EP4677491A1

EP4677491A1 - Methods and apparatus for quantum processing units

Info

Publication number: EP4677491A1
Application number: EP23926560.6A
Authority: EP
Inventors: Jiaying YANG; Per Persson; Simone GASPARINETTI; Axel Martin ERIKSSON
Original assignee: Telefonaktiebolaget LM Ericsson AB
Current assignee: Telefonaktiebolaget LM Ericsson AB
Priority date: 2023-03-06
Filing date: 2023-03-06
Publication date: 2026-01-14
Also published as: WO2024186239A1; EP4677491A4; CN120826687A

Abstract

Methods and apparatus for a quantum processing unit, QPU, include a method in a QPU comprising first and second qubits (QB1, QB2), parametric coupling circuitry coupling the first and second qubits (QB1, QB2), and a waveguide (200) coupled to the second qubit (QB2). The method comprises exciting (310) the first qubit (Q1) to an arbitrary superposition of a ground state and a first excited state, generating (320) an AC flux signal using quadrature signals I(t) = A(t) cos(— 2πθ(t)) and Q(t) = A(t) sin(— 2πθ(t)), and parametrically driving (330) the coupling circuitry with the AC flux signal, to transfer the quantum state of the first qubit to the second qubit and emit a photon encoded with the quantum state into the waveguide. Waveform A(t) shapes the photon emitted into the waveguide (200) and θ(t) is a time-varying phase profile that adjusts the photon's phase so that its propagating mode is time-symmetric with respect to amplitude and phase.

Description

METHODS AND APPARATUS FOR QUANTUM PROCESSING UNITS

TECHNICAL FIELD

The present disclosure is generally related to quantum processing, and is more particularly related to methods and apparatus for coupling qubits to a waveguide for transmitting quantum information.

BACKGROUND

A distributed quantum computing system requires that information can be shared between spatially separated quantum processing units (QPUs). This requires the creation of a quantum communication channel, which can be implemented, for example, by emitting a microwave photon from a sender processor into a microwave waveguide and absorbing it, at the other end of the waveguide, in a receiver processor. For this purpose, deterministically encoding the state of a stationary qubit into a travelling photon so that the quantum state can be reliability transferred to the receiver is of great interest.

The transfer of quantum state via microwave photons transmitted through a microwave waveguide has been described, for example, by P. Kurpiers et al., "Deterministic quantum state transfer and remote entanglement using microwave photons," Nature 558, 264 (2018). In this example, first and second quantum nodes are coupled, respectively, to each end of a coaxial waveguide. Each quantum node includes a superconducting transmon qubit, coupled to one microwave resonator configured for qubit readout and another resonator configured for excitation transfer of photons to/from the coaxial waveguide. At one node, a cavity-assisted Raman process is used to transfer the qubit state of the transmon to a time-symmetric microwave photon emitted into the coaxial waveguide. The reverse process is then used to absorb the photon at the other node, resulting in a transfer of the qubit's quantum state through the quantum channel to a receiver.

Improved structures and techniques for transferring quantum states through a channel are needed.

SUMMARY

An object of the presently disclosed invention is to provide an apparatus and corresponding method for transferring quantum information through a quantum channel, e.g., a waveguide, with high fidelity of the resulting state transfer. This object is achieved with a superconducting circuit comprising one data qubit (QB1) and one emitter qubit (QB2) coupled to the quantum channel, with a coupler between them to parametrically exchange the quantum state. The deterministic transfer of a qubit state into a propagating microwave photon with a single-rail-encoding is thus realized with a simpler approach, based on a transition between two qubits that resembles an iSWAP-like operation. A time-dependent parametric drive is used to shape the temporal profile of the propagating mode to be time-symmetric, so that the absorption process by a receiving node can be implemented as a time-reversed version of the emission, allowing for a high-fidelity transfer of the state of the first qubit through the channel.

To make the photon absorption process a time-reverse process of photon emission, the photon profile is reshaped from exponentially decaying into time-symmetric, by controlling the parametric drive of the coupler to make its modulation amplitude dependent on time. However, because of the parametric drive to the coupler, there is a dispersive shift of the qubit Q.B1 and qubit Q.B2 frequencies, resulting in the phase of the emitted photon being dependent on the modulation amplitude of the parametric drive.

To address this problem, a self-calibrating system that can automatically compensate the varying phase of the emitted photon is introduced. The photon (uncompensated) is measured, and its varying phase is characterized. The photon is emitted again after compensating the varying phase with both quadratures of a drive signal for producing the AC flux signal applied to the coupler between the qubits. This phase compensation protocol is independent on the information in the emitted photon and the system therefore only needs to be calibrated once. The whole calibration can be automatically realized, after which the emitted photon after the phase compensation has a constant or nearly constant phase.

An example embodiment of the disclosed techniques and devices is an apparatus for use in a Q.PU, the apparatus comprising first and second qubits and parametric coupling circuitry coupling the first qubit to the second qubit. The apparatus further comprises driver circuitry connected to the parametric coupling circuitry and configured to drive the parametric coupling circuitry with DC flux and an AC flux signal. The drive circuitry is configured to generate the AC flux signal using quadrature drive signals /(t) and Q(t), according to /(t) = (t) cos(— 2?r0(t)) and <?(t) = (t) sin(— 2?r0(t)), where (t) is a waveform for shaping a photon emitted from the second qubit into a waveguide coupled to the second qubit and where 0(t) is a time-varying phase profile that adjusts the phase of the emitted photon across the photon's temporal profile.

Another example embodiment of the disclosed techniques is a method corresponding to the apparatus summarized above. This example method is transmitting quantum information from a Q.PU including apparatus like that described above, i.e., comprising first and second qubits, parametric coupling circuitry coupling the first qubit to the second qubit, and a waveguide interface connected to the second qubit, for connection to a first end of a waveguide. The example method comprises exciting the first qubit to an arbitrary superposition of a ground state and a first excited state, generating an AC flux signal for driving the parametric coupling circuitry, and parametrically driving the parametric coupling circuitry with DC flux and the generated AC flux signal, to transfer the quantum state of the first qubit to the second qubit, for emission of a photon encoded with the quantum state into the waveguide. The generating of the AC flux signal uses quadrature drive signals /(t) and (t) according to /(t) = (t) cos(— 2?r0(t)) and <?(t) = (t) sin(— 2?r0(t)), where (t) is a waveform for shaping a photon emitted from the second qubit into the waveguide and where 0(t) is a time-varying phase profile that adjusts the phase of the emitted photon across the photon's temporal profile.

Applied properly, these techniques and apparatus result in that the emitted photon is measured with a flat phase, increasing the fidelity of the procedure for transferring the quantum state of a qubit through a quantum channel.

Further details and variations of the above-summarized embodiments are provided below.

BRIEF DESCRIPTION OF THE FIGURES

Figure 1 is a schematic illustration of an example apparatus for use in a quantum processing unit (QPU).

Figure 2A illustrates an example apparatus, according to some embodiments. Figure 2B is a microphotograph illustrating a portion of the apparatus of Figure 2A.

Figure 3 is a process flow diagram illustrating aspects of example methods, according to various embodiments of the presently disclosed techniques.

Figure 4 is an energy level diagram of the quantum system.

Figure 5A and Figure 5B illustrate temporal profiles of an emitted microwave photon, before and after compensation.

Figure 6A and Figure 6B illustrate an example coupler drive signal, for emitting uncompensated and compensated photons, respectively.

Figure 7 illustrates the reconstructed density matrices for two quantum states, using quantum state tomography, for photons compensated according to the techniques described herein.

Figure 8 is a table illustrating improvements in fidelity for each of the quantum states represented in Figure 7. Figure 9 is a schematic diagram illustrating details of an example apparatus according to some embodiments.

DETAILED DESCRIPTION

In the discussion that follows, aspects of the invention are described with respect to a specific implementation of a distributed quantum computing system that includes quantum nodes comprising qubits in the form of superconducting transmons and circuits configured to transfer the quantum state of a qubit from one node to another via microwave photons propagated through a microwave waveguide. It should be understood, however, that the inventive techniques are not limited to circuits employing superconducting transmons or to any particular type of waveguide. Note that the term "waveguide" is used herein in a broad sense, i.e., to refer a physical structure that guides electromagnetic waves by restricting the transmission of energy to an intended direction along the waveguide. This includes coaxial cables, optical fibers, coplanar waveguides, and microwave waveguides in the form of hollow or dielectrically loaded tubes.

The inventive techniques can be demonstrated in the context of a superconducting circuit consisting of a first qubit QB1, which may be referred to as a data qubit, a second qubit QB2, which may be referred to as an emitter qubit and which is coupled to a quantum channel, and a coupler 110 between the qubits to parametrically exchange the quantum state. This structure is illustrated in Figure 1, which shows qubit QB1, which can be regarded as a data qubit that can be set to an arbitrary quantum, and qubit QB2, to which the state of qubit QB1 may be transferred, for emission into waveguide 120 of a microwave photon encoded with the quantum state of qubit QB2. This structure can thus be used to realize the deterministic transfer of a qubit state into a propagating microwave photon with a single-rail-encoding. A time-dependent parametric drive provides a resonant exchange interaction between QB1 and QB2, in a manner similar to an iSWAP operation, and shapes the temporal profile of the propagating mode to be time-symmetric, so that the absorption process by a receiving node can be implemented as a time-reversed version of the emission.

Figure 2A and Figure 2B illustrate physical details of an example device implementing the schematic of Figure 1. Figure 2A illustrates two superconducting transmon bits, qubit QB1 and qubit QB2, each capacitively coupled to a parametric coupler 200 arranged between them. A flux line 210 is inductively coupled to a superconducting quantum interference device (SQUID) loop 220 of the coupler 200, at the position indicated by the bold arrow in Figure 2A, enabling the frequency tunability of the coupler 200. A close-up view of the SQUID loop 220 is shown in Figure 2B. Qubit QB2 is strongly coupled to a coplanar waveguide 230, extending from the right-hand side of Figure 2A, and is designed and fabricated to have a decay rate r of 2TT 8M HZ, SO that once qubit Q.B2 is populated, there will be a microwave photon emitted into the waveguide 230.

To make the photon absorption process at the other end of waveguide 230 a time-reverse process of the photon emission, the photon profile, i.e., the envelope of the photon's waveform, is reshaped from exponentially decaying into a time-symmetric profile. This is done by controlling the parametric drive of the coupler 200 to make its modulation amplitude dependent on time. Shaping of microwave photons is described in Pechal, Marek, "Microwave photonics in superconducting circuits," Diss. ETH Zurich, 2016. However, a problem that arises from such temporal shaping of the photon profile is that the amplitude changes over time of the parametric drive to the coupler 200 cause a dispersive shift of the frequencies of qubit Q.B1 and qubit Q.B2, which in turn results in the phase of the emitted photon varying over time, in dependency on the modulation amplitude of the parametric drive. Thus, instead of the emitted photon having a flat phase, it has a varying phase, which reduces the fidelity of the transfer procedure.

To improve the fidelity of the transfer, it is desired that the photon have a constant phase, in addition to a time-symmetric temporal profile.

Figure 3 is a process flow diagram illustrating, in general terms, the steps for transferring quantum information from a superconducting qubit to a propagating mode in a waveguide, with the emitted photon having a desired temporal profile. Details provided below explain how the apparatus may be driven so that the phase of the photon across its temporal profile is corrected, to be constant or nearly constant. The process is described below in the context of the superconducting transmon structure discussed above, in connection with Figures 1 and 2, but it will be appreciated that the techniques may be more generally applicable.

As shown at block 310, the method includes the step of preparing qubit Q.B1 in an arbitrary superposition state of the ground state | g) and the first excited state | e), i.e., a | g) + | e). As shown at block 320, an AC flux signal is generated for driving the parametric coupler 200; details of this AC flux signal will be provided below.

As shown at block 330, the coupler is parametrically driven, with a DC flux and the AC flux signal, to exchange the population from qubit Q.B1 to qubit Q.B2. This may be seen as similar to the operation of an iSWAP gate for an isolated two-qubit system. However, this operation differs from that of an iSWAP gate in that Q.B2 does not maintain the state, but immediately emits the microwave photon into the waveguide 230. Put another way, qubit Q.B2 is never actually populated - it instead acts to mediate the decay of qubit Q.B1 into the waveguide. The parametric flux modulation applied to the coupler activates a resonant exchange interaction between qubit QB1 and QB2; this interaction is easier to implement than the second-order Ig) <-> |f) transition used in previous systems to control the emission of the photon.

An energy level diagram for the system is shown in Figure 4, where the preparation step from block 310 of Figure 3 is shown at transition 1 in Figure 4 and the iSWAP-like operation described above is shown at transition 2.

Because qubit QB2 is strongly coupled to the waveguide 230 , it immediately decays into the ground state while emitting a microwave photon into the waveguide. This is shown at transition 3 in Figure 4. The emitted photon is in the same superposition state in which qubit QB1 was prepared, a 10) + P 1 1). As a result, the state of qubit QI is transferred to a propagating mode of the waveguide, while qubit QI is left in its ground state. In this step, the emitted photon is also shaped to be time- symmetric and compensated to address the varying phase.

The propagating mode is characterized by temporally matching the profile of the emitted photon. Quantum state tomography may be performed to reconstruct the density matrix of the mode, e.g., to evaluate the fidelity of the state transfer.

Referring back to block 330 of Figure 3, the coupler is parametrically driven with both the DC and AC flux tones: (t) = (PDC + A(t)cos(comt). (1)

The strength of the coupling rate J between the two qubits is tunable by (PDC and the amplitude of A(t), (PAC. (PDC is the DC flux sent to the coupler, which tunes the coupler frequency closer to the qubits to make the coupling rate J larger. The AC tone A(t)cos(com t) is for exchanging the excitation from qubit QB1 to qubit QB2. This AC tone is a sinusoidal pulse with frequency com := | col - co21 , the detuning between the dressed frequencies col and co2 of the qubits, and is applied to the coupler flux line after the state preparation pulse sent to QB1. The AC flux tone has an envelope A(t), which varies much more slowly than com.

Since qubit QB2 is coupled to the waveguide 230, it decays into the ground state while emitting a microwave photon into the waveguide 230. The emitted photon may be read out by measuring the averaged amplitude of the output field vs time, (aout(t)).

As noted above, to facilitate photon reabsorption at the receiver it is desirable that the emitted photon has a time-symmetric envelope. In this case, a receiver identical to the emitter can reabsorb the photon with high efficiency when subjected to a time-reversed control sequence. The photon may be shaped into a temporal profile having the expression l/cosh(t/i), by controlling the coupling rate J between qubits, by defining the envelope A(t) as: where Teff is the effective decay rate. This may be defined, for example, at 0.25T. The emitted photon can be reshaped to have a symmetric temporal mode, as shown in the magnitude curve in Figure 5A, which shows the magnitude and phase of the emitted photon in the case where the photon is shaped, but no phase compensation is performed. It can be seen in Figure 5A that the phase varies considerably across the temporal profile of the shaped photon.

The symmetry s of the photon field (aout(t)) may be calculated according to: (flout. (0/ \ .^dt ■=■<//,

(3) where (aout(t)) contains both real and imaginary part. In this example, the symmetry of the shaped photon before phase compensation is 61.7%, because of the varying phase seen in Figure 5A.

However, to make the photon absorption process a time-reversed process of photon emission, not only should the magnitude profile of the emitted photon be time-symmetric, but the phase of the emitted photon should additionally be constant. Below, the source of the varying phase and techniques for compensating for it are described.

The phase-varying problem is caused by the reshaping of the photon. When a flux AC tone with a constant envelope is applied to the coupler, the time-averaged frequency of the coupler, coc ((PAC), decreases as the flux tone amplitude (PAC increases. This is due to the non-linear dependence between the coupler frequency and the amplitude of the flux applied to it. Consequently, the frequencies of both qubits are slightly pushed down, due to their capacitive coupling to the coupler. Hence, when applying a modulation pulse to the coupler, the frequency of the emitted photon will depend on the amplitude of the coupler drive co2((PAC) = col ((PAC) + com, where col and co2 are the frequencies of qubit Q.B1 and qubit Q.B2, respectively. As long as the coupler amplitude (PAC is constant, so is the frequency (or phase) of the emitted photon. However, when shaping a photon, a flux AC tone with a slowly varying time-dependent envelope A(t) is applied to the coupler, e.g., as in Equation 2. This leads to a time-varying push in frequency on the Q.B1. As a consequence, the phase of the emitted photon has a corresponding time-variation, defined as 0(t) = arg[(aout(t))]. This time-varying phase is seen in Figure 5A.

To compensate for the varying phase 0(t) corresponding to a specific coupler AC tone envelope A(t), 0(t) is first characterized, by measuring the emitted photon resulting from the drive with A(t) using characterization circuitry connected to the waveguide and configured to characterize the amplitude and frequency of the photon emitted into the waveguide. This is shown at block 340 of Figure 3. Importantly, the phase variation is insensitive to which quantum state is emitted. Hence, once 0(t) is characterized for a given profile A(t), the resulting characterization can be used to compensate the varying phase for all quantum states, to high accuracy. Note that any constant shift of the average coupler frequency due to the profile A(t) translates into a corresponding shift of the emitted frequency and can be trivially adjusted for.

Once the phase variation has been characterized, a signal representing this characterized phase can be provided to the drive circuitry, to compensate for and flatten the phase. This is shown at block 350 of Figure 3. To compensate for the varying phase, the drive signal to the coupler is split into two quadratures, on each of which the reversed varying phase is played, according to:

The resulting drive signal can be seen in Figure 6B. Figure 6A shows the signal A(t) as applied to the coupler for shaping the photon, but without phase compensation. Figure 6B, on the other hand, shows the application of the modulation signal in quadrature parts l(t) and Q(t), per Equation (4), to realize the phase compensation.

This protocol simplifies the calibration and operation process of the distributed communication system, but still corrects the varying phase to a high extent. And, this protocol can be realized automatically.

After applying this phase compensation process, a constant or nearly constant phase may be achieved for the emitted photon while keeping its envelope unchanged, as shown in the example illustrated in Figure 5B. The calculated symmetry of the emitted photon shown in Figures 5A and 5B increases from 61.7% to 98.1% after compensating for the varying phase. The approach described here was tested by performing a comparison between the two cases with and without phase compensation. Quantum state tomography was performed for states 10) + 11) and 11), under each of the two cases. Figure 7 shows the resulting reconstructed density matrices for states 10) + 11) and 11), where the bars show the measured result and the frame shows the expected result, under the two cases: (a) shaped photon, but without phase compensation, and (b) shaped photon and with phase compensation. By initially preparing qubit QB1 in states | g) + | e) and | e), exchanging the states to qubit QB2 and operating quantum state tomography to the emitted photon, the reconstructed density matrices for the two states illustrated in Figure 7 were obtained.

The fidelity of the reconstructed density matrix (compared to the ideal case) is shown in Figure 8. It can be seen that, after phase compensation, there are fidelity improvements for both states | 0) + | l) and | 1).

Figure 9 is a schematic diagram of an example apparatus in which the above techniques may be applied. The apparatus includes a microwave transceiver part 910, and a superconducting circuit part 950. The superconducting circuit part 920 includes a qubit QB1, a qubit QB2, and a parametrically driven coupler 200 arranged between them. The parametric coupler 200 includes a SQUID device 220, for applying the flux to the coupler, arranged in Figure 9 at roughly the midpoint of coupler 200, but its details are not illustrated in this figure. A waveguide interface 955 couples qubit QB2 to waveguide 230.

The parametric coupler 200 is driven by a signal from arbitrary waveform generator (AWG) 915, which forms part of the microwave transceiver part 910. Likewise, qubit QB1 is excited to a desired superposition state with another signal from AWG 915. Microwave photons emitted into waveguide 230 are passed through microwave circulator 960 and microwave isolators 965, as well as appropriate filters and amplifiers, and supplied to vector analyzer 920 and analog-to-digital converter 925, for measurement and digitization. The digitized characterization of the varying phase 0(t) for an uncompensated photon, for a given envelope profile A(t), can then be supplied to AWG 915, for use in generating the quadrature drive signals l(t) and Q(t) for generating the AC flux signal applied to the parametric coupler 200. Note that the three outputs from the AWG 915 and the two inputs to the ADC 925 each have two ports, corresponding to the in-phase (I) and quadrature (Q) components of the respective signals, but only one port is illustrated in the figure, for simplicity.

It will be appreciated that the apparatus shown in Figure 9 is but one example of an apparatus in which the techniques and concepts described herein may be employed - in this case utilizing two superconducting transmons with a parametric coupler between them. (The term "parametric coupler" refers to a tunable coupler, i.e., a coupling circuit or structure where the coupling can be adjusted in a controlled fashion, e.g., by application of a flux as in examples described herein.) The techniques may be applied more generally, with structures where the coupling strength between two quantum processing components, such as qubits, resonators, etc., changes as a function of a time-varying waveform driving the coupling. Thus, some embodiments of the presently disclosed invention can be considered as comprising first and second quantum processing components, the second quantum processing component being configured to emit quantum-encoded particles into a waveguide coupled to the second quantum processing component, and parametric coupling circuitry coupling the first quantum processing component to the second quantum processing component. In the examples detailed herein, these first and second quantum processing components are first and second superconducting transmon qubits, but other embodiments might comprise a resonator and a qubit, two realizations of quantum-encoded photons, one or more other sorts of qubits such as fluxonium-based qubits, etc.

This more generalized apparatus further comprises driver circuitry connected to the parametric coupling circuitry and configured to drive the parametric coupling circuitry with a time-varying waveform configured to vary the coupling between the first and second quantum processing components. The drive circuitry is configured to generate the time-varying waveform using quadrature drive signals /(t) and Q(t), according to /(t) = (t) cos(— 2?r0(t)) and <?(t) = (t) sin(— 2?r0(t)), where (t) is a waveform for shaping the amplitude of a quantum-encoded particle emitted from the second quantum processing component into the waveguide, and where 0(t) is a time-varying phase profile that adjusts the phase of the emitted particle across its temporal profile.

With this arrangement, the waveform (t) can be configured to shape the amplitude of the emitted particle into a time-symmetric temporal profile. The phase change 0(t) across this temporal profile induced by a non-linear response of the parametric coupler to the waveform (t) can be characterized, by receiving and measuring an emitted particle, and then used in the driver circuitry as shown above, to reverse these effects and produce emitted particles having time-symmetric profiles with respect to both amplitude and phase. As discussed above, this improves the fidelity of quantum information transfer through the waveguide.

In the apparatuses and techniques based on superconducting transmons described above, the photon profile is reshaped from exponentially decaying into time-symmetric, so that the photon absorption process in a receiver can be a time-reverse process of the photon emission process, by controlling the parametric drive of the coupler to make its modulation amplitude dependent on time. As explained above, however, because of the parametric drive to the coupler there is a dispersive shift of the qubit QB1 and qubit QB2 frequencies, resulting in the phase of the emitted photon being dependent on the modulation amplitude of the parametric drive.

To address this problem, a self-calibrating system that can automatically compensate the varying phase of the emitted photon was described. According to this process, a photon (uncompensated) is measured for a first time, and its varying phase is characterized. The photon is emitted again after compensating the varying phase with both quadratures of a drive signal for producing the AC flux signal applied to the coupler between the qubits. This phase compensation protocol is independent on the information in the emitted photon and is the system therefore only need to be calibrated once. The whole calibration can be automatically realized, after which the emitted photon after the phase compensation has a constant or nearly constant phase.

At the heart of the disclosed techniques and apparatus is the use of two conjugating quadratures to prepare the time-dependent coupler drive, instead of using a single quadrature. This allows the varying phase to be effectively encoded into the customized coupler drive, keeping the frequency of the drive constant over time. The entire phase compensation process can be realized automatically, solving the problem of the varying phase and increase the fidelity of quantum state transfer of the emitted photon.

ABBREVIATIONS

Abbreviation Explanation

QPU Quantum processing unit

QB1 The first qubit, data qubit

QB2 The second qubit, emitter qubit lg) Ground state of the qubit le) First excited state of the qubit

10) Vacuum state of the emitted photon

11) Fock state 1 of the emitted photon REFERENCES

1. P. Kurpiers, P. Magnard, T. Walter, B. Royer, M. Pechal, J. Heinsoo, Y. Salath'e, A. Akin, S. Storz, J.- C. Besse, et al., "Deterministic quantum state transfer and remote entanglement using microwave photons," Nature 558, 264 (2018). 2. Pechal, Marek, "Microwave photonics in superconducting circuits," Diss. ETH Zurich, 2016.

3. M. Ganzhorn, G. Salis, D. Egger, A. Fuhrer, M. Mergenthaler, C. M uller, P. M uller, S. Paredes, M. Pechal, M. Werninghaus, et al., "Benchmarking the noise sensitivity of different parametric two- qubit gates in a single superconducting quantum computing platform," Physical Review Research 2, 033447 (2020).

Claims

CLAIMS What is claimed is:

1. An apparatus for use in a quantum processing unit (Q.PU), the apparatus comprising: a first qubit (Q.B1) and a second qubit (Q.B2); parametric coupling circuitry (230) coupling the first qubit to the second qubit; driver circuitry connected to the parametric coupling circuitry and configured to drive the parametric coupling circuitry (230) with DC flux and an AC flux signal, wherein the drive circuitry is configured to generate the AC flux signal using quadrature drive signals /(t) and Q(t), according to /(t) = (t) cos(— 2?r0(t)) and <?(t) = (t) sin(— 2?r0(t)), where (t) is a waveform for shaping a photon emitted from the second qubit into a waveguide (200) coupled to the second qubit, and where 0(t) is a time-varying phase profile that adjusts the phase of the emitted photon across the emitted photon's temporal profile.

2. The apparatus of claim 1, wherein the first qubit (Q.B1) and second qubit (Q.B2) are superconducting transmons.

3. The apparatus of claim 1 or 2, wherein the AC flux signal is based on a frequency selected to be approximately equal to or equal to a difference between frequencies of the first and second qubits.

4. The apparatus of any one of claims 1-3, further comprising: a waveguide (200) coupled to the second qubit; and characterization circuitry connected to the waveguide (200), the characterization circuitry being configured to receive photons emitted into the waveguide and to measure phase variation across the temporal profile of a received photon; and circuitry configured to provide a signal characterizing the measured phase variation to the driver circuitry for use in generating the AC flux tone.

5. The apparatus of any one of claims 1-4, wherein (t) is shaped to cause the photon emitted from the second qubit into the waveguide to have an envelope with a symmetric temporal profile.

6. A method for transmitting quantum information from a quantum processing unit (Q.PU) that comprises first and second qubits, parametric coupling circuitry coupling the first qubit to the second qubit, and a waveguide coupled to the second qubit, the method comprising: exciting (310) the first qubit to an arbitrary superposition of a ground state and a first excited state; generating (320) an AC flux signal for driving the parametric coupling circuitry, using quadrature drive signals /(t) and <?(t) according to /(t) = (t) cos(— 2?r0(t)) and <2 (t) = A(t) sin(— 2?r0(t)), where (t) is a waveform for shaping a photon emitted from the second qubit into the waveguide and where 0(t) is a time-varying phase profile that adjusts the phase of the emitted photon across the emitted photon's temporal profile; and parametrically driving (330) the parametric coupling circuitry with DC flux and the generated AC flux signal, to transfer the quantum state of the first qubit to the second qubit, for emission of a photon encoded with the quantum state into the waveguide.

7. The method of claim 6, wherein the first and second qubits are superconducting transmons.

8. The method of claim 6 or 7, wherein the AC flux signal is based on a frequency selected to be approximately equal to or equal to a difference between frequencies of the first and second qubits.

9. The method of any one of claims 6-8, the method further comprising: in characterization circuitry connected to the waveguide, receiving a photon emitted into the waveguide and measuring phase variation across the temporal profile of the received photon; and providing a signal characterizing the measured phase variation for use in generating the AC flux signal with the quadrature drive signals.

10. The method of any one of claims 6-9, wherein (t) is shaped to cause the photon emitted from the second qubit into the waveguide to have an envelope with a symmetric temporal profile.