Patent Publication Number: US-2023153669-A1

Title: Quantum computing device using two gate types to prevent frequency collisions in superconducting quantum computers

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with Government support contract W911NF-14-1-0124 awarded by the Army Research Office (ARO). The Government has certain rights to this invention. 
    
    
     BACKGROUND 
     The currently claimed embodiments of the present invention relate to superconducting quantum computers, and more specifically, to superconducting quantum computers that combine two gate types to avoid frequency collisions. 
     Quantum computation is based on the reliable control of quantum bits (referred to herein throughout as qubits). The fundamental operations required to realize quantum algorithms are a set of single-qubit operations and two-qubit operations which establish correlations between two separate quantum bits. The realization of high fidelity two-qubit operations may be desirable both for reaching the error threshold for quantum computation and for reaching reliable quantum simulations. 
     Currently for superconducting qubits single-qubit gates are implemented with microwave controls. There are three main types of two-qubit gates: gates based on tunable frequency qubits, gates based on microwave-driven qubits (e.g., cross-resonance, flick fork. Bell Rabi, MAP, sideband transitions, and gates based on geometric phases (e.g., resonator-induced phase gate, holonomic gates)). 
     For gates based on tunable frequency qubits, the qubits themselves are tuned in frequency to activate a resonance interaction. These gates essentially have two operating points: an ‘off’-position with essentially zero coupling and an ‘on’ position when the qubits have a strong two-qubit interaction. These gates have a very good on-off ratio, but because the qubits are tunable via externally applied magnetic flux, they can be limited by 1/f noise which limits the coherence of the qubits to a few microseconds. 
     For gates based on microwave-driven qubits, the qubits can be designed to be fixed in frequency so they are immune to flux noise. However, microwave pulses are required to activate the gate. The problems with these gates are that they have a low on/off ratio and are very hard to address the gate of interest without activating unwanted interactions. 
     Gates based on geometric phases are based on the path of the quantum state in its state space and the acquired quantum phase associated with its excursion. Adiabatic geometric gates are robust against certain types of noise, but are generally slow and require controls to adiabatic. Non-adiabatic gates can be faster and potentially share the noise-resilience of their adiabatic cousins. 
     Superconducting Josephson junction qubits are a promising technology for building a quantum computer. The transmon-type of superconducting qubits is operated at a relatively high ratio of Josephson energy to charging energy that allows the transmon (transmission-line shunted plasma oscillation) qubit to operate at reduced charge noise sensitivity while still allowing coupling between qubits and between the qubits and the transmission line or bus. This allows transmon qubits to be coupled in a 2D arrangement with nearest neighbor interactions (for example in a two-dimensional lattice) via fixed-frequency microwave resonators. Fixed frequency transmon qubits (single-junction transmons) are highly coherent (i.e., essentially having a single resonance frequency). Therefore, there is a need to enable interactions between qubits. Cross-Resonance (CR) interaction between qubits can be used to couple the qubits. In Cross-Resonance, the qubits are driven with microwave tones at the frequency of the neighboring qubits to establish interaction between the qubits. To enable CR interaction between a plurality of qubits, the qubits are relatively closely spaced in frequency (for example less than 200 MHz). However, this leads to frequency collision issues and crosstalk between the qubits. 
     As future quantum computers may use a large number of qubits (hundreds to thousands, or more), it may be desirable to limit frequency collision or crosstalk between qubits. While CR may provide benefits in allowing coupling between neighboring qubits, CR may not be adequate for establishing coupling between farther away located qubits. Hence, a need remains for a solution that cures the frequency collision issue and crosstalk when using CR to couple qubits. 
     SUMMARY 
     An aspect of the present invention is to provide a quantum computing device including a first plurality of qubits having a first resonance frequency and a second qubit having a second resonance frequency, the second resonance frequency being different from the first resonance frequency. The quantum computing device also includes a first fixed frequency bus configured to couple the first plurality of qubits, the first plurality of qubits being configured to interact via cross-resonance through the first fixed frequency bus. The quantum computing device further includes a first tunable frequency bus configured to couple at least one of the first plurality of qubits to the second qubit. 
     In an embodiment, the quantum computing device further includes a second plurality of qubits having the second resonance frequency; and a second fixed frequency bus configured to couple the second plurality of qubits, the second plurality of qubits being configured to interact via cross-resonance through the second fixed frequency bus. In an embodiment, the first plurality of qubits are arranged in one of a first row or in a first column and are configured to interact via cross-resonance through the first fixed frequency bus, and the second plurality of qubits are arranged in one of a second row or in a second column and are configured to interact via cross-resonance through the second fixed frequency bus. 
     In an embodiment, the quantum computing device further includes a third plurality of qubits having the first resonance frequency, the third plurality of qubits being arranged in one of a third row or in a third column; and a third fixed frequency bus configured to couple the third plurality of qubits, the third plurality of qubits being configured to interact via cross-resonance through the third fixed frequency bus. 
     In an embodiment, the first plurality of qubits arranged in the first row or in the first column are configured to interact with the second plurality of qubits arranged in the second row or the second column via the first tunable frequency bus and the second plurality of qubits arranged in the second row or in the second column are configured to interact with the third plurality of qubits arranged in the third row or the third column via the first tunable frequency bus. In an embodiment, the second plurality of qubits are arranged between the first plurality of qubits and the third plurality of qubits so as to prevent cross-resonance interaction between the first plurality of qubits and the third plurality of qubits. 
     In an embodiment, the quantum computing device further includes a third plurality of qubits having a third resonance frequency, the third plurality of qubits being arranged in one of a third row or in a third column, the third resonance frequency being different from the first resonance frequency and different from the second resonance frequency; a third fixed frequency bus configured to couple the third plurality of qubits, the third plurality of qubits being configured to interact via cross-resonance through the third fixed frequency bus; and a second tunable frequency bus configured to couple at least one of the third plurality of qubits to the second plurality of qubits. 
     In an embodiment, the first plurality of qubits arranged in the first row or in the first column are configured to interact with the second plurality of qubits arranged in the second row or the second column via the first tunable frequency bus, and the second plurality of qubits arranged in the second row or in the second column are configured to interact with the third plurality of qubits arranged in the third row or the third column via the second tunable frequency bus. 
     Another aspect of the present invention is to provide another quantum computing device. The quantum computing device includes a first plurality of qubits having a first resonance frequency and a second qubit having a second resonance frequency, the second resonance frequency being different from the first resonance frequency. The quantum computing device further includes a first tunable frequency bus configured to couple the first plurality of qubits to the second qubit. 
     In an embodiment, the first plurality of qubits are configured to interact with each other through the first tunable frequency bus via cross-resonance. In an embodiment, at least one of the first plurality of qubits and the second qubit are configured to interact through a parametric iSWAP gate. 
     In an embodiment, the first plurality of qubits comprise three qubits, the three qubits being configured to interact with each other through the first tunable frequency bus via cross-resonance, and the second qubit being configured to interact with the three qubits through the first tunable frequency bus via a parametric iSWAP gate. 
     In an embodiment, the quantum computing device further includes a second plurality of qubits having the second resonance frequency and a third qubit having the first resonance frequency; and a second tunable frequency bus configured to couple the third qubit to the second plurality of qubits. In an embodiment, the second plurality of qubits are configured to interact with each other through the second tunable frequency bus via cross-resonance. In an embodiment, the third qubit is prevented from interacting via cross-resonance with the first plurality of qubits. 
     In an embodiment the quantum computing device further includes a second plurality of qubits having the second resonance frequency and a third qubit having a third resonance frequency, the third resonance frequency being different from the first resonance frequency and the second resonance frequency; and a second tunable frequency bus configured to couple the third qubit to the second plurality of qubits. 
     Another aspect of the present invention is to provide a method of producing a quantum computing device. The method includes producing a first plurality of qubits having a first resonance frequency and a second qubit having a second resonance frequency on a qubit chip, the second frequency being different from the first frequency; at least one of producing a first fixed frequency bus on the qubit chip or attaching the qubit chip to a chip comprising the first frequency bus so as to enable interaction between the first plurality of qubits via cross-resonance; and at least one of producing a first tunable frequency bus on the qubit chip or attaching the qubit chip to a chip comprising the first tunable frequency bus so as to enable coupling at least one of the first plurality of qubits to the second qubit using the first tunable frequency bus. 
     Yet another aspect of the present invention is to provide a method of producing a quantum computing device. The method includes producing a first plurality of qubits having a first resonance frequency and a second qubit having a second resonance frequency on a qubit chip, the second frequency being different from the first frequency; and at least one of producing a first tunable frequency bus on the qubit chip or attaching the qubit chip to a chip comprising the first tunable frequency bus so as to enable coupling at least one of the first plurality of qubits to the second qubit using the first tunable frequency bus. 
     The frequency of a qubit corresponds to the transition energy between two states of the qubit being used, for example between the ground state and the first excited state. A qubit has two quantum states (e.g., ground state and first excited state) that are sufficiently separated in energy to form separated energy levels and/or decoupled from any additional quantum states so that the qubit is approximately a two-quantum state structure under operation conditions. The transition energy between the two states defines the resonance frequency of the qubit. For some qubits, the transition energy can be modified, for example, by the application of a magnetic field. 
     The term “resonance frequency” of a qubit as used in this specification corresponds to the frequency resonant with the energy separation between two energy levels of the qubit (e.g., ground state energy level and excited state energy level). The resonance frequency defines the energy transition from one energy level to another energy level. Each “energy level” of a qubit corresponds to an energy level defined within a relatively narrower bandwidth of energies or frequencies. The term “narrower” is used herein to mean that the bandwidth is smaller than the transition energy between the energy states of the qubit. For example, the production of a plurality of qubits that are intended to have the same resonance frequency may actually result in qubits with small differences in resonance frequency, but still within the bandwidth. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure, as well as the methods of operation and functions of the related elements of structure and the combination of parts and economies of manufacture, will become more apparent upon consideration of the following description and the appended claims with reference to the accompanying drawings, all of which form a part of this specification, wherein like reference numerals designate corresponding parts in the various figures. It is to be expressly understood, however, that the drawings are for the purpose of illustration and description only and are not intended as a definition of the limits of the invention. 
         FIG.  1    is a schematic diagram of a quantum computing device having a plurality of qubits, according to an embodiment of the present invention; 
         FIG.  2 A  is a schematic diagram of a quantum circuit including two qubits interacting via a first fixed frequency bus (CR bus), according to an embodiment of the present invention; 
         FIG.  2 B  is a schematic diagram of a quantum circuit including two qubits interacting via a first tunable frequency bus, according to an embodiment of the present invention; 
         FIG.  3    is a schematic diagram of a quantum computing device having a plurality of qubits, according to another embodiment of the present invention; 
         FIG.  4 A  shows a schematic diagram of a quantum circuit including four qubits interacting via a first tunable frequency bus, according to an embodiment of the present invention; 
         FIG.  4 B  shows a schematic diagram of four qubits interacting via the first tunable frequency bus, according to embodiment of the present invention; 
         FIG.  4 C  is an image of a quantum computing device on a chip including four qubits and a tunable frequency bus, according to an embodiment of the present disclosure; 
         FIG.  5 A  is an energy diagram showing the first excited energy levels of four qubits Q 1 , Q 2 , Q 3  and Q 4 , according to an embodiment of the present invention; 
         FIG.  5 B  is a plot of the energy or frequency of the first excited energy levels of the four qubits Q 1 , Q 2 , Q 3  and Q 4 , according to an embodiment of the present invention; and 
         FIG.  6    is a plot of the population in qubit Q 3  as a function of the frequency modulating the tunable bus versus time, in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
       FIG.  1    is a schematic diagram of a quantum computing device having a plurality of qubits, according to an embodiment of the present invention. The quantum computing device  10  includes a first plurality of qubits  12  having a first resonance frequency. The quantum computing device  10  further includes a second qubit or a second plurality of qubits  14  having a second resonance frequency. The second resonance frequency is different from the first resonance frequency. For example, the second resonance frequency of the second qubit or the second plurality of qubits can be relatively higher than the first resonance frequency of the first plurality of qubit. For example, the first plurality of qubits may have a first resonance frequency in the range 4.8-5.0 GHz and the second qubit or qubits may have a second resonance frequency in the range 5.6-5.8 GHz. 
     Although four first qubits are shown in  FIG.  1   , as it must be appreciated two or more qubits can be used. Similarly, although four second qubits are shown in  FIG.  1   , as it must be appreciated one, two or more second qubits can be used. However, in practical implementations, often more than two first qubits and more than two second qubits are used. 
     As shown in  FIG.  1   , the quantum computing device  10  further includes a first fixed frequency bus  16  (depicted as a dotted line) configured to couple the first plurality of qubits  12 . The first plurality of qubits  12  are configured to interact via cross-resonance (CR) through the first fixed frequency bus  16 . The quantum computing device  10  further includes a first tunable frequency bus  18  configured to couple at least one of the first plurality of qubits  12  to a second qubit  14 . In an embodiment, the first tunable frequency bus  18  is configured to couple the first plurality of qubits  12  (for example four as shown in  FIG.  1   ) to a second plurality of qubits  14  (for example four as shown in  FIG.  1   ). In an embodiment, the first tunable frequency bus  18  is configured to couple the at least one of the first plurality of qubits  12  and the second qubit  14  through a parametric iSWAP gate. In an embodiment, the first tunable frequency bus  18  is configured to couple the first plurality of qubits  12  and the second plurality of qubits  14  through the parametric iSWAP gate. 
     In an embodiment, the quantum computing device  10  further includes a second fixed frequency bus  17  configured to couple the second plurality of qubits  14 . The second plurality of qubits  14  are configured to interact via cross-resonance (CR) through the second fixed frequency bus  17 . 
       FIG.  2 A  is a schematic diagram of a quantum circuit including two qubits interacting via the first fixed frequency bus (CR bus), according to an embodiment of the present invention. As shown in  FIG.  2 A , first qubit Q 1  in the first plurality of qubits  12  and second qubit Q 2  in the first plurality of qubits  12  interact via first fixed frequency bus  16  (CR bus). In an embodiment, the first fixed frequency bus  16  is a microwave transmission line connecting qubit Q 1  to qubit Q 2 . Similarly, although not illustrated in  FIG.  2 A , first qubit Q 1  in the second plurality of qubits  14  and second qubit Q 2  in the second plurality of qubits  14  interact via second fixed frequency bus (CR bus)  17 . In an embodiment, the second fixed frequency bus  17  is a microwave transmission line connecting qubit Q 1  to qubit Q 2 . In the qubits Q 1  and Q 2 , the vertical lines “∥” correspond to a capacitance and the “X” symbol corresponds to a Josephson junction. Although the qubit is shown having a single Josephson Junction (JJ) and a single capacitance, any number of Josephson junctions and any number of capacitances can be used. In addition, other elements can also be included in the circuit of the qubits Q 1  and Q 2  and first fixed frequency bus  16  (CR bus) and/or second fixed frequency bus  17  (CR bus). 
       FIG.  2 B  is a schematic diagram of a quantum circuit including two qubits interacting via the first tunable frequency bus  18 , according to an embodiment of the present invention. As shown in  FIG.  2 B , first qubit Q 1  in the first plurality of qubits  12  and second qubit Q 2  in the first plurality of qubits  12  interact via first tunable frequency bus  18  (for example via a parametric iSWAP gate). In an embodiment, as shown in  FIG.  2 B , the first tunable frequency bus  18  includes a gate (e.g., a parametric iSWAP gate) “G” that connects qubit Q 1  to qubit Q 2  via a transmission line. The gate “G” is activated by applying a magnetic field “B”. Similar to  FIG.  2 A , in the qubits Q 1  and Q 2 , the vertical lines “∥” correspond to a capacitance and the “X” symbol corresponds to a Josephson junction. Although the qubits Q 1  and Q 2  are shown having a single Josephson junction and a single capacitance, any number of Josephson junctions and any number of capacitances can be used. In addition, other elements can also be included in the circuit of the qubits Q 1  and Q 2 . In addition, although the first tunable frequency bus  18  is shown having gate “G” (e.g., parametric iSWAP gate) using two Josephson junctions, as it can be appreciated, other types of gates can be used that use another number of Josephson junctions. In an embodiment, the first tunable frequency bus  18  includes a superconducting quantum interference device (SQUID). The term parametric iSWAP gate is understood in the superconducting quantum computing art as being a gate that performs a swap in the phase matrix through the imaginary number “i”. 
     As shown in  FIG.  1   , the first plurality of qubits  12  are arranged a column. However, the first plurality of qubits  12  can also be arranged in a first row. The first plurality of qubits  12  are configured to interact via cross-resonance (CR) through the first fixed frequency bus  16 . As also shown in  FIG.  1   , the second plurality of qubits  14  are arranged in a second column. However, the second plurality of qubits  14  can also be arranged in a second row. The second plurality of qubits  14  are configured to interact via cross-resonance (CR) through the second fixed frequency bus  17 . 
     As also shown in  FIG.  1   , in an embodiment, the quantum computing device  10  also includes a third plurality of qubits  13  having the first resonance frequency similar to the first plurality of qubits  12 . As shown in  FIG.  1   , the third plurality of qubits  13  are arranged in a third column. However, the third plurality of qubits  13  can also be arranged in a third row. The quantum computing device further includes a third fixed frequency bus  19  configured to couple the third plurality of qubits  13 . The third plurality of qubits  13  are configured to interact via cross-resonance (CR) through the third fixed frequency bus  19 . 
     As also shown in  FIG.  1   , in an embodiment, the quantum computing device  10  also includes a fourth plurality of qubits  15  having the second resonance frequency similar to the second plurality of qubits  14 . As shown in  FIG.  1   , the fourth plurality of qubits  15  are arranged in a fourth column. However, the fourth plurality of qubits  15  can also be arranged in a fourth row. The quantum computing device  10  further includes a fourth fixed frequency bus  21  configured to couple the fourth plurality of qubits  15 . The fourth plurality of qubits  15  are configured to interact via cross-resonance (CR) through the fourth fixed frequency bus  21 . 
     In an embodiment, the first plurality of qubits  12  arranged in the first row or in the first column are configured to interact with the second plurality of qubits  14  arranged in the second row or the second column via the first tunable frequency bus  18 . In an embodiment, the second plurality of qubits  14  arranged in the second row or in the second column are configured to interact with the third plurality of qubits  13  arranged in the third row or the third column via the first tunable frequency bus  18 . In an embodiment, the third plurality of qubits  13  arranged in the third row or in the third column are configured to interact with the fourth plurality of qubits  15  arranged in the fourth row or the fourth column via the first tunable frequency bus  18 . 
     In an embodiment, the second plurality of qubits  14  are arranged between the first plurality of qubits  12  and the third plurality of qubits  13  so as to prevent cross-resonance interaction, and thus substantially reduce or eliminate frequency collision and crosstalk, between the first plurality of qubits and the third plurality of qubits  13 . Similarly, the third plurality of qubits  13  are arranged between the second plurality of qubits  14  and the fourth plurality of qubits  15  so as to prevent cross-resonance interaction, and thus substantially reduce or eliminate frequency collision and crosstalk, between the second plurality of qubits  14  and the fourth plurality of qubits  15 . 
     In an embodiment, the third plurality of qubits  13  have a third resonance frequency. The third plurality of qubits  13  are arranged in a third column, as shown in  FIG.  1   . However, the third plurality of qubits  13  can also be arranged in a third row. In this embodiment, however, the third resonance frequency is different from the first resonance frequency of the first plurality of qubits  12  and different from the second resonance frequency of the second plurality of qubits  14 . In an embodiment, the third fixed frequency bus  19  is configured to couple the third plurality of qubits  13 . The third plurality of qubits  13  are configured to interact via cross-resonance (CR) through the third fixed frequency bus  19 . In an embodiment, the quantum computing device includes a second tunable frequency bus  22  that is configured to couple at least one of the third plurality of qubits  13  to the second plurality of qubits  14 . 
     In an embodiment, the first plurality of qubits  12  arranged in the first row or in the first column are configured to interact with the second plurality of qubits  14  arranged in the second row or the second column via the first tunable frequency bus  18 . In an embodiment, the second plurality of qubits  14  arranged in the second row or in the second column are configured to interact with the third plurality of qubits  13  arranged in the third row or the third column via the second tunable frequency bus  22 . 
     In an embodiment, the first, the second, the third and/or fourth plurality of qubits can be transmon qubits. 
       FIG.  3    is a schematic diagram of a quantum computing device having a plurality of qubits, according to another embodiment of the present invention. The quantum computing device  30  includes a first plurality of qubits  32  having a first resonance frequency and a second qubit  34  having a second resonance frequency. The second resonance frequency is different from the first resonance frequency. In an embodiment, the second frequency is higher than the first frequency, for example. In an embodiment, for example the second frequency can be in the range 4.8-5.0 GHz and the first frequency can be in the range 5.6-5.8 GHz. The quantum computing device further includes a first tunable frequency bus  36  configured to couple the first plurality of qubits  32  to the second qubit  34 . 
     In an embodiment, the first plurality of qubits  32  are configured to interact with each other through the first tunable frequency bus  36  via cross-resonance (CR). 
       FIG.  4 A  shows a schematic diagram of a quantum circuit including four qubits interacting via the first tunable frequency bus  36 , according to an embodiment of the present invention.  FIG.  4 B  shows a schematic diagram of four qubits interacting via the first tunable frequency bus  36 , according to embodiment of the present invention.  FIG.  4 C  is an image of a quantum computing device on a chip including four qubits and a tunable frequency bus, according to an embodiment of the present disclosure. As shown in  FIG.  4 A , qubits Q 1 , Q 2  and Q 4  of the first plurality of qubits  32  interact with each other and with the qubit Q 3  corresponding to the second qubit  34  via the first tunable frequency bus  36  (for example, via a parametric iSWAP gate). 
     In an embodiment, as shown in  FIG.  4 A , the first tunable frequency bus  36  includes a gate (e.g., a parametric iSWAP gate) “G” that connects qubits Q 1 , Q 2 , Q 3 , and Q 4  using a transmission lines connected to quantum gate “G”. The gate “G” is activated by applying a magnetic field “B”. In the qubits Q 1 , Q 2 , Q 3 , and Q 4 , the vertical lines “∥” correspond to a capacitance and the “X” symbol corresponds to a Josephson junction. Although the qubits Q 1 , Q 2 , Q 3 , and Q 4  are shown having a single Josephson junction and a single capacitance, any number of Josephson junctions and any number of capacitances can be used. In addition, other elements can also be included in the circuit of the qubits Q 1 , Q 2 , Q 3 , and Q 4 . In addition, although the first tunable frequency bus  36  is shown having gate “G” (e.g., parametric iSWAP gate) using two Josephson junctions, as it can be appreciated, other types of gates can be used that use another number of Josephson junctions. In an embodiment, the first tunable frequency bus  36  includes a superconducting quantum interference device (SQUID). Therefore, in an embodiment, the first plurality of qubits  32  and the second qubit  34  are configured to interact the first tunable frequency bus  36  (via a parametric iSWAP gate). 
     In an embodiment, as shown in  FIGS.  3 ,  4 A and  4 B , the first plurality of qubits  32  comprise three qubits (for example, Q 1 , Q 2  and Q 4 ). The three qubits  32  (Q 1 , Q 2  and Q 4 ) are configured to interact with each other through the first tunable frequency bus  36  via cross-resonance (CR). The second qubit  34  (Q 3 ) is configured to interact with the three qubits  32  (Q 1 , Q 2  and Q 4 ) through the first tunable frequency bus  36  (via a parametric iSWAP gate). 
       FIG.  5 A  is an energy diagram showing the first excited energy levels of the four qubits Q 1 , Q 2 , Q 3  and Q 4 , according to an embodiment of the present invention. The first excited energy level of qubit Q 1  is denoted as |0001&gt;, the first excited energy level of qubit Q 2  is denoted as |0010&gt;, the first excited energy level of qubit Q 4  is denoted as |1000&gt;, and the first excited energy level of qubit Q 3  is denoted as |0100&gt;. The frequencies noted above each excited energy state correspond to a transition energy (or frequency) between the ground state (energy or frequency equal to zero) and the first excited energy level, respectively, 4.605 GHz for Q 1 , 4.685 GHz for Q 2 , 4.727 for Q 4  and 5.478 for Q 3 . As shown in  FIG.  5 A , the excited energy levels |0001&gt;, |0010&gt; and |1000&gt; of qubits Q 1 , Q 2 , and Q 4 , respectively, are close to each other with a difference in energy between |0001&gt; and |0010&gt; of 80 MHz, between |0010&gt; and |1000&gt; of 42 MHz, and between |0001&gt; and |1000&gt; of 122 MHz. This allows qubits Q 1 , Q 2 , and Q 4  to interact via CR. On the other hand, the difference in energy between the excited energy level |0100&gt; of qubit Q 3  and energy levels |0001&gt; of qubit Q 1 , between the excited energy level 10100&gt; of qubit Q 3  and energy level |0010&gt; of qubit Q 2  and between the excited energy level |0100&gt; of qubit Q 3  and energy level |1000&gt; of qubit Q 4  is, respectively, 873 MHz, 793 MHZ and 751 MHz. which is greater than any of 80 MHz, 42 MHz and 122 MHz. As a result, in this case interaction between qubits Q 1 , Q 2 , Q 4  and Q 3  is enabled by the first tunable frequency bus  36  (using for example the parametric iSWAP gate). The inventors implemented a quantum computing device with the four qubits Q 1 , Q 2 , Q 3  and Q 4  coupled to a tunable frequency bus using a parametric iSWAP gate. The inventor reported results that show successful implementation of the interaction between the various qubits. The inventors plotted the results. 
       FIG.  6    is a plot of the population in qubit Q 3  as a function of the frequency modulating the tunable bus versus time, in accordance with an embodiment of the present invention. Regions where population in qubit Q 3  decreases correspond to the difference in frequency (energy) between the excited energy levels. The plot shows the difference in energy (or frequency) between the excited energy level |0100&gt; of qubit Q 3  and energy levels |0001&gt; of qubit Q 1 , between the excited energy level |0100&gt; of qubit Q 3  and energy level |0010&gt; of qubit Q 2 , and between the excited energy level |0100&gt; of qubit Q 3  and energy level |1000&gt; of qubit Q 4  is, respectively, 873 MHz, 793 MHZ and 751 MHz. The plot shows that the interaction is stable over at least a time frame of 5 μs and gives typical timescales for the interaction (less than about 1 μs). 
       FIG.  5 B  is a plot of the energy or frequency difference of the first excited energy levels of the four qubits Q 1 , Q 2 , Q 3  and Q 4 , according to an embodiment of the present invention. The three lower horizontal lines in the plot that are below about 200 MHZ correspond to transitions between the three states |0001&gt;, |0010&gt; and |1000&gt; of qubits Q 1 , Q 2  and Q 4 . These three are located within a relatively narrower bandwidth of energies or frequencies. The term “narrower” is used herein to mean that the bandwidth (about 122 MHz in this case) is smaller than the difference in energy (around 800 MHz) between the energy states of each of three qubits and the fourth qubit. The three upper horizontal lines in the plot that are between about 750 MHz and about 900 MHz correspond to the excitation manifold that includes the excited energy level |0100&gt; of qubit Q 3 . Transitions out of the qubit space (e.g., a transition between state |1001&gt; and state |0002&gt;), lie between the lower horizontal lines and the upper horizontal lines in the plot. This plot demonstrates that the tunable bus interaction is not sufficient on its own because transitions between similar frequency qubits are within the same bandwidth as transitions out of the qubit subspace. 
     Returning to  FIG.  3   , the quantum computing device  30  further includes a second plurality of qubits  38  having the second resonance frequency and a third qubit  40  having the first resonance frequency. The quantum computing device  30  further includes a second tunable frequency bus  42  configured to couple the third qubit  40  to the second plurality of qubits  38 . In an embodiment, the second plurality of qubits  38  include the second qubit  34 . 
     In an embodiment, the second plurality of qubits  38  are configured to interact with each other through the second tunable frequency bus  42  via cross-resonance (CR). In an embodiment, the third qubit  40  and the second plurality of qubits  38  are configured to interact through the second tunable frequency bus  42  (via parametric iSWAP gate). In an embodiment, the third qubit  40  is prevented from interacting via cross-resonance with the first plurality of qubits  32 . 
     In an embodiment, the quantum computing device  30  further includes a second plurality of qubits  38  having the second resonance frequency and a third qubit  40  having a third resonance frequency. In this embodiment, the third resonance frequency is different from the first resonance frequency of the first plurality of qubits  32  and from the second resonance frequency of the second plurality of qubits  38 . The second tunable frequency bus  42  is configured to couple the third qubit  40  to the second plurality of qubits  38 . In an embodiment, the second plurality of qubits  38  are configured to interact with each other through the second tunable frequency bus  42  via cross-resonance (CR). In an embodiment, the third qubit  40  and the second plurality of qubits  38  are configured to interact through the second tunable frequency bus  42  via a parametric iSWAP gate. 
     In an embodiment, the second plurality of qubits  38  comprise three qubits. The three qubits are configured to interact with each other through the second tunable frequency bus  42  via cross-resonance (CR), and the third qubit  40  is configured to interact with the three qubits  38  through the second tunable frequency bus  42  via a parametric iSWAP gate. 
     As it can be appreciated from the above paragraphs there is also provided a method of producing a quantum computing device (e.g., the quantum computing device  10 ), according to an embodiment of the present invention. The method includes producing a first plurality of qubits  12  having a first resonance frequency and a second qubit  14  having a second resonance frequency on a qubit chip (for example as shown  FIG.  4 C ). The second frequency is different from the first frequency. The method further includes at least one of producing a first fixed frequency bus  16  on the qubit chip or attaching the qubit chip to a chip comprising the first fixed frequency bus  16  so as to enable interaction between the first plurality of qubits  12  via cross-resonance (CR). The method also includes at least one of producing a first tunable frequency bus  18  on the qubit chip or attaching the qubit chip to a chip comprising the first tunable frequency bus  18  so as to enable coupling at least one of the first plurality of qubits  12  to the second qubit  14  using the first tunable frequency bus  18 . 
     According to an embodiment of the present invention, as it can be appreciated from the above paragraphs, there is further provided a method of producing a quantum computing device (for example, quantum computing device  30 ). The method includes producing a first plurality of qubits  32  having a first resonance frequency and a second qubit  34  having a second resonance frequency on a qubit chip (for example the qubit chip shown in  FIG.  4 C ). The second frequency is different from the first frequency. The method also includes at least one of producing a first tunable frequency bus  36  on the qubit chip or attaching the qubit chip to a chip comprising the first tunable frequency bus  36  so as to enable coupling at least one of the first plurality of qubits  32  to the second qubit  34  using the first tunable frequency bus  36 . 
     The descriptions of the various embodiments of the present invention have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.