Cavity-mediated quantum gate

Abstract

A cavity-mediated quantum gate, such as a quantum controlled phase gate, may be formed by concurrently driving a first qubit and a second qubit at or near a resonant frequency of an oscillator via which the first qubit is coupled to the second qubit.

Description

CAVITY-MEDIATED QUANTUM GATECROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Patent Application No. 18 / 979,298, filed December 12, 2024, entitled “Cavity-Mediated Quantum Gate”, which claims the benefit and priority to U.S. Provisional Patent Application No. 63 / 696,817, entitled, “Cavity- Mediated Quantum Gate”, filed on September 19, 2024, the disclosures of both are hereby incorporated herein in their entireties.BACKGROUND

[0002] In classical computing, transistors, typically formed in a semiconductor material, can be combined to form various logical gates. One of the ongoing challenges in quantum computing is that of building reliable, low-error, quantum logic gates from which more complicated quantum circuits and associated computations can be built.BRIEF DESCRIPTION OF THE DRAWINGS

[0003] Certain features of the subject technology are set forth in the appended claims. However, for the purpose of explanation, several embodiments of the subject technology are set forth in the following figures. However, for purposes of explanation, several aspects of the subject technology are depicted in the following figures.

[0004] FIG. l is a block diagram depicting a system including a quantum computer according to aspects of the subject technology.

[0005] FIG. 2 is a block diagram illustrating a quantum computing system having a quantum processor according to aspects of the subject technology.

[0006] FIG. 3 is a block diagram illustrating a cavity -mediated quantum phase gate according to aspects of the subject technology.

[0007] FIG. 4 is a phase space diagram illustrating a time evolution of a quantum state according to aspects of the subject technology.

[0008] FIG. 5 is another phase space diagram illustrating time evolution of a quantum state according to aspects of the subject technology.

[0009] FIG. 6 is a timing diagram illustrating applications of various pulses to one or more qubits that form a quantum gate according to aspects of the subject technology.

[0010] FIG. 7 is a diagram illustrating a time evolution under H of a quantum state in phase space in the lab frame according to aspects of the subject technology.

[0011] FIG. 8 is a diagram illustrating a time evolution under H of a quantum state in phase space in the lab frame according to aspects of the subject technology.

[0012] FIG. 9 is a phase space diagram illustrating a “flower” of the “flowers” approach according to aspects of the subject technology.

[0013] FIG. 10 is a phase space diagram illustrating a photonic state returning to where it started according to aspects of the subject technology.

[0014] FIG. 11 is a phase space diagram illustrating a photonic state returning to where it started for N = 6 evolution for x = 0 according to aspects of the subject technology.

[0015] FIG. 12 is a phase space diagram illustrating an N = 6 flower with x not equal to zero according to aspects of the subject technology.

[0016] FIG. 13 is a phase space diagram illustrating an N = 4 example for T~ = T / 2, according to aspects of the subject technology.

[0017] FIG. 14 is a phase space diagram illustrating an N = 6 example for T~ = T / 2 according to aspects of the subject technology.

[0018] FIG. 15 is a phase space diagram illustrating aN = 4 flower according to aspects of the subject technology.

[0019] FIG. 16 is a phase space diagram illustrating an antisymmetric mode flower for m = 1 according to aspects of the subject technology.

[0020] FIG. 17 is a phase space diagram illustrating an antisymmetric mode flower for m = 2 according to aspects of the subject technology.

[0021] FIG. 18 is a phase space diagram illustrating an antisymmetric mode flower for m = 3 according to aspects of the subject technology.

[0022] FIG. 19 is a phase space diagram illustrating an m = 1 flower with the introduction of an imperfection xsaccording to aspects of the subject technology.

[0023] FIG. 20 illustrates an example two-dimensional qubit architecture according to aspects of the subject technology.

[0024] FIG. 21 is a timing diagram illustrating the sign of the effective always-on ZZ interaction according to aspects of the subject technology.

[0025] FIG. 22 is a block diagram illustrating another cavity-mediated quantum phase gate according to aspects of the subject technology.

[0026] FIG. 23 is a block diagram illustrating yet another cavity-mediated quantum phase gate according to aspects of the subject technology.

[0027] FIG. 24 illustrates a flow diagram of an exemplary process, in accordance with one or more embodiments of the subject technology.

[0028] FIG. 25 illustrates an example electronic device in which aspects of the subject technology may be used, in accordance with one or more embodiments of the subject technology.DETAILED DESCRIPTION

[0029] The description set forth below describes various configurations of the subject technology and is not intended to represent the only configurations in which the subject technology may be practiced. The appended drawings are incorporated herein and constitute a part of the description. The description includes specific details for the purpose of providing an understanding of the subject technology. However, the subject technology is not limited to the specific details set forth herein and may be practiced using one or more other embodiments of the subject technology. In one or more embodiments of the subject technology, structures and components are shown in block diagram form in order to avoid obscuring the concepts of the subject technology.

[0030] Aspects of the subject disclosure relate generally to a controlled quantum phase gate (e.g., a controlled-Z (CZ) gate), such as a cross-resonance gate, that is formed from two qubits (or transmons or other anharmonic systems) coupled together by an oscillator (e.g., a cavity or other resonator). The quantum gate can be generated by driving both qubits at a frequency close to the resonance frequency of the oscillator, and without directly driving the oscillator. This can cause the oscillator to be significantly populated (excited), and can generate the desired quantum gate using the two-qubit / oscillator system. Additional features of the disclosure help to mitigate gate errors due to, for example, the quantum Stark shift, and the always-on ZZ interaction.

[0031] FIG. l is a block diagram depicting a system that may incorporate a quantum computer according to aspects of the subject technology. Not all of the depicted components may be required, however, and one or more implementations may include additional components not shown in the figure. Variations in the arrangement and type of the components may be made without departing from the spirit or scope of the claims as set forth herein. Depicted or described connections and couplings between components are not limited to direct connections or direct couplings and may be implemented with one or more intervening components unless expressly stated otherwise.

[0032] As depicted in FIG. 1, a system 100 may include one or more client devices 102, one or more servers 106, and / or one or more quantum computing systems 108. As shown, the client devices 102 may be communicatively coupled to the server(s) 106 via a network 104. For example, the network 104 may represent one or more local area networks (LANs) and / or one or more wider area networks, such as the Internet. In one or more implementations, the server(s) 106 may implement one or more web services that can be accessed by one or more of the client devices 102 via the network 104. In one or more implementations, the web services provided by the servers 106 may include quantum computing services. For example, the server(s) 106 may facilitate access to one or more of the quantum computing systems 108, such as from a client device 102.

[0033] In the example of FIG. 1, the quantum computing systems 108 include a first quantum computing system, QCS1, a second quantum computing system, QCS2, and a third quantum computing system, QCS3. However, it is appreciated that this is merely illustrative, and the system 100 may include more or fewer than three quantum computing systems in other implementations. In various examples, the multiple quantum computing systemsQCS1, QCS2, and QCS3 may represent multiple instances of a same or similar type of quantum computing system, or implementations of multiple different types of quantum computing systems. For example, different types of quantum computing systems may implement different types of qubit architectures, as will be discussed in further detail hereinafter.

[0034] Although FIG. 1 illustrates an example in which one or more quantum computing systems can be accessed by a web services provider, in one or more implementations, one or more of the quantum computing systems 108 may also, or alternatively, be accessible directly (e.g., locally at the quantum computing system). For example, FIG. 1 also illustrates how a quantum computing system 108 may be accessed directly from a client interface 109 that is directly communicatively coupled to the quantum computing system 108. For example, the client interface 109 may include one or more input / output components, such as keyboards, display screens, touch interfaces, etc. that can be operated by a user to control an associated quantum computing system 108. In one or more implementations, the client interface 109 and the quantum computing system 108 can be integrated into a common package (e.g., as a standalone quantum computer).

[0035] FIG. 2 is a block diagram depicting an example of a quantum computing system according to aspects of the subject technology. In the example of FIG. 2, the quantum computing system 108 includes both classical computer 201 and a quantum computer 203. As shown, the classical computer 201 may include one or more classical processors, such as a classical processor 200. As shown, the quantum computer 203 may include one or more quantum processors, such as a quantum processor 202. In this example, the classical computer 201 is coupled to the quantum computer 203 via an interface 204 over which control signals can be sent from the classical processor 200 to the quantum processor 202. In this example, the quantum computing system 108 may include one or more input / output components 206, such as keyboards, display screens, touch interfaces, etc. that can be operated by a user to control the classical processor 200, such as for generating commands and / or associated control signals for the quantum processor 202.

[0036] In the example of FIG. 2, the quantum computing system 108 may include drive circuitry 208 for the quantum processor 202. For example, the drive circuitry 208 may include one or more electromagnetic pulse generators (e.g., microwave pulse generators, such as one more lasers and / or other light sources), and / or corresponding electrical and / or opticalpathways for guiding electromagnetic pulses to one or more of the qubits 210 of the quantum processor, for controlling operation of the quantum processor 202. In the example of FIG. 2, the drive circuitry 208 is depicted as being arranged along an edge of the quantum processor 202. However, this is merely illustrative, and, in various implementations, the drive circuitry 208 may be otherwise arranged with respect to, and or integrated within, the classical computer 201 or the quantum processor 202.

[0037] As illustrated in FIG. 2, the classical processor 200 may include transistors 209 (e.g., thousands, millions, billions, or trillions of transistors 209). For example, the transistors 209 may be formed in a silicon substrate using one or more semiconductor processing operations (e.g., etching, lithography, etc.). The classical processor 200 may include classical logic gates (e.g., thousands, millions, billions, or trillions of classical logic gates), each formed by combinations of one or more of the transistors 209 and / or other circuit elements formed in the silicon substrate.

[0038] Although the example of FIG. 2 depicts a quantum computing system 108 that includes a classical computer 201 with a classical processor 200, it is appreciated that, in one or more implementations, a quantum computing system 108 may be provided without a classical computer 201. For example, as indicated by the dashed lines in FIG. 2, in one or more implementations, input / output (VO) components 206 may be directly coupled to the quantum processor and / or the drive circuitry 208 for controlling operation of the quantum processor 202 without involvement of a classical processor 200.

[0039] As shown, the quantum processor 202 may include qubits 210 (e.g., tens, hundreds, thousands, millions, billions, or trillions of qubits 210). Each qubit 210 may represent a single physical qubit, or may represent a logical qubit, which may be formed from one, two, or more than two physical qubits. In various implementations, qubits 210 may be implemented as trapped-ion qubits (e.g., including trapped ionized ytterbium atoms), superconducting qubits (e.g., transmons), Rydberg atom qubits, tunable superconducting qubits, quantum dot qubits, topological qubits, photonic qubits, Nuclear Magnetic Resonance (NMR) qubits, diamond nitrogen-vacancy (NV) center qubits, and / or any other suitable qubit architectures and / or quantum-scale anharmonic systems. In some implementations, qubits 210 may be implemented on, within, and / or located with respect to a substrate (e.g., a silicon, gallium arsenide, graphene, sapphire, aluminum nitride, and silicon carbide, or other substrate). For example, some qubits may be implemented using a superconducting loop on asilicon substrate. In other implementations, qubits may be implemented separately from a substrate of the quantum processor 202.

[0040] Qubits 210 may be used, by the quantum processor 202, individually and / or in various combinations, for storage of information, and / or for implementation of one or more quantum logic gates as discussed in further detail hereinafter. As examples, FIGS. 3, 22, and 23 illustrate examples of quantum gates that may be implemented using two qubits 210 coupled via an oscillator 300 (e.g., a cavity or other resonator such as a microwave transmission line resonator), in accordance with aspects of the subject disclosure.

[0041] For example, FIG. 3 illustrates a quantum gate 301 (e.g., a controlled phase gate, such as a controlled-Z (CZ) gate, such as a cavity-mediated cross-resonance gate), implemented by concurrently driving (e.g., by concurrently applying an electromagnetic pulse 302 to) a first qubit 210A and second qubit 210B that are coupled via the oscillator 300 at a frequency (e.g., b) that corresponds (e.g., approximately) to an oscillator frequency (e.g., v) of an oscillator 300. For example, the electromagnetic pulse(s) 302 may have a frequency (e.g., b) at or near (e.g., different by a detuning, e, from) a resonance frequency (e.g., v) of the oscillator 300. The oscillator 300 may be implemented as a cavity (e.g., a cavity formed in or on a substrate, and / or an optical cavity), or other resonator.

[0042] For example, consider two transmons treated within a two-level (e.g., qubit) approximation, coupled via a resonator (e.g., oscillator 300), and both driven at the same frequency near resonance with the resonator. The resulting Hamiltonian is:

[0043] Herecreates an excitation in the resonator, which has frequencyandare Pauli matrices acting on qubit i = 1,2 in the { 11 ), |0)} basis and having frequency an. cr is the raising operator of transmon j. Q, is the Rabi frequency driving transmon z at frequency ~ v. In Eq. 1, g is the coupling between each of the transmons and the resonator, and h.c. stands for the Hermitian conjugate of the preceding term.

[0044] Moving into the interaction picture with respect toandmaking the rotating-wave approximation (RWA), we arrive at

[0045] whereis the (negative of the) detuning between the drive frequency and the resonator andis the (negative of the) detuning between the drive frequency and the qubit.

[0046] To understand how the cross-resonance gate works, let us change the local basis to diagonalize each qubit. To do this, we rotate the Hamiltonian with the unitarywhereso that the new Hamiltonian is

[0047] Assumingand moving into the interaction picture with respect towe arrive at

[0048] To solve this Hamiltonian, we can think ofanumber (rather than an operator) that takes one of four possible values depending on what eigenstates of Zt we are in. Then we calculate time evolution of the harmonic oscillator b under a simple Hamiltonian, something that can be done analytically. It is convenient to go into an interaction picture with respect towhich gives:

[0049] Time evolution under this time-dependent Hamiltonian is given by the following unitary:If we choose our parameters such thatwhere n is an integer, the unitary dramatically simplifies towhich, up to commuting single-qubit terms, is equivalent to evolution for time t underSince(whereis the projector on the +1 eigenstate of Z), if we further choose our parameters such thatwe get a gate that, up to single-qubit rotations, is equivalent to a CZ gate. Eliminating e from the two constraints and solving for t, we get

[0050] The fastest gate occurs for n = 1 and takes time

[0051] To understand better how the gate works, consider the Heisenberg evolution of b under H in Eq. 9:

[0052] Taking the expectation value of both sides in some initial state, we plot the resulting time evolution in phase space in FIG. 4. As shown in FIG. 4, (b(f)) takes timeto make a n counterclockwise circles of radius and, thus, enclose an area ofWe can interpret the total phase picked in Eq. 12 as

[0053] The reason for this factor of 2 in front of the area is in the conversion factors between and on the one hand and positionand momentumon the other. For example, a 1 * 1 square (which has area 1) in our figures corresponds to a rectangle in the (x,p) plane with sidesandwhich has areaThis factor of 2 is the reason we multiply the areas in our figures by 2 to get the correct phase.

[0054] Notice that, under the conditions of Eq. 11, the unitary that takes us from the rotating frame to the lab frame isi.e. it is an identity. In other words, the whole picture is simultaneously rotating at rate e around the origin and undergoes n full rotations. In particular, if we take general time t (e.g., that doesn’t necessarily satisfy Eq. 11) then the rotating-frame Eq. 17 turns into the lab-frame equation

[0055] A convenient way to think about this evolution is to first evolve in the rotating frame as in FIG. 4, and then rotate the whole picture by 6 = rf clockwise around the origin. The result is shown in FIG. 5. Since the angle (0 = ef) by which we rotate counterclockwise within the circle 500 in the rotating frame is the same as the counterclockwise angle the whole picture rotates, once we rotate the circle 500 to the lab frame, the tangent to it at the current location in phase space (the end of the arrow 501) is always pointing directly down (as indicated by arrow 502).

[0056] The dominant corrections to the Hamiltonian in Eq. 8 is the quantum Stark shift, which we now derive. Suppose. This impliesIn this case, the term proportional to g in Eq. 7 is just the original interaction term in Eq. 1 :

[0057] The effect of these terms is suppressed because |A,| » g, which makes them highly off-resonant. The lowest order correction can be obtained in second-order (in g) perturbation theory:where f are coefficients that, for infinite anharmonicity, scale as / is the quantum Stark shift. The name comes from the fact that the qubit experiences a Stark- like shift that is proportional to the number of photons in the oscillator, much like the regular Stark shift, which is also proportional to the intensity. We see that this effect arises even in the limit of infinite anharmonicity, but is especially important when taking into account higher levels of the transmons because higher levels of the transmon give additional constraints on the choice of parameters and make the gate slower. Adding this correction to our Hamiltonian, we obtain

[0058] The evolution under this Hamiltonian is again exactly solvable using the same approach (treatingas numbers). The difference is that, not onlybut also the frequency of the oscillatornow also depends onGoing now into a rotating frame relative to we get exactly the same unitary as before, except e is replaced with

[0059] The unitary that we use for going into the lab frame from the rotating frame isso the full evolution in the lab frame is

[0060] This discussion also generalizes to transmons. Here we show how the cavity- mediated cross-resonance gate is modified in the presence of finite (i.e. not infinite) anharmonicity. The Hamiltonian isWe now make the rotating-wave approximation and go to a rotating frame with respect toto getwhere we again use a reverse notation for detuning for consistency. Let’s now diagonalize each transmon assuming that > « A, / / . The lowest two new eigenstates are then with energiesWe therefore get a cross-resonance gate given by the Hamiltonianwhereso that at vanishing anharmonicity we have no gate, while at infinite anharmonicity, we recover

[0061] Now let’s use a similar calculation to derive the quantum Stark shift in the limitIn this limit, thedressing is weak, so the dressed states and energies are approximately the same as the undressed ones. We can then compute the quantum Stark shift using second order perturbation theory in(here the index on the transmon is dropped, since the quantum Stark shift is computed separately for each transmon).Starting in statewe get the termStarting from state |1), we get the termSo, we can write the quantum Stark shift aswhereThis again reduces to zero in the case of zero anharmonicity and isin the regime of infinite anharmonicity (which is the regime discussed above).

[0062] So, in the case of transmons, these modified expressions may be used for the Stark shift Hamiltonian and for the Stark-shift-free Hamiltonian.

[0063] In accordance with aspects of the disclosure, a quantum computing system (e.g., a quantum computing system 108) is provided that includes a first qubit (e.g., qubit 210A), an oscillator (e.g., oscillator 300), and a second qubit (e.g., qubit 210B) coupled to the first qubit via the oscillator, in which the quantum computing system is configured to concurrently drivethe first qubit and the second qubit at a frequency (e.g., 5) that corresponds to an oscillator frequency (e.g., v) of the oscillator (e.g., as illustrated in FIG. 3, and as described in further detail herein in connection with, for example, Eq. 1). The quantum computing system (e.g., the classical computer 201, the drive circuitry 208, and / or a quantum processor 202 implementing the first qubit, the second qubit, and the oscillator) may be configured to drive the first qubit and the second qubit without directly driving the oscillator (e.g., as illustrated in FIG. 3). For example, the first qubit, the oscillator, and the second qubit may form a controlled phase gate when the first qubit and the second qubit are driven at the frequency. In one or more implementations, the controlled phase gate may form a cavity-mediated crossresonance gate that forms a controlled-Z (CZ) gate when the first qubit and the second qubit are driven at the frequency. Driving the first qubit and the second qubit at the frequency without directly driving the oscillator may populate the oscillator with a number of excitations greater than 0.1, in one or more implementations.

[0064] In one or more implementations, driving the first qubit and the second qubit at the frequency may include applying an electromagnetic pulse (e.g., electromagnetic pulse 302) having the frequency (e.g., 5) concurrently to each of the first qubit and the second qubit. For example, applying the electromagnetic pulse having the frequency concurrently to each of the first qubit and the second qubit may include applying the electromagnetic pulse concurrently to each of the first qubit and the second qubit for an amount of time (e.g., a time, Z, as given by Eq. 16) that is inversely proportional to a square root of a product of a first ratio and a second ratio, the first ratio (e.g., Qy / Ay) between a Rabi frequency of the first qubit and a detuning (e.g., Ai) between a resonance frequency (e.g., my) of the first qubit and the frequency (e.g., <5) that corresponds (e.g., approximately) to the oscillator frequency (e.g., v), the second ratio (e.g., Q2 / A2) between a Rabi frequency of the second qubit and a detuning (e.g., A2) between a resonance frequency (e.g., 02) of the second qubit and the frequency (e.g., <5) that corresponds (e.g., approximately) to the oscillator frequency (e.g., v).

[0065] In one or more implementations, the qubits may be superconducting qubits (e.g., transmons), and applying the electromagnetic pulse having the oscillator frequency concurrently to each of the first qubit and the second qubit may include applying the electromagnetic pulse concurrently to each of the first qubit and the second qubit for an amount of time (e.g., a time, Z, as derived by substituting the expression in Eq. 36 for thesin(6) in Eq. 16) that is based on an anharmonicity (e.g., z / i) of the first qubit and an a harmonicity (7 / 2) of the second qubit.

[0066] Aspects of the subject technology relate to various approaches to mitigating an effect of a quantum Stark shift, including a “three integers” approach, and a “flowers” approach as described in further detail hereinafter.

[0067] With respect to the “three integers” approach, to make sure the unitary U(f) in Eq. 25 above is simple (and, as before, doesn’t entangle qubits with the oscillator), we now may ensure thatfor all four choices ofNotice that this condition not only dramatically simplifies U(f), but also ensures thatso we don’t have to worry about going from the rotating frame to the lab frame at the end.

[0068] The condition in Eq. 40 is equivalent to requiring thatwhere andare integers (e.g., so thatsatisfying two additional conditions: (a) that is even and (b) that for all 4 combinationsof signs. Note that a special case of this situation is whenand ni are all even. Another special case is whenin which caseand so condition (a) above reduces to a requirement thatis even, therefore recovering the old condition thatis an integer. It is also noted that, under these conditions, the (in general, potentially undesirably entangling) transformation to the rotating frame relative to fb^b is conveniently equal to identity.

[0069] Once these conditions are satisfied, the unitary simplifies toDefining four phases(where, the condition for implementing a gate that is, up to single-qubit rotations, equivalent to a CZ gate is

[0070] Various parameters may be used as tuning knobs to satisfy this condition, while simultaneously obtaining integer values for, andIf gi,and v are not tunable, we only have two knobs (Z and r>) to obtain three integer valuesandTherefore, we need an additional knob, which can be realized by makingor v tunable. Finally, Eq. 45 is can be satisfied since this equation has two additional tuning knobs,and

[0071] Much like in the construction without quantum Stark shifts, where it is important to have stability to ensure that) remains very close to an integer despite all fluctuations, we will now similarly provide stability to ensure thatand f t remain very close tointeger values.

[0072] It is also noted that, typically, fl « c, which means that ne » 1, which in turn means that this approach will typically lead to a sqrt( / / ,) slowdown of the gate.

[0073] In one or more implementations, the quantum computing system (e.g., a quantum computing system 108) may be configured to mitigate an effect of a quantum Stark shift of one or both of the first qubit or the second qubit (e.g., using the “three integers” approach discussed above) by driving the first qubit and the second qubit for an amount of time (e.g., time, Z, in Eq. 41 above) that is proportional to a first integer (e.g., ri,) and inversely proportional to a detuning (e.g., e) between the frequency and the oscillator frequency. The amount of time may be further proportional to a second integer (e.g., rii) and inversely proportional to a first expansion coefficient (e.g., / 7) of a Hamiltonian (e.g., Hstark of Eq. 22) of the quantum Stark shift (e.g., as in Eq. 42 above), and proportional to a third integer (e.g., ri2) and inversely proportional to a second expansion coefficient (e.g., / 2) of the Hamiltonian of the quantum Stark shift (e.g., as in Eq. 43 above).

[0074] Turning now to the “flowers” approach, the quantum Stark shift may be (e.g., exactly) cancelled by strategically applying it pulses simultaneously on both qubits 210A and 210B. The following discussion shows how this approach works in the case where N= 4 pairs of pulses are applied (e.g., wherein may be smallest TV that works). Defining xthe Hamiltonian with the quantum Stark shift iswhereWe apply this Hamiltonian (e.g., by applying an electromagnetic pulse to the qubits) for timeWe then apply n pulses on both qubits, which changesfor both i. This effectively changes the signs of x and AT to givewhich we apply for the same amount of time r. We then again applypulses, thus going back to which we again apply for time r. We then again applypulses, thus going back to which we again apply for time r. FIG. 6 illustrates an example of this pulse sequence.

[0075] In one or more implementations, the quantum computing system(s) disclosed herein may be configured to mitigate (e.g., using the “flowers” approach) an effect of a quantum Stark shift of one or both of the first qubit 210A or the second qubit 21 OB by, in addition to applying the electromagnetic pulse having the frequency concurrently to each of the first qubit 210A and the second qubit 21 OB for a first amount of time (e.g., 4T in FIG. 6), applying, during the first amount of time, at least four pairs of additional pulses, each of the at least four pairs of additional pulses applied for a second amount of time (e.g., tpi, as shown and described in connection with FIG. 6) shorter than the first amount of time. For example, each of the additional pulses may include a 7t-pulse corresponding to a phase rotation of a corresponding qubit by one hundred eighty degrees (e.g., as described above).

[0076] Evolution under H for time t in the lab frame can be directly read out from Eq. 20:The intuition is the same as in FIG. 5, excepti.e. we first rotate (in the rotating frame) counterclockwise by an anglet around the circle 500 whose center \s M Z (1 + x)) to the right of the initial point. And then the whole picture is rotated clockwise around the origin by the same angle 0, so that the tangent is pointing directly down.

[0077] Now the evolution under H_ for the time t in the lab frame is

[0078] This evolution is shown in FIG. 7. Defining e = <r(l-x), we first rotate (in the rotating frame) counterclockwise by an angle 0 = ( I around the circle 700 whose center is ML' to the left of the initial point. And then the whole picture is rotated clockwise around the origin by the same angle 0, so that the tangent 702 is pointing directly up.

[0079] Let’s now consider an evolution for some time t+ under H. and then for some time t_ under H_. Evolving under H and bringing the corresponding circle 800 to the lab frame, the tangent to it in the current location is pointing down, as shown in FIG. 8. This means that, if we now switch to 77_, the corresponding rotating frame evolution will proceed along a circle to the left, and the direction of travel in phase space will be exactly reversed, i.e. we will be initially traveling directly up. If we were to now bring the whole picture to the lab frame, the tangent to the H_ circle would be pointing directly up. So, if we were to switch again to H, we would again exactly reverse the direction of travel and would travel and would travel (still counterclockwise) around a new rotating-frame circle.

[0080] Now that we understand how the evolution works, it is convenient to work in the rotating frame where both H and H_ have the b>'b term rotated away:

[0081] For the case x = 0, FIG. 9 shows that, after the four evolutions for time r, the photonic state in phase spaces comes back to where it started. Starting again at the origin and first evolving under H counterclockwise along the bottom right circle of radius r = M / c for an angle er = 3x12. We then evolve under H_ counterclockwise along the upper right circle of the same radius for the same angle. Then we evolve along upper left circle, and then along the bottom left circle. We call this approach the “flowers” approach because FIG. 9 looks like a 4-petal flower 900.

[0082] As shown in FIG. 10, the photonic state does come back to where it started even for x A 0. Starting again at the origin and first evolving under H counterclockwise along the right small circle of radius n = (M / cy(l + x) for an angle e(7 + x)r. An evolution may then be performed under H_ counterclockwise along the upper large circle of radius r2~ = (M / e) / (J^x) for an angle e(7-x)r, and then along the left small circle, and then along the bottom large circle.Intuitively, the photonic state comes back to where it started because the over-rotation by rex in rr (7 + x) cancels with the under-rotation by rex in TC(7 -x).

[0083] To be more precise, we will now show that the centers of the four circles sit at the vertices of a rhombus. The effect of switching the sign of M is to flip the instantaneous direction of travel, but keep the counterclockwise sense in which the circle is traced (assuming for this sentence that x < 7); so the line connecting the centers of neighboring circles goes through the point at which the n pulse induces the switch from one circle to the next.

[0084] Since we go along a small circle for angle e(7 + x)r, half of the remaining angle is

[0085] Similarly, half of the angle left after going along a large circle isSincethe corresponding triangle is a right triangle, so the four lines connecting neighboring circles form a rhombus, which immediately fixes the entire picture.

[0086] We will now show that all flowers with even N > 4 work. We apply N pairs of TT pulses at time intervalsIn FIG. 11, we show that, for the case x = 0, the N = 6 evolution takes the photonic state to where it started. FIG. 12 shows the N = 6 flower 1200 forand pThe key is that P i +pi is independent of x (i.e. over-rotation and under-rotation by exr cancel), so, like in the original x = 0 protocol, the third angle in the triangle is

[0087] So, we have N congruent triangles making up an TV-sided polygon.

[0088] The phase of the resulting (up to single-qubit rotations) controlled-phase gate can be computed from the areas of the flowers. In particular, the phase accumulated for a particular initial state of Ziis twice the area of the corresponding flower. The area of the flower can be computed by calculating separately the areas of the TV-sided polygon 1202 and the parts of circles of flower 1200 outside the polygon. The area of the polygon is

[0089] whereThe area of the relevant parts of the circles is

[0090] The accumulated phase is 2(AP+ Ac). The phase is the same for (Zi,Z2) = (1, 1) andSimilarly, the phase is the same for (Zi,Z2) = (1,-1) and (Zi,Z2) = (—1, 1). However, the values of x and AT are different between these two groups. We will denote these values byTo get a gate equivalent (up to single-qubit rotations) to a CZ gate, we use the following conditionThis translates to

[0091] The equation for the total phase TT can be solved for e, which can then be plugged into the formula for the total duration of the gate:Assuming that x is small, the formula for Ztotwill be approximately the same as for x = 0, in which case Ztot grows monotonically with N. Since the smallest N for which the cancellation works is N = 4, we may use N= 4 to get the fastest gate. The time this gate will take is thenwhich is slower than the ideal gate (without quantum Stark shifts) by a factor of only 1.45. If 7T pulses can be applied well, this approach may be more promising than the “three integers” approach, which may be much slower, unless all 3 integers can be made small, which may not be possible in cases in which ft « p as discussed above.

[0092] N > 4 may also be desirable in use cases in which we are interested in canceling Z noise on the qubits. Although the gate duration increases monotonically with N, the gatedoesn’t really get appreciably slower at higher N. Indeed, even in the limit A — > co, the gate is slower than the ideal gate only by TT / 2 ~ 1.57, which is not that different from the N = 4 factor of 1.45.

[0093] We can slightly generalize the flowers approach as follows. Recall that above we apply N pairs of TT pulses at equal time intervalsThis means that the last pair of pulses is applied at the very end of the time evolution, e.g., no more time evolution takes place after this last pair of pulses, which are just used to return us to the original rotating frame. We can generalize the flowers approach by making the first time interval (e.g., the wait time before the first pair of pulses) shorter, say T~ < T instead of r. The remaining pulses may then be applied at the usual time intervals r. And then, after the last pair of pulses, we evolve for timeWhat this does is rotates the entire flower in phase space. FIG. 13 shows the N= 4 example forwhile FIG. 14 shows the N= 6 example for

[0094] Takingturns our sequence of TT pulses into a CPMG (equivalently CP) pulse sequence for canceling phase (Z) noise on the qubits.

[0095] One potential disadvantage of using CPMG (or any other sequence with T~ < r) over the original pulse sequence is that we may apply all N pairs of pulses during the gate. On the other hand, in the original pulse sequence, the last pair of pulses occurs at the very end, so it simply brings us to the original rotating frame and may be absorbed into subsequent gates. So, the original pulse sequence uses N - 1 pairs of pulses instead of N.

[0096] To alleviate imperfections in the implementation of the TT pulses themselves, we can alternate TT pulses around X axis and Y axis. Since we only care about flipping Z to -Z, this doesn’t affect the ideal flowers discussed above.

[0097] In one or more implementations, the quantum computing system (e.g., quantum computing system 108) may also include an additional oscillator between the first qubit 210A and the second qubit 210B. For example, in an implementation that includes two transmons coupled via two oscillators, the quantum Stark shift may be canceled in the case where the resulting (e.g., cross-resonance) gate is mediated by more than one mode.

[0098] For example, consider two transmons with annihilation operators c, coupled each to its own oscillator bi, where the two oscillators are coupled to each other. In this example, the Hamiltonian is

[0099] Consider the situation where gi, g2, and g~ can all be treated as small parameters proportional to g, in which case the always-on ZZ interaction appears at sixth order in g. We define annihilation operatorsandfor the symmetric and antisymmetic modes, respectively. Equivalently we can write this asandor equivalently The Hamiltonian then becomes

[0100] We now add the cross-resonance drive

[0101] We do the rotating wave approximation and go into the rotating frame to obtain the combined Hamiltonian

[0102] We now diagonalize each transmon, restrict the Hamiltonian to the lowest two states, and drop the off-resonant terms, which giveswhere the anglesatisfiesWe can now drop the single-qubit terms since they commute with the rest of the Hamiltonian and slightly change the rotating frame to obtainwhere where

[0103] We see that Hsand Hacommute with each other, so we are implementing two commuting cross-resonance gates in parallel, one via the symmetric mode and one via the anti-symmetric mode. Let’s suppose that we want to tune closer to the symmetric mode, so that we make only one circle in the corresponding phase space:while we allow potentially for any integer number of circles in the phase space corresponding to the antisymmetric mode:Once these two conditions are satisfied, the spin unitary takes the formSo, to get a gate equivalent (up to single-qubit rotations) to a CZ gate, we use

[0104] This formula makes sense at least in part because ifwe should not be able to get any coupling as the formula confirms. Let’s assume thatand then check this condition later. In that case, the condition to get the CZ gate and the conditioncan be used to eliminate Gto obtain (like in the case of a single-mode cross-resonance gate, except the couplings gi are reduced byand

[0105] Assuming for simplicity that gi = gi = g andwe haveg sin# « g since we typically work at sin# « 1. If we further assume thatthis means that which means as desired. By tuning the parameters such thatsfor some integer naand further tuning the parameters slightly so that the CZ phase is exactly we get the CZ gate, which is only a factor of V2 slower than the single-mode gate. Notice that, because G is negative and comes with a minus sign in the formula for the total phase, the two modes actually contribute constructively.

[0106] As we apply the N= 4 flower to the symmetric mode, the antisymmetric mode will also come back to the origin provided nais odd. With the quantum Stark shift, the Hamiltonian is wherewherefor coefficients

[0107] LetwhereLet’s first understand why theflower works in the caseThe following uses an example of the CPMG version of the flower where we first wait for time, followed by three time intervals of durationfollowed by one more time interval of duration r / 2.

[0108] For m = 0, the antisymmetric mode undergoes the simple CPMG N = 4 flower shown in FIG. 15. In FIG. 15, we start at the origin and follow the arrow 1500 until we get to the square, then we follow the arrow 1502, then the arrow 1504, etc.., until we come back to the origin.

[0109] Forthe antisymmetric mode undergoes the following flower shown in FIG. 16. The first angle (arrow 1600), instead of being 3TT / 4 is nowso we make a full circle and another. The evolution then starts down the arrow 1602 and goes for twice as large of an angle, i.e.i.e. makes two full circles plus The evolution then starts down the arrow 1604 and goes for the same angle, etc... until coming back to the origin.

[0110] For m = 2, the antisymmetric mode undergoes the flower in FIG. 17. The first angle (arrow 1700) isSO makes a full circle and anotherThe evolution then starts down the arrow 1702 and goes for twice as large of an angle, i.ei.e. makes 3 full circles plus 3TT / 2. The evolution then starts down the arrow 1704 and goes for the same angle, etc... until coming back to the origin.

[0111] Forthe antisymmetric mode undergoes the flower in FIG. 18. The first angle (arrow 1800) isSO makes 2 full circles and anotherThe evolution then starts down the arrow 1802 and goes for twice as large of an angle, i.ei.e. makes 5 full circles plus The evolution then starts down the arrow 1804 and goes for the same angle, etc... until coming back to the origin.

[0112] Finally, for m = 4, the evolution follows exactly the same flower as for m = 0, except we also go 6 times around each of the four circles. The flowers then repeat based on what m is modulo 4.

[0113] Now let’s introduce the imperfectionFor, it will work exactly like the regular N = 4 flower. For m = 1, the flower is shown in FIG. 19. As before, there are circles of two radii:and. And, as before, the smaller circles are over-rotating, while the larger circles are under-rotating. The case of nonzero xsworks similarly for other values of m.

[0114] To calculate the total phase, we compute the areas of both the symmetric-mode flowers (for various values of Msand xs) and the anti-symmetric mode flowers (for various values of Msand xf) and appropriately combine them taking into account that in reality (a< 0 (while the above-noted figures are for positive ea) but also that Zz comes with a negative sign in Ma.

[0115] In one or more implementations, the quantum computing system (e.g., quantum computing system 109) may also include an additional oscillator coupled to the second qubit. For example, in one or more implementations, a cross-resonance gate may be implemented along one bond, while the two transmons participating in the gate are also coupled to other oscillators along bonds that we are not currently using for a gate. These other oscillators will be coupled during the gate to the transmons via the usual coupling and via the quantum Stark shift, and if these couplings are not negligible (e.g., if the corresponding oscillators are not too far in frequency from the oscillator participating in the gate), we would also want to cancel them.

[0116] To see how to make this cancellation work, suppose transmon 1 with annihilation operator ci is coupled to transmon 2 with annihilation operator cz via a cross-resonance gate mediated by an oscillator with annihilation operator bz. Furthermore, suppose that ci is also coupled to another oscillator with annihilation operator b\. The Hamiltonian is then very similar to the one analyzed above for the case where two transmons are coupled via two modes. The difference is in coupling coefficients and in the fact that cz is not coupled to bi. The Hamiltonian iswherewhere

[0117] Here H2mediates the cross-resonance gate, while Hi describes the parasitic coupling to mode bi. The two Hamiltonians commute. So, the problem is very similar to the problem we solved for the case of the cross-resonance gate mediated by two modes. We use H2to implement a cross-resonance gate via an N = 4 flower with e2t = 2n. And then, as in the above two-mode mediated gate, we set at = limi for an odd integer m. This will help to ensure that the oscillator bi comes back to the initial state, without a substantial residual effect on the qubit (e.g., not even a phase gate because both qubit states undergo a flower of the same area).

[0118] It is worth pointing out that, if each bond consists of two oscillators, then we will cancel a lot of such couplings (e.g., two couplings per bond). So, if, in a square lattice, coupling two qubits via a gate mediated by two modes is performed, and then each of the qubits is connected to three more bonds (with two modes per bond), if all modes involved have different frequencies, all fourteen modes may be brought back to where they started from, which involves substantial tuning. The number of conditions can be reduced if some of the modes have the same frequency, but one has to be careful in that case to avoid issues if we want to drive only one of these modes but not the other. The number of conditions can also be reduced if the adjacent modes are very far away in frequency from the mode that is used for the gate: in that case the parasitic coupling to such adjacent modes will be negligible.

[0119] In one or more implementations, the quantum computing system (e.g., quantum computing system 108) may also include a third qubit, and using the flowers scheme to implement a gate between the first qubit and the second qubit can also (e.g., including by applying, during the first amount of time, the at least four pairs of additional pulses) may cancel an always-on ZZ interaction between at least the second qubit and the third qubit. As an example, consider a 2D architecture as shown in FIG. 20, where an are frequencies of qubits at the sites of the square lattice and v, and ~v, are frequencies of oscillators on the bonds. In an example, we are implementing two-qubit gates simultaneous along all V3 and ~V3 bonds 2006. Remaining bonds 2004 are also shown.

[0120] In an example, we are implementing the cross-resonance gate using the N= 4 flowers scheme. As explained herein, this gate may be implemented in at least the following two ways: (a) the original N = 4 flowers approach waits for time r = 3?r / (2e) before applying the first pair of TT pulses, then applies the rest of the pulses at the same time interval r; or (b) a CPMG-like modification of this pulse sequence is to wait for time r / 2 = 3?r / (4e) before applying the first pair of TT pulses, and then applying the rest of the pulses still at the time interval r. Pulse sequence (1) may then be used on the qubits 2002 and pulse sequence (2) on the qubits 2000. It is appreciated that the bonds that we do not want to couple along during this gate (e.g., bonds 2004) couple a qubit 2002 and a qubit 2000. Because the two pulse sequences are r / 2 out of sync with each other, the sign of the effective always-on ZZ interaction flips every r / 2, which results in the cancellation of always-on ZZ interactions along all bonds 2004. This result is illustrated in FIG. 21.

[0121] In the example of FIG. 21, the horizontal axis is time. The first row 2100 shows the four time intervals of duration r corresponding to the pulses from sequence (1) applied on transmons 2002. The sign is the sign of the effective Z. The third row 2106 shows the time intervals corresponding to the pulses from the CPMG-like sequence (2) applied on transmons 2000. Finally, the middle row 2004 shows the resulting effective sign of the ZZ interaction between a transmon 2002 and a transmon 2000. We see that the ZZ interaction is echoed away.

[0122] In the above scheme, if we want to implement a gate not on all qubit 2002 to qubit 2002 bonds and qubit 2000 to qubit 2000 bonds, but only on some, we can still keep applying the pulses on all transmons. This will cancel always-on ZZ interactions on qubit 2002 to qubit 2000 bonds, but will not cancel the ZZ interactions on the qubit 2002 to qubit 2002bonds and qubit 2000 to qubit 2000 bonds that are kept turned off. The solution to this may be to implement gates on all qubit 2002 to qubit 2002 bonds and all qubit 2000 to qubit 2000 bonds, but tune the total nonlinear phase on some bonds to be zero.

[0123] A potential issue with the above 2D scheme is that when we drive two V3 bonds that are in the same column and separated by one vi bond, the two qubits sharing the vi may also end up coupled via an unintentional vimediated cross-resonance gate. One way to resolve this problem is to define two difference frequencies V3 and alternate them along a column.

[0124] In one or more implementations, the quantum computing system (e.g., quantum computing system 108) may also include a plurality of additional qubits and an array of coupled additional cavities (e.g., a metamaterial), at least a subset of the additional cavities coupled to at least one of the plurality of additional qubits. In these implementations, the quantum computing system may be configured to apply the at least four pairs of additional pulses to one or more pairs of the additional qubits that are coupled to the array of coupled additional cavities.

[0125] For example, a gate implemented using the qubit driving operations described herein works in the case of N qubits (or other anharmonic systems) coupled via an array of coupled oscillators (e.g., a metamaterial). In particular, the gate can allow for the simultaneous application of the cross-resonance gate between multiple pairs of distant qubits.

[0126] For a chain of N qubits connected by ~N metamaterial cavities, simultaneity qualitatively changes the situation only if we can do ~N gates at once. So let’s assume we want to do all N / 2 pairs of gates at once.

[0127] The Hamiltonian iswhere bk are annihilation operators for the eigenmodes (of energy Vk) of the metamaterial, and qubit z is driven with its own frequency 3, and Rabi frequency Q,. The qubit-metamaterial couplings are, where gi is the coupling of the qubit to the corresponding cavity andis the matrix that diagonalizes the metamaterial. Assuming that the nearest-neighbor coupling in the metamaterial is J, we have ~ N eigenmodes with frequencies betweenand so the modes are separated by Suppose we want to make every pair of qubits do a cross-resonance gate via a particular eigenmode in such a way that the effect of other eigenmodes (other than the one a given pair of qubits is talking to) can be neglected.

[0128] In a first example, two qubits may be provided along with ~N metamaterial modes. Ignoring the quantum Stark shift, we getwhereNotice that the cross-resonance gate reduces the structure of the coupling to a simple Zt coupling, which dramatically simplifies gate analysis (since one can simply solve the evolution for different values of Zt and treat Ztas numbers). In this sense, the crossresonance gate enables the implementation with superconducting qubits of what can be done directly with trapped ions. With trapped ions, one can separately engineer a o a coupling and a coupling which together can give a coupling of the formOn the other hand, with superconducting qubits, engineering the termwhich coherently excites both the transmon and the cavity is very challenging. The cross-resonance gate works by tuning close to one of the modes (e.g., the corresponding a is small). It is clear that the condition for neglecting the other modes iswhereis how far the nearest mode is (if we had long range interactions, then one could get a mode that is split from neighboring modes my much more thanSince the gate takes time T ~ l / (gi,k sinft), we haveNotice that, if we use the metamaterial in real (rather than virtual) way, we get the speed limit(assuming we don’t populate the metamaterial with many photons and taking advantage of collective enhancement).

[0129] Once we are in the limitand each pair of qubits talks through one mode (while other modes can be neglected), both the three integers approach and the flowers approach described above for canceling the quantum Stark shift work. In cases where othermodes cannot be neglected, we can use the generalization of the integers approach and the flowers approach to multiple modes, as described above.

[0130] FIG. 22 illustrates another example of a quantum gate 401 (e.g., a controlled phase gate, such as a controlled-Z (CZ) gate, which may be implemented as a cavity- mediated cross-resonance gate), implemented by concurrently driving the first qubit 210A and the second qubit 21 OB, in an implementation in which the qubits 210A and 21 OB are implemented as spin-locked qubits.

[0131] In accordance with aspects of the disclosure, a quantum computing system (e.g., quantum computing system 108) may include a first spin-locked qubit (e.g., qubit 210A in the spin-locked state of FIG. 22), an oscillator (e.g. oscillator 300), and a second spin-locked qubit (e.g., qubit 210B in the spin-locked state of FIG. 22) coupled to the first spin-locked qubit via the oscillator. For example, a first qubit and a second qubit may be driven (e.g., to spin lock the qubits, such as after first driving the bare first and second qubits at their bare resonant frequencies to establish coherent superposition states) resonantly with Rabi frequency Qsi. Upon driving the bare first and second qubits resonantly with Rabi frequenciessi, the resulting spin-locked qubits can be thought of as having frequencysi. Thus, in one or more implementations, the frequency of the first spin-locked qubit may be a Rabi frequency of the first spin-locked qubit, the frequency of the second spin-locked qubit may be a Rabi frequency of the second spin-locked qubit, and the first spin-locked qubit, the oscillator, and the second spin-locked qubit may form a controlled phase gate (e.g., CZ gate) between the two qubits when the first spin-locked qubit is driven at each of two additional (e.g., near resonant) frequencies and the second spin-locked qubit is driven at each of two further additional (e.g., near resonant) frequencies.

[0132] The Hamiltonian for such as system may be expressed as:For example, the quantum computing system of FIG. 22 may be configured to (e.g., to form the quantum gate between the first spin-locked qubit and the second spin-locked qubit) concurrently drive (i) the first spin-locked qubit (e.g., with electromagnetic pulses 400A and 402A, which may represent two concurrent pulses having two respective frequencies, or a single pulse having two frequencies) at each of two frequencies (e.g., two separate cross-resonance drive frequencies, as shown in the term of Eq. 97 above, with two different frequenciesand (ii) the second spin-locked qubit (e.g., with electromagnetic pulses 400B and 402B, which may represent two concurrent pulses having two respective frequencies, or a single pulse having two frequencies) at each of two additional frequencies. For example, the two frequencies (e.g., a first frequency of the electromagnetic pulse 400A and a second, different, frequency of the electromagnetic pulse 402A) may be symmetrically above and below a frequency (e.g., Qsi) of the first spin-locked qubit (e.g., and may be applied with Rabi frequencies Qcrof equal amplitude) and the two additional frequencies (e.g., a first frequency of the electromagnetic pulse 400B and a second, different, frequency of the electromagnetic pulse 402B) may be symmetrically above and below a frequency (e.g., Qsi) of the second spin-locked qubit (e.g., and may be applied with Rabi frequencies Qcrof equal amplitude).

[0133] In this case, in both of the regimes,andspin locking reduces the always-on interaction by a small factor (Q^A and A / Q^z, respectively, where A = v - coo in this case) while keeping the desired interaction strength unchanged. In the case of transmons (e.g., a transmon implementation of the qubits 210A and 21 OB of FIG. 22), we get a suppression relative to the qubit case not by 77 / A, but by ( / / / A)2. So, in the spin-locked case, the ratio of good to bad interactions gets better as we switch from qubits to transmons. This is in contrast to the bare case in which both the good and the bad interactions get multiplied by 77 / A (as we switch from qubits to transmons) keeping the good-to-bad ratio unchanged. This improvement may be provided by the near-resonant two-frequency driving described herein, which allows us to get only 771suppression (and not z / 2) for the good interaction.

[0134] In the example of FIG. 22, the first qubit 210A and the second qubit 21 OB are implemented as spin-locked qubits. In one or more other implementations, the first qubit 210A and the second qubit 21 OB may be implemented as dual -rail qubits.

[0135] For example, FIG. 23 illustrates another example of a quantum gate 501 (e.g., a controlled phase gate, such as a controlled-Z (CZ) gate, which may be implemented as a cavity-mediated cross-resonance gate), implemented by driving the first qubit 210A and the second qubit 210B (e.g., without driving the oscillator 300), in an implementation in which the qubits 210A and 210B are implemented as dual -rail qubits. In accordance with aspects of the disclosure, a quantum computing system (e.g., quantum computing system 108) may include a first dual-rail qubit (e.g., qubit 210A of FIG. 23), an oscillator (e.g., oscillator 300),and a second dual-rail qubit (e.g., qubit 21 OB of FIG. 23) coupled to the first dual-rail qubit via the oscillator, in which the quantum computing system is configured to drive the first dual-rail qubit with at least two concurrent first electromagnetic pulses (e.g., electromagnetic pulses 500A and 502A) and to concurrently drive the second dual-rail qubit with at least two concurrent second electromagnetic pulses (e.g., electromagnetic pulses 500B and 502B), to cause the first dual-rail qubit, the oscillator, and the second dual-rail qubit to form a quantum gate (e.g., as shown in FIG. 23).

[0136] For example, the Hamiltonian for driving the first qubit 210A in such a system may be expressed aswhere, out of four driving terms, only the two that give near-resonant driving are shown.

[0137] In one or more implementations, the first dual-rail qubit and the second dual-rail qubit each include a pair of qubits (e.g., QUBIT1-1 and QUBIT1-2 of the qubit 210A, and QUBIT2-1 and QUBIT2-2 of the qubit 21 OB) coupled to each other and to the oscillator, and the quantum computing system may be configured to drive the first dual-rail qubit with the at least two concurrent first electromagnetic pulses by driving a first one (e.g., QUBIT1-1) of the pair of qubits of the first dual-rail qubit concurrently at two frequencies symmetrically above and below (e.g., by an amount 2g~, see, e.g., Eq. 98 above) a frequency (e.g., -c + J) corresponding to a difference between a frequency (e.g., cob) of the first dual-rail qubit and a frequency (e.g., v) of the oscillator. For example, the electromagnetic pulse 500A may have a frequency of -c + A + 2g~, and the electromagnetic pulse 502A may have a frequency of -£ + A - 2g~ in some implementations.

[0138] In one or more implementations, the first dual-rail qubit and the second dual-rail qubit each include a pair of qubits (e.g., QUBIT1-1 and QUBIT1-2 of the qubit 210A, and QUBIT2-1 and QUBIT2-2 of the qubit 210B) coupled to each other and to the oscillator, and the quantum computing system may be configured to drive the first dual-rail qubit with the at least two concurrent first electromagnetic pulses by driving (e.g., with the electromagnetic pulse 500A) a first qubit (e.g., QUBIT1-1) of the pair of qubits of the first dual-rail qubit (e.g., at a Rabi frequency of the first qubit of the pair) with a first phase, and concurrently driving (e.g., with the electromagnetic pulse 502A) a second qubit (e.g., QUBIT1-2) of the pair ofqubits of the first dual-rail qubit (e.g., at a Rabi frequency of the second qubit of the pair) and with a second phase out of phase (e.g., one hundred eighty degrees out of phase) with the first phase.

[0139] FIG. 24 illustrates a flow diagram of an exemplary process 2400, in accordance with one or more embodiments of the subject technology. For explanatory purposes, the process 2400 is primarily described herein with reference to FIGS. 1-23. However, the process 2400 is not limited to the items shown in FIGS. 1-23, and one or more blocks (or operations) of the process 2400 may be performed by one or more other components of other suitable devices. Further, for explanatory purposes, the blocks of the process 2400 are described herein as occurring serially or linearly. However, multiple blocks of the process 2400 may occur in parallel. In addition, the blocks of the process 2400 need not be performed in the order shown and / or one or more blocks of the process 2400 need not be performed and / or may be replaced by other operations.

[0140] At block 2402, a controlled phase gate (e.g., quantum gate 301, 401, or 501 as described herein) may be generated by concurrently driving a first qubit (e.g., qubit 210A) and a second qubit (e.g., qubit 210B). For example, driving the first qubit and the second qubit may include driving the first qubit and the second qubit at a frequency (e.g., 5) corresponding to an oscillator frequency (e.g., v) of an oscillator (e.g., oscillator 300) that couples the first qubit to the second qubit, without directly driving the oscillator. For example, driving the first qubit and the second qubit at the frequency may include applying an electromagnetic pulse (e.g., electromagnetic pulse 302) having the frequency concurrently to the first qubit and the second qubit.

[0141] As another example, the first qubit may be a first spin-locked qubit, the second qubit may be a second spin-locked qubit, and driving the first qubit and the second qubit may include driving (i) the first spin-locked qubit at each of two frequencies that are symmetrically above and below a frequency of the first spin-locked qubit and (ii) the second spin-locked qubit at each of two additional frequencies that are symmetrically above and below a frequency of the second spin-locked qubit. As another example, the first qubit may be a first dual-rail qubit, the second qubit may be a second dual-rail qubit, and driving the first qubit and the second qubit may include driving the first dual-rail qubit with at least two concurrent first electromagnetic pulses and concurrently driving the second dual-rail qubitwith at least two concurrent second electromagnetic pulses to cause the first dual-rail qubit, the oscillator, and the second dual-rail qubit to form the controlled phase gate.

[0142] At block 2404, one or more quantum computing operations may be performed using the controlled phase gate.

[0143] Quantum computers may require gates between qubits. In accordance with aspects of the disclosure, various implementations of a two-qubit gate (e.g., quantum gate 301, 401, and / or 501) are disclosed.

[0144] While a cross-resonance gate between two qubits or other anharmonic systems has been studied both theoretically and experimentally, it has only been studied for the case where the two anharmonic systems are coupled directly, or via an ancillary system which is not being populated significantly during the operation of the gate. For example, prior crossresonance gates have been formed by driving one qubit at the frequency of another qubit, without driving the other qubit. Aspects of the subject technology provide a cross-resonance gate for two qubits (or other anharmonic systems) that are coupled through an oscillator (e.g., oscillator 300) or a set of oscillators (a metamaterial), where the coupled system (e.g., the oscillator 300 or set of oscillators) plays a significant role during the gate operation and is actively and significantly populated.

[0145] In summary, the basic operation of a quantum gate in accordance with the subject disclosure is described. Four approaches for increasing the fidelity of the gate are described herein. The first two approaches include the “three integers” approach and the “flowers” approach described herein, and may be used to cancel or reduce the dominant error called the quantum Stark shift. It is then described herein how these two approaches can be used for the case of coupling via two oscillators, to cancel coupling to adjacent cavities, to cancel always- on ZZ interactions, and / or for the case of coupling via a collection of oscillators (e.g., a metamaterial). The third approach disclosed herein includes using spin-locked qubits (e.g., as in the example of FIG. 22). The fourth approach disclosed herein includes using dual-rail qubits are used (e.g., as in the example of FIG. 23).

[0146] In accordance with aspects of the subject technology, a cavity -mediated crossresonance gate is provided. For example, two nonlinear superconducting elements may be coupled via a cavity and both driven (e.g., as in the example of FIG. 3) at a frequency that is near resonant with the frequency of the cavity, as shown in Eq. 1. As a result, a quantum gatemay be implemented between the two nonlinear superconducting elements. In general, a cavity-mediated gate between nonlinear superconducting elements may be provided in which the cavity is not driven (e.g., only the superconducting elements are driven) and in which the cavity is substantially populated (e.g. with number of excitations in the cavity above 0.1).

[0147] In one or more implementations, two qubits may be coupled via a cavity and both of them driven at the same frequency near the cavity frequency as shown in Eq. 1. Evolving under this Hamiltonian of Eq. 1 for, for example, the time given in Eq. 16, produces a CZ gate between the two qubits. In the common limit Qy« Ay, we have sin6( = Q / A,. In the case of transmons (e.g., a superconducting qubit), the evolution time may be modified so that Q / A / is replaced with the expression in Eq. 36.

[0148] In one or more implementations, a “three integers” approach may be performed for canceling the bad interaction, as described herein. For example, the gate may be implemented as in FIG. 3 and as described in Eq. 1, but the evolution time and other gate parameters may be chosen to satisfy Eqs. 41-43.

[0149] In one or more implementations, a “flowers” approach may be applied for canceling the bad interaction (as described herein). For example, the gate may be implemented as in FIG. 3 and as described Eq. 1, and N additional 7r-pulses (e.g., around the A or Y axis) may be applied simultaneously on both qubits at time intervals r = n / e + 2jr / (7Ve), where e is the detuning between the drive frequency and the cavity. N can be any even number greater than or equal to two. Provided e is set according to Eq. 61, the resulting gate between the two qubits is a CZ gate. In one or more implementations, the first time interval may be shorter, e.g., T~ < T instead of r, and there may be an additional time interval of duration r - T~ at the end. A special case of this is r~ = T / 2, which corresponds to a CPMG-like pulse sequence.

[0150] As discussed herein, generalizations of the “three integers” and “flowers” approaches described herein can be applied to the case of coupling via two oscillators, to cancel coupling to adjacent cavities, to cancel always-on ZZ interactions, and for the case of coupling via a collection of oscillators, e.g., a metameterial.

[0151] For example, as discussed herein, the cross-resonance gate may include qubits that are coupled via two oscillators instead of one. Supposeis the detuning of the drives from the symmetric mode andis the detuning from the antisymmetric mode. Suppose thedriving is close to resonance with the symmetric mode, e.g., The time and detuningsmay then be set such thatand, where nais an odd integer. As the N = 4 flower is applied to the symmetric mode, the antisymmetric mode will also come back to the origin provided nais odd.

[0152] As discussed herein, the cross-resonance gate (with detuning 62 between the cavity and the drive) may include one of the qubits also coupled to another adjacent cavity (with detuning ei between the cavity and the drive). The time and detunings may be set such thatand = 2im\ with m an odd integer. Then the N= 4 flower applied to the crossresonance gate automatically decouples the adjacent cavity.

[0153] As described herein, the flowers approach can also be sometimes used to cancel the always-on ZZ interaction. As an example, consider an array of qubits (or transmons) as shown in FIG. 22, where the cavities are shown as bonds 2004 and 2006. Suppose we are simultaneously implementing cross-resonance gates along all bonds 2006 of FIG. 22. Then if we use the regular N = 4 flower on the qubits 2002, and the CPMG-like N = 4 flower on the qubits 2000, all always-on ZZ couplings along bonds 2004 (which couple a qubit 2000 and a qubit 2002) will be canceled.

[0154] As discussed herein, N qubits may be coupled via a metamaterial made up of ~ N coupled cavities. Dividing (e.g., arbitrarily) qubits into N / 2 pairs, for each pair, a different eigenmode of the metamaterial may be selected and a cross-resonance gate mediated by this eigenmode may be implemented. In implementations in which the coupling gt,k si nft between qubit z and mode k, for all z and k, is much smaller than the frequency spacing between the modes (which is the same as the gate time being much longer than the inverse of the frequency spacing between the modes), then the three-integers approach and the flowers approach for canceling the quantum Stark shift work for each of the N / 2 simultaneously implemented cross-resonance gates. In cases where off-resonant modes cannot be neglected, the generalization of the integers approach and the flowers approach can be applied.

[0155] In accordance with aspects of the subject technology, a cavity -mediated crossresonance gate between spin-locked qubits or transmons may also be provided (e.g., as described in connection with FIG. 22). In one or more implementations, a two-frequency drive may be applied, as described herein. As in the example of FIG. 22, two qubits (or transmons) may be coupled to an oscillator. In this example, each qubit may be drivenresonantly with Rabi frequencyand also with an additional cross resonance driveas shown in Eq. 97 . While Eq. 97 doesn’t show it explicitly, two separateterms may be included with two different frequenciesand with Rabi frequenciesof equal amplitude.

[0156] In accordance with aspects of the subject technology, a cavity -mediated crossresonance gate between dual-rail qubits or transmons may be provided (e.g., as described in connection with FIG. 23). In one or more implementations, a two-frequency drive may be applied. As in the example of FIG. 23, each qubit (or transmon) may be replaced with two coupled qubits (or transmons). Each pair of qubits may be coupled to each other and to the cavity as expressed in the following Hamiltonian(e.g., temporarily ignoring the drive for the purpose of computing the bad always-on interaction). One of the qubits on each side may be driven with two frequencies, such as frequencies that are approximately equal to -e+A±2g~ (see Eq. 98, which shows the drive by one of these frequencies).

[0157] In one or more other implementations, both qubits on a given side can be driven with Rabi frequencies TT out of phase with each other. The resulting good interaction between a dual-rail logical qubit and the cavity is of the form(e.g., where A is the logical Ain the dual-rail basis) which can be used to implement a crossresonance gate. The protocol works in essentially the same way for the case of two transmons (e.g., instead of two qubits).

[0158] In summary, the asymptotic results in the regime of large A for bare qubits / transmons, dual-rail qubits / transmons, and spin-locked qubits / transmons are summarized in the table below. A peculiar feature of these tables is that the bad / good ratio drops (e.g., improves) by a factor of / / / A as we go from qubits to transmons in both the dualrail case and the spin-locked case. On the other hand, in the bare case, the bad / good ratio isunchanged as we go from qubits to transmons. So, for the dual-rail case and the spin-locked case, transmons in some sense are better than qubits (even though the overall strength of the good interaction decreases as we go from qubits to transmons).

[0159] FIG. 25 illustrates an example electronic system 2500 in which aspects of the classical computing systems of the present disclosure may be implemented, in accordance with one or more embodiments of the subject technology. The electronic system 2500 may be, and / or may be a part of, a computing device (e.g., client devices 102, client device 109, server(s) 106, classical computer 201). The electronic system 2500 may include various types of computer-readable media and interfaces for various other types of computer-readable media. The electronic system 2500 may include a bus 2510, a storage device 2502, a system memory 2504, an input device interface 2506, an output device interface 2508, a ROM 2512, a network interface 2514, and a processing unit 2516, or subsets and variations thereof. Not all depicted components may be used in all embodiments, however, and one or more embodiments may include additional or different components than those shown in the figure. Variations in the arrangement and type of the components may be made without departing from the spirit or scope of the claims as set forth herein. Additional components, different components, or fewer components may be provided.

[0160] Network interface 2514 may be configured to allow data to be exchanged between the electronic system 2500 and devices attached to a network or networks (e.g., network 104), such as other computer systems or devices. In various embodiments, network interface 2514may support communication via any suitable wired or wireless general data networks, such as types of Ethernet networks, for example. Additionally, network interface 2514 may support communication via telecommunications / telephony networks, such as analog voice networks or digital fiber communications networks, via storage area networks such as Fiber Channel SANs (storage area networks) or via any other suitable type of network and / or protocol.

[0161] The bus 2510 collectively represents all system, peripheral, and chipset buses that communicatively connect the numerous internal devices of the electronic system 2500. In one or more embodiments, the bus 2510 communicatively connects the processing unit 2516 with the other components of the electronic system 2500 (e.g., the ROM 2512, the system memory 2504, and the persistent storage device 2502). From various memory units, the processing unit 2516 retrieves instructions to execute and data to process in order to execute the operations of the subject disclosure. The processing unit 2516 may be a controller and / or a single- or multi-core processor or processors in various embodiments.

[0162] The ROM 2512 may store static data and instructions that are needed by the one or more processing unit(s) 2516 and other modules of the electronic system 2500. The storage device 2502, on the other hand, may be a read-and-write memory device. The storage device 2502 may be a non-volatile memory unit that stores instructions and data (e.g., static and dynamic instructions and data) even when the electronic system 2500 is off. Data may include one or more long-term data stores (e.g., databases). In one or more embodiments, a mass-storage device (such as a magnetic or optical disk and its corresponding disk drive) may be used as the storage device 2502. In one or more embodiments, a removable storage device (such as a flash drive, and its corresponding disk drive) may be used as the storage device 2502. Generally speaking, a computer-accessible medium may include non-transitory storage media or memory media, such as magnetic or optical media.

[0163] Like the storage device 2502, the system memory 2504 may be a read-and-write memory device. However, unlike the storage device 2502, the system memory 2504 may be a volatile read-and-write memory, such as random-access memory. The system memory 2504 may store any of the instructions and data that one or more processing unit 2516 may need at runtime to perform operations. Data may include one or more short-term data stores (e.g., caches and buffers). In one or more embodiments, the processes of the subject disclosure are stored in the system memory 2504 and / or the storage device 2502. From these variousmemory units, the one or more processing unit 2516 retrieves instructions to execute and data to process in order to execute the processes of one or more embodiments, discussed below.

[0164] Embodiments within the scope of the present disclosure may be partially or entirely realized using a tangible computer-readable storage medium (or multiple tangible computer-readable storage media of one or more types) encoding one or more instructions. The tangible computer-readable storage medium also may be non-transitory in nature.

[0165] The computer-readable storage medium may be any storage medium that may be read, written, or otherwise accessed by a general-purpose or special-purpose computing device, including any processing electronics and / or processing circuitry capable of executing instructions. For example, without limitation, the computer-readable medium may include any transitory semiconductor memory (e.g., the system memory 2504), such as RAM, DRAM, SRAM, T-RAM, Z-RAM, and TTRAM. The computer-readable medium also may include any non-transitory semiconductor memory (e.g., the storage device 2502), such as ROM, SSD, PROM, EPROM, EEPROM, NVRAM, flash, nvSRAM, FeRAM, FeTRAM, MRAM, PRAM, CBRAM, SONOS, RRAM, NRAM, racetrack memory, FJG, and Millipede memory.

[0166] Further, the computer-readable storage medium may include any nonsemiconductor memory, such as optical disk storage, magnetic disk storage, magnetic tape, other magnetic storage devices, or any other medium capable of storing one or more instructions. In one or more embodiments, the tangible computer-readable storage medium may be directly coupled to a computing device, while in other embodiments, the tangible computer-readable storage medium may be indirectly coupled to a computing device, e.g., via one or more wired connections, one or more wireless connections, or any combination thereof.

[0167] Instructions may be directly executable or may be used to develop executable instructions. For example, instructions may be realized as executable or non-executable machine code or as instructions in a high-level language that may be compiled to produce executable or non-executable machine code. Further, instructions also may be realized as or may include data. Computer-executable instructions also may be organized in any format, including routines, subroutines, programs, data structures, objects, modules, applications, applets, functions, etc. As recognized by those of skill in the art, details including, but notlimited to, the number, structure, sequence, and organization of instructions may vary significantly without varying the underlying logic, function, processing, and output.

[0168] While the above discussion primarily refers to microprocessors or multi-core processors that execute software, one or more embodiments are performed by one or more integrated circuits, such as ASICs or FPGAs. In one or more embodiments, such integrated circuits execute instructions that are stored on the circuit itself.

[0169] The bus 2510 also connects to the input device interface 2506 and output device interface 2508. The input device interface 2506 enables the system to receive inputs. For example, the input device interface 2506 allows a user to communicate information and select commands on the electronic system 2500. The input device interface 2506 may be used with input devices such as keyboards, mice, dials, switches, sliders, and other interfaces (physical or virtual) for a user to supply information to the electronic system 2500. The output device interface 2508 may be used with output devices such as displays, speakers, and other interfaces (physical or virtual) for the computing electronic system 2500 to provide information. One or more embodiments may include devices that function as both input and output devices, such as a touchscreen.

[0170] The bus 2510 also couples the electronic system 2500 to one or more networks and / or to one or more network nodes through the network interface 2514. The network interface 2514 may include one or more interfaces that allow the electronic system 2500 to be a part of a network of computers (e.g., a local area network (LAN), a wide area network (WAN), or a network of networks (the Internet)). For example, the network interface 2514 may include a network interface card (NIC).

[0171] A network set up by an entity, such as a company or a public sector organization, to provide one or more web services (such as various types of cloud-based computing or storage) accessible via the Internet and / or other networks to a distributed set of clients may be termed a provider network. Such a provider network may include numerous data centers hosting various resource pools, such as collections of physical and / or virtualized computer servers, storage devices, networking equipment and the like, needed to implement and distribute the infrastructure and web services offered by the provider network. The resources may in some embodiments be offered to clients in various units related to the web service, such as an amount of storage capacity for storage, processing capability for processing, asinstances, as sets of related services and the like. A virtual computing instance may, for example, comprise one or more servers with a specified computational capacity (which may be specified by indicating the type and number of CPUs, the main memory size and so on) and a specified software stack (e.g., a particular version of an operating system, which may in turn run on top of a hypervisor).

[0172] A compute node, which may be referred to also as a computing node, may be implemented on a wide variety of computing environments, such as commodity-hardware computers, virtual machines, web services, computing clusters and computing appliances. Any of these computing devices or environments may, for convenience, be described as compute nodes.

[0173] A number of different types of computing devices may be used singly or in combination to implement the resources of the provider network in different embodiments, for example computer servers, storage devices, network devices and the like. In some embodiments a client or user may be provided direct access to a resource instance, e.g., by giving a user an administrator login and password. In other embodiments the provider network operator may allow clients to specify execution requirements for specified client applications and schedule execution of the applications on behalf of the client on execution platforms (such as application server instances, Java™ virtual machines (JVMs), general- purpose or special-purpose operating systems, platforms that support various interpreted or compiled programming languages such as Ruby, Perl, Python, C, C++ and the like or high- performance computing platforms) suitable for the applications, without, for example, requiring the client to access an instance or an execution platform directly. A given execution platform may utilize one or more resource instances in some embodiments; in other embodiments, multiple execution platforms may be mapped to a single resource instance.

[0174] In many environments, operators of provider networks that implement different types of virtualized computing, storage and / or other network-accessible functionality may allow customers to reserve or purchase access to resources in various resource acquisition modes. The computing resource provider may provide facilities for customers to select and launch the desired computing resources, deploy application components to the computing resources and maintain an application executing in the environment. In addition, the computing resource provider may provide further facilities for the customer to quickly and easily scale up or scale down the numbers and types of resources allocated to the application,either manually or through automatic scaling, as demand for or capacity requirements of the application change. The computing resources provided by the computing resource provider may be made available in discrete units, which may be referred to as instances. An instance may represent a physical server hardware platform, a virtual machine instance executing on a server or some combination of the two. Various types and configurations of instances may be made available, including different sizes of resources executing different operating systems (OS) and / or hypervisors, and with various installed software applications, runtimes, and the like. Instances may further be available in specific availability zones, representing a logical region, a fault tolerant region, a data center or other geographic location of the underlying computing hardware, for example. Instances may be copied within an availability zone or across availability zones to improve the redundancy of the instance, and instances may be migrated within a particular availability zone or across availability zones. As one example, the latency for client communications with a particular server in an availability zone may be less than the latency for client communications with a different server. As such, an instance may be migrated from the higher latency server to the lower latency server to improve the overall client experience.

[0175] In some embodiments the provider network may be organized into a plurality of geographical regions, and each region may include one or more availability zones. An availability zone (which may also be referred to as an availability container) in turn may comprise one or more distinct locations or data centers, configured in such a way that the resources in a given availability zone may be isolated or insulated from failures in other availability zones. That is, a failure in one availability zone may not be expected to result in a failure in any other availability zone. Thus, the availability container of a resource instance is intended to be independent of the availability container of a resource instance in a different availability zone. Clients may be able to protect their applications from failures at a single location by launching multiple application instances in respective availability zones. At the same time, in some embodiments inexpensive and low latency network connectivity may be provided between resource instances that reside within the same geographical region (and network transmissions between resources of the same availability zone may be even faster).

[0176] As set forth above, content may be provided by a content provider to one or more clients. The term content, as used herein, refers to any presentable information, and the term content item, as used herein, refers to any collection of any such presentable information. Acontent provider may, for example, provide one or more content providing services for providing content to clients. The content providing services may reside on one or more servers. The content providing services may be scalable to meet the demands of one or more customers and may increase or decrease in capability based on the number and type of incoming client requests. Portions of content providing services may also be migrated to be placed in positions of reduced latency with requesting clients. For example, the content provider may determine an “edge” of a system or network associated with content providing services that is physically and / or logically closest to a particular client. The content provider may then, for example, “spin-up,” migrate resources or otherwise employ components associated with the determined edge for interacting with the particular client. Such an edge determination process may, in some cases, provide an efficient technique for identifying and employing components that are well suited to interact with a particular client, and may, in some embodiments, reduce the latency for communications between a content provider and one or more clients.

[0177] As used in this specification and any claims of this application, the terms “base station,” “receiver,” “computer,” “server,” “processor,” and “memory” all refer to electronic or other technological devices. These terms exclude people or groups of people. For the purposes of the specification, the terms “display” or “displaying” means displaying on an electronic device.

[0178] The predicate words “configured to,” “operable to,” and “programmed to” do not imply any particular tangible or intangible modification of a subject but, rather, are intended to be used interchangeably. In one or more embodiments, a processor configured to monitor and control an operation or a component may also mean the processor being programmed to monitor and control the operation or the processor being operable to monitor and control the operation. Likewise, a processor configured to execute code may be construed as a processor programmed to execute code or operable to execute code.

[0179] Phrases such as an aspect, the aspect, another aspect, some aspects, one or more aspects, an implementation, the implementation, another implementation, some embodiments, one or more embodiments, an embodiment, the embodiment, another embodiment, some embodiments, one or more embodiments, a configuration, the configuration, another configuration, some configurations, one or more configurations, the subject technology, the disclosure, the present disclosure, other variations thereof and alikeare for convenience and do not imply that a disclosure relating to such phrase(s) is essential to the subject technology or that such disclosure applies to all configurations of the subject technology. A disclosure relating to such phrase(s) may apply to all configurations or one or more configurations. A disclosure relating to such phrase(s) may provide one or more examples. A phrase such as an aspect or some aspects may refer to one or more aspects and vice versa, which applies similarly to other foregoing phrases.

[0180] The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” or as an “example” is not necessarily to be construed as preferred or advantageous over other embodiments.Furthermore, to the extent that the term “include,” “have,” or the like is used in the description or the claims, such term is intended to be inclusive in a manner similar to the phrase “comprise” as “comprise” is interpreted when employed as a transitional word in a claim.

[0181] All structural and functional equivalents to the elements of the various aspects described throughout this disclosure that are known or later come to be known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the claims. Moreover, nothing disclosed herein is intended to be dedicated to the public, regardless of whether such disclosure is explicitly recited in the claims.

[0182] The previous description is provided to enable any person skilled in the art to practice the various aspects described herein. Various modifications to these aspects will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other aspects. Thus, the claims are not intended to be limited to the aspects shown herein but are to be accorded the full scope consistent with the language claims, wherein reference to an element in the singular is not intended to mean “one and only one” unless specifically so stated, but rather “one or more.” Unless specifically stated otherwise, the term “some” refers to one or more. Headings and subheadings, if any, are used for convenience only and do not limit the subject disclosure.

Claims

CLAIMSWhat is claimed is:

1. A quantum computing system, comprising: a first qubit; an oscillator; and a second qubit coupled to the first qubit via the oscillator, wherein the quantum computing system is configured to concurrently drive the first qubit and the second qubit at a frequency that corresponds to an oscillator frequency of the oscillator.

2. The quantum computing system of claim 1, wherein the quantum computing system is configured to drive the first qubit and the second qubit without directly driving the oscillator.

3. The quantum computing system of claim 2, wherein the first qubit, the oscillator, and the second qubit form a controlled phase gate when the first qubit and the second qubit are driven at the frequency.

4. The quantum computing system of claim 3, wherein the controlled phase gate comprises a cavity-mediated cross-resonance gate that forms a controlled-Z (CZ) gate when the first qubit and the second qubit are driven at the frequency, and wherein driving the first qubit and the second qubit at the frequency without directly driving the oscillator populates the oscillator with a number of excitations greater than 0.1.

5. The quantum computing system of claim 1, wherein driving the first qubit and the second qubit at the frequency comprises applying an electromagnetic pulse having the frequency concurrently to each of the first qubit and the second qubit.

6. The quantum computing system of claim 5, wherein applying the electromagnetic pulse having the frequency concurrently to each of the first qubit and the second qubit comprises applying the electromagnetic pulse concurrently to each of the first qubit and the second qubit for an amount of time that is inversely proportional to a square root of a product of a first ratio and a second ratio, the first ratio between a Rabi frequency ofthe first qubit and a detuning between a resonance frequency of the first qubit and the frequency that corresponds to the oscillator frequency, the second ratio between a Rabi frequency of the second qubit and a detuning between a resonance frequency of the second qubit and the frequency that corresponds to the oscillator frequency.

7. The quantum computing system of claim 5, wherein the qubits are superconducting qubits, and wherein applying the electromagnetic pulse having the oscillator frequency concurrently to each of the first qubit and the second qubit comprises applying the electromagnetic pulse concurrently to each of the first qubit and the second qubit for an amount of time that is based on an anharmonicity of the first qubit and an anharmonicity of the second qubit.

8. The quantum computing system of claim 1, wherein the quantum computing system is configured to mitigate an effect of a quantum Stark shift of one or both of the first qubit or the second qubit by driving the first qubit and the second qubit for an amount of time that is proportional to a first integer and inversely proportional to a detuning between the frequency and the oscillator frequency.

9. The quantum computing system of claim 8, wherein the amount of time is further proportional to a second integer and inversely proportional to a first expansion coefficient of a Hamiltonian of the quantum Stark shift, and proportional to a third integer and inversely proportional to a second expansion coefficient of the Hamiltonian of the quantum Stark shift.

10. The quantum computing system of claim 5, wherein the quantum computing system is configured to mitigate an effect of a quantum Stark shift of one or both of the first qubit or the second qubit by, in addition to applying the electromagnetic pulse having the frequency concurrently to each of the first qubit and the second qubit for a first amount of time, applying, during the first amount of time, at least four pairs of additional pulses, each of the at least four pairs of additional pulses applied for a second amount of time shorter than the first amount of time.

11. The quantum computing system of claim 10, wherein each of the additional pulses comprises a pi-pulse corresponding to a phase rotation of a corresponding qubit by one hundred eighty degrees.

12. The quantum computing system of claim 11, further comprising an additional oscillator between the first qubit and the second qubit.

13. The quantum computing system of claim 11, further comprising an additional oscillator coupled to the second qubit.

14. The quantum computing system of claim 11, further comprising a third qubit, wherein applying, during the first amount of time, the at least four pairs of additional pulses mitigates an always-on ZZ interaction between at least the second qubit and the third qubit.

15. The quantum computing system of claim 11, further comprising a plurality of additional qubits and an array of coupled additional cavities, wherein at least a subset of the coupled additional cavities is coupled to at least one of the additional qubits.

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