,  425 ,  426 . Especially, when the failure of the qubit relates to Z control, the resonance frequencies of the qubits  401 ,  402 ,  403 ,  404 ,  405 ,  406  are not controllable. In this case, the faulty qubit may remain coupled to the neighboring qubits. 
       FIG. 6 a    shows a schematic which illustrates qubits arranged in a 2-dimensional array with references to  FIG. 2 .  FIG. 6  shows a 2-dimensional grid  600  of 9 qubits, each represented by a circle, which belong to a 2-dimensional array which contains more than 9 qubits. The lines of  FIG. 6 a    represent the coupling relations among the qubits  601 ,  602 ,  603 ,  604 ,  605 , namely that the two qubits on either side of each line are coupled to each other. The lines do not correspond to any of the control or data lines  22 ,  24 ,  26 . 
     In some implementations, any two qubits may be coupled to each other capacitively. Alternatively, any two qubits may be coupled to each other via any other mechanism than capacitive coupling, which will facilitate entangling the two qubits. In case the qubits are coupled to each other capacitively, the shape of each qubit  601 ,  602 ,  603 ,  604 ,  605  may be arranged such that the neighbouring qubits are coupled to one another capacitively due to proximity between conducting parts of the two qubits. For example, the shape of each qubit  601 ,  602 ,  603 ,  604 ,  605  may be cross-shaped such that only the nearest qubits are directly coupled and the degree of coupling between next nearest qubits are negligible for the scheme of computation. 
     In some implementations, once the qubits  601 ,  602 ,  603 ,  604 ,  605  are arranged in the 2-dimensional grid  600 , the degree of capacitive coupling between any two qubits are determined by the shape of each qubit  601 ,  602 ,  603 ,  604 ,  605  and the distance between them. The degree of coupling between any two qubits  601 ,  602 ,  603 ,  604 ,  605  may be further controlled by the resonance frequencies via the Z control signals to the qubits  601 ,  602 ,  603 ,  604 ,  605 . 
     In the example of  FIGS. 6 a  and 6 b   , it will be assumed that only the neighboring qubits are coupled to one another and the degree of coupling between next nearest qubits will be assumed to be negligible for the scheme of computation. In the example of  FIGS. 6 a  and 6 b   , the qubit  603  will be assumed to be faulty, in other words, unusable for the computation because the resonance frequency of the qubit  603  is not controllable by control magnetic flux bias CZ provided by the Z control line  22  attached to the qubit  603 . 
     As discussed above, the faulty qubit  603  may arise due to the failure of the qubit Z control line  22 , the first attenuator  31 , or the Z drive qubit circuit element  106 ,  421 ,  422 ,  423 ,  424 ,  425 ,  426 . Each Z control line  22  may be faulty at various failure points such as faulty cabling, loosened connectors, errors in PCB packaging, wirebonds and lithography. 
     Since the resonance frequency of the faulty qubit  603  is not controllable, the resonance frequency of the faulty qubit  603  will stay in the first point  201 , the flux insensitive point. When the resonance frequencies of the coupled qubits  601 ,  602 ,  604 ,  605  are brought within the operation dynamic range  206 , and therefore away from the flux insensitive point  201 , there may still exist a degree of coupling and this may persist. This residual coupling with the faulty qubit  603  may render the neighboring qubits  601 ,  602 ,  604 ,  605  unusable for the purpose of the computation. Therefore, one faulty qubit  603  may lead to exclusion of the qubit  603  itself and all of the coupled qubits  601 ,  602 ,  604 ,  605 , in total five qubits, from contributing to the computation, in case of 2-dimensional grid geometry of qubits. 
     This issue may be addressed if the faulty qubit  603  could be selectively decoupled from the neighboring qubits  601 ,  602 ,  604 ,  605 . Then only the faulty qubit  603  itself can be excluded from the operation, the neighboring qubits  601 ,  602 ,  604 ,  605  may still take part in the operation. This may be achieved by bringing only the faulty qubit  603  to the fourth point  204 , such that the interaction with the qubits  601 ,  602 ,  604 ,  605  within the dynamic range can be regarded as substantially negligible or turned off. This will be explained in more detail below and in  FIG. 6   b.    
       FIG. 6 b    shows a schematic diagram which shows the procedure of selectively decoupling the faulty qubit  603  from the 2-dimensional grid of qubits  601 ,  602 ,  603 ,  604 ,  605 . 
     Each graph corresponds to the plot  200  provided in  FIG. 2  with the horizontal axis representing the magnetic flux and with the vertical axis representing the resonance frequency of the qubit  601 ,  602 ,  603 ,  604 ,  605 . Three panels a first panel  610 , a second panel  620 , and a third panel  630 , each comprising five graphs for the qubits  601 ,  602 ,  603 ,  604 ,  605  represents different conditions for Z magnetic flux bias applied to the qubits  601 ,  602 ,  603 ,  604 ,  605 . In particular, the magnetic flux in this example is proportional to the combination, for example a vector sum, of the control magnetic flux bias CZ and the global magnetic flux bias GZ. 
     The operation dynamic range  646  is represented by the two dotted lines in all of the graphs of  FIG. 6 b   . The operation dynamic range  646  of  FIG. 6 b    corresponds to the operation dynamic range  206  explained in  FIG. 2 . The beginning point and the ending point of the operation dynamic range  646  corresponds to the second point  202  and the third point  203  explained in  FIG. 2 . The first point  641  in all of the graphs of  FIG. 6 b    corresponds to the first point  201  explained in  FIG. 2 . The fourth point  644 , as indicated in the graphs for the qubit  603  in a second panel  620  and a third panel  630  corresponds to the fourth point  204  explained in  FIG. 2 . 
     The first panel  610  shows five plots, representing the states of the qubits  601 ,  602 ,  603 ,  604 ,  605  when neither the control magnetic flux bias CZ nor the global magnetic flux bias GZ is provided. As explained in  FIG. 2 , all of qubits  601 ,  602 ,  603 ,  604 ,  605  remain at the first point  641 , the flux insensitive point. 
     The second panel  620  shows five plots representing the states of the qubits  601 ,  602 ,  603 ,  604 ,  605  when only the global magnetic flux bias GZ is provided. The global magnetic flux bias GZ is set such that all of the qubits  601 ,  602 ,  603 ,  604 ,  605  are moved to the fourth point  644 . As discussed in  FIG. 2  above, the fourth point  644  is far removed from the operation dynamic range  646  such that any qubit whose frequency is at the fourth point  644  is decoupled from the qubits whose frequencies are within the dynamic range  646 . 
     The third panel  630  shows five plots representing the states of the qubits  601 ,  602 ,  603 ,  604 ,  605  when the control magnetic flux bias CZ is provided for the neighboring qubits  601 ,  602 ,  604 ,  605  such that the combination of the global magnetic flux bias GZ and the control magnetic flux bias CZ brings the neighboring qubits  601 ,  602 ,  604 ,  605  to the operational dynamic range  646 . In this case, the faulty qubit  603  is decoupled from the neighboring qubits  601 ,  602 ,  604 ,  605 . 
     Therefore, by using the control magnetic flux bias CZ and the global magnetic flux bias GZ, only the faulty qubit  603  can be isolated from the network of qubits  601 ,  602 ,  603 ,  604 ,  605  and the qubits  601 ,  602 ,  604 ,  605  directly coupled to the faulty qubit  603  can be made to take part in the computation. 
     The method described in  FIG. 6 b    may be employed if the all of the qubits  601 ,  602 ,  603 ,  604 ,  605  in a given area of the 2-dimensional grid  600  take part in the calculation either as data qubits or as ancillary qubits, and none of the qubits  601 ,  602 ,  603 ,  604 ,  605  are used as tunable couplers. 
     The 2-dimensional grid  600  of qubits  601 ,  602 ,  603 ,  604 ,  605  as shown in  FIG. 6 a   , is merely an example of the arrangement of the qubits  601 ,  602 ,  603 ,  604 ,  605 . Any other geometry of arrange the qubits  601 ,  602 ,  603 ,  604 ,  605  may be possible. For example, a honeycomb geometry in 2-dimensional array or a 3-dimensional array of qubits may be possible. The general concept of the selective exclusion of faulty qubit using global magnetic flux bias GZ applies to any geometry of qubit arrays. 
       FIG. 7  shows a flowchart which illustrates the procedure for isolating the faulty qubit  603  within the network of coupled qubits  601 ,  602 ,  603 ,  604 ,  605  with references to  FIGS. 2, 6   a  and  6   b.    
     In step S 710 , a first group of qubits are identified whose resonance frequencies are not controllable. As discussed above, the fault can reside at any point from the qubit control electronics  20  and the qubits  601 ,  602 ,  603 ,  604 ,  605  such that the resonance frequency of the qubits  601 ,  602 ,  603 ,  604 ,  605  cannot be controlled with the qubit control magnetic flux bias CZ. Also, any of the qubits  601 ,  602 ,  603 ,  604 ,  605  themselves may not be operable due to defects arising from fabrication process. In the example of  FIGS. 6 a  and 6 b   , the first group of qubits corresponds to the faulty qubit  603 . 
     In step S 720 , a second group of qubits are identified which will be used for computation. The second group of qubits may be all of the rest of the available qubits excluding the first group of qubits identified in step S 710 . Alternatively, the second group of qubits may not be all of the rest of the available qubits depending on the algorithm or the calibration to be performed modified in view of the first group of qubits. In the example of  FIGS. 6 a  and 6 b   , the second group of qubits corresponds to the neighboring qubits  601 ,  602 ,  604 ,  605 . 
     In step S 730 , the global magnetic flux bias GZ may be set such that the resonance frequencies of all of the qubits  601 ,  602 ,  603 ,  604 ,  605  are outside the operational dynamic range  646  and at the fourth point  644 . This is as shown in the second panel  620  of  FIG. 6   b.    
     In step S 740 , the control magnetic flux biases CZ of the second group of qubits may be set such that the resonance frequencies of the second group of qubits are brought into the operational dynamic range  206 . This is as shown in the third panel  630  of  FIG. 6   b.    
     Implementations of the quantum subject matter and quantum operations described in this specification can be implemented in suitable quantum circuitry or, more generally, 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.