Method of operating a parametric amplifier, method of operating a qubit, qubit system

Abstract

The present disclosure relates to methods of operating a parametric amplifier. In one arrangement, a parametric amplifier is provided that comprises a nanostructure suspended between an input electrode and an output electrode. The nanostructure is spaced apart from a control electrode and configured such that a vibration of the nanostructure varies a distance between at least a portion of the nanostructure and the control electrode. A single-electron transistor is formed that comprises the nanostructure as a conducting island of the single-electron transistor. An oscillatory input signal is applied to the input electrode. An oscillatory control signal is applied to the control electrode. The control signal comprises a DC component that defines a potential of the island and an electrostatic tensioning of the nanostructure and an oscillatory pump component that provides an amplified version of the input signal in an output signal provided at the output electrode.

Description

[0001] METHOD OF OPERATING A PARAMETRIC AMPLIFIER, METHOD OF OPERATING A QUBIT, QUBIT SYSTEM

[0002] The present disclosure relates to methods of operating a parametric amplifier, methods of operating a qubit, and a qubit system. The disclosure is particularly applicable to operating the parametric amplifier in a cryogenic regime and to spin qubits.

[0003] Parametric amplifiers have had a significant impact on quantum technologies, especially on superconducting qubits as they can allow for near quantum-limited amplification for robust measurement of the quantum state. Such parametric amplifiers are typically based on superconducting elements such as Josephson junctions, which makes them inherently susceptible to magnetic fields. It can therefore be challenging to use these amplifiers for spin qubits which require externally applied magnetic fields.

[0004] A parametric amplifier is implemented as a mixer. A strong pump signal at twice the resonance frequency of the oscillator is used to change the spring constant of the resonator. A weaker signal, at the resonance frequency, is then added to the system and that weaker signal experiences parametric amplification. In essence, the idea of the parametric amplifier is to transfer energy from a “strong” signal to the “weak” signal of interest.

[0005] Suspended carbon nanotubes combine electrical and mechanical degrees of freedom. Several parametric amplifiers using carbon nanotubes have been developed, as described in the following three publications:

[0006] B J Aleman et al. “A Carbon Nanostructure-based NEMS Parametric Amplifier for Enhanced Radio Wave Detection and Electronic Signal Amplification”. In: Journal of Physics: Conference Series 302 (July 2011), p. 012001,

[0007] Alexander Eichler et al. “Parametric Amplification and Self-Oscillation in a Nanostructure Mechanical Resonator”. In: Nano Letters 11.7 (May 2011), pp. 2699-2703, and

[0008] Chung-Chiang Wu and Zhaohui Zhong. “Parametric amplification in single-walled carbon nanostructure nanoelectromechanical resonators”. In: Applied Physics Letters 99.8 (Aug. 2011), p. 083110. All three publications describe arrangements operating at temperatures above 100 K. This is intentional as at lower temperatures (e.g., < 1 K), quantum dots (i.e., where a single electron is confined) form in the carbon nanotube. Operating at these high temperatures leads to high dissipation within the carbon nanotube. Non-linear damping limits the parametric gain. The parameter that is changed for parametric amplification in these implementations is the spring constant of the carbon nanostructure, which is determined by the electrostatic tension. The electrostatic tension can be changed by changing a voltage applied to a gate electrode. To achieve a higher gain, a higher change in gate voltage is required. This requires a larger signal, which may cause unwanted heat dissipation.

[0009] It is an object of the disclosure to provide an alternative and / or improved parametric amplifier or method of operating a parametric amplifier.

[0010] According to an aspect of the invention, there is provided a method of operating a parametric amplifier, wherein: the parametric amplifier comprises a nanostructure suspended between an input electrode and an output electrode, the nanostructure being spaced apart from a control electrode and configured such that a vibration of the nanostructure varies a distance between at least a portion of the nanostructure and the control electrode; and the method comprises: forming a single-electron transistor comprising the nanostructure as a conducting island of the single-electron transistor; applying an oscillatory input signal to the input electrode; and applying an oscillatory control signal to the control electrode, the control signal comprising a DC component that defines a potential of the island and an electrostatic tensioning of the nanostructure and an oscillatory pump component that provides an amplified version of the input signal in an output signal provided at the output electrode.

[0011] Thus, a method is provided in which a nanostructure-based parametric amplifier is used in a low temperature regime in which a conducting island of a single-electron transistor is formed by the nanostructure. The island may comprise a quantum dot. The nanostructure may be a nanotube, optionally a carbon nanotube. The quantum dot modifies the electrical conductance of the nanostructure and electron tunnelling is coupled to the mechanical motion. The inventors have found that while these effects complicate implementation of the parametric amplifier they can be leveraged to achieve higher gains at cryogenic temperatures. High gains make it possible to operate the amplifier with smaller changes in a control signal (including a pump signal) than would be required at higher temperatures where a single-electron transistor (and / or quantum dot) is not formed. This makes the method more compatible with use in low temperature environments, such as in dilution fridges or other cryogenic devices.

[0012] Operating the amplifier at cryogenic temperatures also lowers the noise temperature. In contrast to alternative amplifiers that use superconducting components, the amplifier of the present disclosure can be operated in high magnetic fields. The amplifier can be placed in close proximity to components that need to be exposed to high magnetic fields such as spin qubits. Amplifiers of the present disclosure can, for example, be positioned in a cryostat in close proximity to a spin qubit to amplify signals from the spin qubit. This provides efficient amplification of the signals from spin qubits, without introducing excessive heat load (due to the high gain) and while benefiting from the lowest possible noise temperature.

[0013] The intrinsically nanoscale dimensions of the parametric amplifier advantageously provide a small footprint, further facilitating integration next to small devices such as qubit devices (e.g., spin qubits).

[0014] Extremely low dissipation can be achieved where the nanostructures are carbon nanotubes. Carbon nanotubes can have low disorder leading to very high quality factors (up to 5 million), which means they have extremely low dissipation.

[0015] In an embodiment, a variation as a function of the control signal of an electrical conductance through the nanostructure comprises a Coulomb peak. The DC component of the control signal may be selected to be within two standard deviations of the maximum of the Coulomb peak. Arranging for the DC component of the control signal to be near to the Coulomb peak ensures high gain.

[0016] In an embodiment, the DC component of the control signal and a magnitude of the oscillatory pump component of the control signal are such that the control signal remains exclusively on one side of the maximum of the Coulomb peak. This approach ensures that gain is positive for the whole cycle of the oscillatory pump component, thereby promoting high average gain and / or easier interpretation of the output signal from the method. In an embodiment, the DC component of the control signal and a magnitude of the oscillatory pump component of the control signal are such that the oscillations of the oscillatory pump component pass through, and are optionally substantially centred on, a steepest portion of the Coulomb peak. This approach promotes high average gain.

[0017] According to an aspect of the invention, there is provided a qubit system, comprising: a qubit configured to generate a qubit output; a magnetic field controller configured to control the qubit using a magnetic field; and a parametric amplifier configured to amplify the qubit output, the parametric amplifier comprising: a nanostructure; an input electrode; an output electrode; and a control electrode, wherein: the nanostructure is suspended between the input electrode and the output electrode; the nanostructure is spaced apart from the control electrode and configured such that a vibration of the nanostructure varies a distance between at least a portion of the nanostructure and the control electrode; and the parametric amplifier is configured to: apply the qubit output as an oscillatory input signal to the input electrode; and apply an oscillatory control signal to the control electrode, the control signal comprising a DC component to define a potential of a conducting island of a single-electron transistor formed by the nanostructure and an electrostatic tensioning of the nanostructure and an oscillatory pump component to provide an amplified version of the qubit output in an output signal provided at the output electrode.

[0018] Embodiments of the disclosure will be further described by way of example only with reference to the accompanying drawings.

[0019] Figure 1 is a schematic side sectional view of an example parametric amplifier showing example inputs to an input electrode, an output electrode, and a control electrode.

[0020] Figure 2 is a magnified view of the amplifier of Figure 1 showing how mechanical motion of a nanostructure of the amplifier changes a capacitance between the nanostructure and a control electrode.

[0021] Figure 3 is a graph depicting a Coulomb peak in a variation of current as a function of a voltage applied to the control electrode (gate voltage).

[0022] Figure 4 is a flow chart depicting example components of a method of operating a parametric amplifier.

[0023] Figure 5 schematically depicts an example qubit system. The present disclosure provides a method of operating a parametric amplifier. The method involves use of a parametric amplifier 10 in which a single-electron transistor is formed at cryogenic temperatures. The single-electron transistor may comprise a nanostructure as a conducting island of the transistor. The island may comprise a quantum dot. Example configurations and operation principles of the parametric amplifier 10 are described below with reference to Figures 1-3. An outline of the method is depicted in the flow chart of Figure 4. The parametric amplifier 10 is particularly useful in the context of amplifying signals from qubits that are controlled by magnetic fields, such as spin qubits, where alternative approaches relying on superconducting components cannot be used or require complex magnetic shielding.

[0024] Figure 1 depicts an example parametric amplifier 10. The parametric amplifier 10 comprises a nanostructure 2. The nanostructure 2 may comprise a nanotube. The nanotube may be a carbon nanotube. Any reference herein to a nanostructure may be understood to encompass the case where the nanostructure 2 is a carbon nanotube.

[0025] The parametric amplifier 10 comprises an input electrode 4, an output electrode 6, and a control electrode 8. The nanostructure 2 is suspended between the input electrode 4 and the output electrode 6. Thus, a portion of the nanostructure 2 is in contact with the input electrode 4 at an input attachment portion and is mechanically constrained by the input electrode 4 (e.g., by being rigidly attached thereto). A portion of the nanostructure 2 at the opposite end is in contact with the output electrode 6 at an output attachment portion and is mechanically constrained by the output electrode 6 (e.g., by being rigidly attached thereto). A portion of the nanostructure 2 between the input attachment portion and the output attachment portion, which may be referred to as a suspended segment, is not directly attached to any other component and / or or otherwise mechanically constrained. The suspended segment may have a range of lengths. The suspended segment may, for example, have a length in the range of lOOnm to 2000nm, optionally in the region of about lOOOnm. The portion of the nanostructure 2 between the input attachment portion and the output attachment portion is able to vibrate. Two example positions of such vibration are depicted in Figures 1 and 2 by the two depicted instances of the nanostructure 2. The convex instance of the nanostructure 2 represents a form of the nanostructure at a time when the nanostructure is relatively far from the control electrode 8 and the concave instance of the nanostructure 2 represents a form of the nanostructure 2 at a different time when the nanostructure 2 is relatively close to the control electrode 8. Double ended arrow 5 schematically indicates up and down movement of a central portion of the nanostructure 2 during the vibration. The nanostructure 2 is thus spaced apart from the control electrode 8 and configured such that a vibration of the nanostructure 2 varies a distance between at least a portion of the nanostructure 2 and the control electrode 8. The distance between the nanostructure 2 and the control electrode 8 may be referred to as a z distance.

[0026] In step SI of the method of operating a parametric amplifier 10 of Figure 4, the method may comprise forming a single-electron transistor. The single-electron transistor may comprise the nanostructure 2 as a conducting island of the transistor. The island may comprise a quantum dot. The forming of the single-electron transistor may comprise cooling the nanostructure 2 to a sufficiently low temperature. The forming of the singleelectron transistor (and / or quantum dot) may, for example, comprise cooling the nanostructure 2 to a temperature below about IK, optionally below about 500mK, optionally below about 200mK, optionally below lOOmK.

[0027] In steps S2-S4 of the method, an input signal Vinis applied to the input electrode 4 (step S2), a control signal is applied to the control electrode 8 (step S3), and an output signal Voutis received at the output electrode 6 (step S4).

[0028] The combination of the suspended nanostructure 2, input electrode 4, output electrode 6, and control electrode 8 may be configured to operate as a transistor. The input electrode 4 and output electrode 6 may thus be referred to as a source electrode and a drain electrode. The control electrode may be referred to as a gate electrode. The transistor may be a single-electron transistor.

[0029] Referring to Figure 1, the control signal may comprise a DC component, VG. The DC component of the control signal may be referred to as a gate voltage. The DC component of the control signal defines a potential of the island (e.g., quantum dot). The DC component therefore controls an electrical degree of freedom of the nanostructure 2. The DC component also defines an electrostatic tensioning of the nanostructure. The DC component therefore also controls a mechanical degree of freedom of the nanostructure 2. The control signal also comprises an oscillatory pump component, Vpump. The pump component is added to the DC component. The resulting time-varying control signal may be referred t

[0030] The input signal is desirably arranged to have a dominant (e.g., sole) oscillatory component at a frequency that is substantially equal to a mechanical resonance frequency of the nanostructure 2. The input signal may, for example take the form Vin= Vincos( jL>mt), where Vinis the amplitude of the input signal, a>mis the frequency of the input signal, and t is time. The input signal may be a signal of interest to be amplified.

[0031] The pump component of the control signal is desirably arranged to have a dominant oscillatory component at a frequency that is substantially twice the mechanical resonance frequency of the nanostructure 2. The pump component may, for example, take the form Vpump=Vpumpcos (i)m^ + ), where Vpumpis the amplitude of the pump component and is a phase difference between the pump component and the input signal.

[0032] Applying the input signal and the control signal results in an output signal of the form Vout= Voutcos(amt), where Voutis the amplitude of the output signal.

[0033] As depicted in Figure 2, mechanical motion of the nanostructure 2 changes a capacitance between the nanostructure 2 and the control electrode 8. The island of the single-electron transistor (e.g., quantum dot) formed in the nanostructure 2 causes single electron tunnelling. As depicted in Figure 3, a variation as a function of the control signal ^(t) of the electrical conductance (and therefore current IDCthrough the nanostructure 2) comprises a Coulomb peak. The DC component V^cis desirably arranged to be near to the Coulomb peak. The DC component VQCmay, for example, be selected so as to be within two standard deviations of the maximum of the Coulomb peak. In an example implementation, the DC component VQCis selected to be substantially equal to a voltage at which the gradient of the Coulomb peak is highest. The DC component V^cof the control signal and a magnitude of the oscillatory pump component t^ump(t) of the control signal may be arranged to be such that the control signal remains exclusively on one side of the maximum of the Coulomb peak. This is depicted schematically in Figure 3. The approach ensures that gain is positive for the whole cycle o ), thereby promoting high average gain and / or easier interpretation of the output signal from the method. To maximise gain, the DC component VGcof the control signal and a magnitude of the oscillatory pump component of the control signal may be selected such that the oscillations of the oscillatory pump component pass through, and are optionally substantially centred on, a steepest portion of the Coulomb peak. The DC component VGcalso controls the frequency of the mechanical resonance due to electrostatic tensioning. The frequency of the mechanical resonance can be coarsely tuned in this way to a desired frequency regime and a suitable Coulomb peak in that regime should then be used to implement the method.

[0034] The mechanical motion of the nanostructure 2 results in a change of capacitance cGbetween the control electrode 8 and the nanostructure 2. For small enough oscillations, this change in capacitance cGis approximately equivalent to applying an effective oscillating voltage with the same frequency as the mechanical motion. The oscillating pump component Vpumpof the control signal can therefore be used to modulate the energy level of the island of the single-electron transistor (e.g., quantum dot) causing an oscillating current / (t) through the nanostructure 2, as depicted schematically in Figure 3. Thus, mechanical oscillations provide capacitance oscillations, the capacitance oscillations provide gate voltage oscillations, the gate voltage oscillations provide island (e.g., quantum dot) energy oscillations, and the island (e.g., quantum dot) energy oscillations provide oscillating current through the nanostructure.

[0035] A more detailed explanation of the transduction of mechanical motion is given below, with the following two assumptions being made.

[0036] Firstly, it is assumed that the conductance is only a function of the charge q is on the nanostructure 2. Therefore, any instantaneous value of control signal VGyields a specific instantaneous value of current, i.e. q(t) -> ^(t) -> / (t).

[0037] Secondly, it is assumed that the electron transport is much faster than the mechanical motion. This is a good approximation for carbon nanostructure devices fabricated using typical techniques.

[0038] Adopting the above assumptions, the conductance for any particular instance of time can be written as follows:

[0039] The instantaneous charge can be approximated as:

[0040] The first term represents a change in charge due to electrostatic effects. The second term represents a change in charge due to the mechanical oscillations. For small enough changes, the first term can be neglected. q = CGVQG(3) cG' is the derivative of cGwith respect to z. Ignoring the first term as it is the electrostatic contribution and substituting this in equation (1):

[0041] In terms of the parametric amplification, <5z(t) has the same frequency as that of the control signal (gate drive) while FSDis the input signal (also referred to as Vin) that is to be amplified. Note that FSDand Vtrefer to the same thing and are used interchangeably within the context of the present disclosure. This leads to mixing of the two signals, with the resulting signal having frequency components at both the sum of and the difference between the two input frequencies.

[0042] As an illustration, a simple case of how the parametric amplification works is now described. Consider the equation of a simple harmonic oscillator: where z is the displacement, ft is the damping, m is the renormalized frequency and E(t) is the external driving force.

[0043] Let the damping be ? = and driving force be E(t) = £’osin(rnmt). We add a parametric drive h0sin(2rnmt) that changes the spring constant:

[0044] An approximate solution of this equation is

[0045] As hQincreases, the response z(t) increases, which causes the parametric amplification of the input signal £'0cos(mmt).

[0046] The gain of the parametric amplifier may be written as:

[0047] Here, (p is the phase difference between the pump signal and the input signal, VPis the magnitude of pump signal and VP Cis the critical pump voltage above which the amplification is expected to diverge due to instability. It can be seen that maximum gain is attained when

[0048] In this expression, VPchanges the spring constant. In Eichler et al., mentioned above, this is achieved by a pump signal connected to the gate which causes the spring constant to change due to electrostatic tensioning of the carbon nanotube. This is the most common method and similar techniques are used in other carbon nanotube-based parametric amplifiers.

[0049] However, when a carbon nanotube is cooled down to milli-Kelvin temperatures, an island of a single-electron transistor (e.g., quantum dot) is formed. The electron tunnelling through the island (e.g., quantum dot) is strongly coupled to the mechanical motion. This coupling causes a rapid change in the spring constant of the carbon nanotube (a rapid change as a function of gate voltage or control signal applied at the control electrode), i.e. a small change in gate voltage (control signal voltage) causes a big change in the spring constant (due to the back-action of single electron tunnelling on the mechanical oscillation). This can be used for parametric pumping. The gain for this parametric amplifier in comparison to earlier works can be written as: where a is a gain in change of spring constant with respect to the gate voltage (control signal) due to the coupling between electron tunnelling and mechanical motion. This also makes it possible to use a smaller pump signal. As the pump signal needs to be applied to the device at cryogenic temperatures, this is especially helpful as at low temperatures where thermal budgets are restricted and any extra heat load (such as from the pump signal) is undesirable. Additionally, since smaller pump signals are required, the gains are not limited due to non-linear damping, which is also lower at cryogenic temperatures.

[0050] This mechanism works at the edges of the Coulomb peaks. This is fortunate, as the nanostructure 2 is conducting at the edge of the Coulomb peaks and equation (8) is valid d G

[0051] (outside Coulomb peaks = 0 and there is no signal).

[0052] Thus, a nanostructure such as a carbon nanotube can be used as the basis for a parametric amplifier, with the parametric aspect coming from modulation of the spring constant of the nanostructure via a control signal (gate drive) from a control electrode (gate electrode). The control signal is desirably at 2rnm, where a>mis the resonance frequency of the nanostructure. An input signal Vincan then be amplified at the resonance frequency

[0053] Due to the lack of any magnetic field sensitive materials in this system, this amplifier can be used for cryogenic applications that require magnetic fields, such as spin qubits.

[0054] The presence of coupling between the island (e.g., quantum dot) and the mechanical motion is used for the parametric excitation.

[0055] Carbon nanotubes can have low disorder leading to very high quality factors (up to 5 million). This means they have extremely low dissipation.

[0056] The nanoscale dimensions of the device allow a small footprint and facilitates integration next to small devices such as spin qubit devices.

[0057] The non-linear modulation leads to parametric self-oscillations. Applications

[0058] A parametric amplifier 10 according to any of the embodiments of the disclosure described above may be used in a method of operating a qubit. The parametric amplifier 10 may be particularly advantageously used in the context of a qubit that is controlled using a magnetic field, such as a spin qubit. This is because alternative approaches relying on superconducting components may be disrupted by the presence of the magnetic field or may require complex magnetic shielding. A method may, for example, be provided in which a qubit output from the qubit is provided as the input signal to the parametric amplifier 10 and the parametric amplifier 10 provides an amplified version of the qubit output as the output signal from the parametric amplifier 10.

[0059] Figure 5 schematically depicts an example qubit system 30. The system 30 comprises a qubit 14 configured to generate a qubit output 15. The qubit 14 may comprise a spin qubit. The system 30 comprises a magnetic field controller 16 configured to control the qubit 14 using a magnetic field. The system 30 comprises a parametric amplifier 10 configured to amplify the qubit output 15. The parametric amplifier 10 may take any of the forms described above with reference to Figures 1-4. The parametric amplifier 10 may be configured to apply the qubit output 15 as an input signal to the input electrode 4 of the parametric amplifier 10. The parametric amplifier 10 may be configured to apply a control signal to the control electrode 8. The control signal may comprise a DC component and an oscillatory pump component, as described above. The output 17 from the output electrode 6 may comprise an amplified version of the qubit output 15. The parametric amplifier 10 and qubit 14 may optionally be integrated together on the same chip 20.

[0060] Cross reference to related applications

[0061] This application claims priority from GB2400221.4 filed on 8 January 2024, the contents of which are hereby incorporated by reference.

Claims

CLAIMS1. A method of operating a parametric amplifier, wherein: the parametric amplifier comprises a nanostructure suspended between an input electrode and an output electrode, the nanostructure being spaced apart from a control electrode and configured such that a vibration of the nanostructure varies a distance between at least a portion of the nanostructure and the control electrode; and the method comprises: forming a single-electron transistor comprising the nanostructure as a conducting island of the single-electron transistor; applying an oscillatory input signal to the input electrode; and applying an oscillatory control signal to the control electrode, the control signal comprising a DC component that defines a potential of the island and an electrostatic tensioning of the nanostructure and an oscillatory pump component that provides an amplified version of the input signal in an output signal provided at the output electrode.

2. The method of claim 1, wherein the island comprises a quantum dot.

3. The method of claim 1 or 2, wherein the nanostructure is a carbon nanotube.

4. The method of any preceding claim, wherein the input signal and pump component have dominant frequencies substantially equal to a mechanical resonance frequency of the nanostructure.

5. The method of claim 4, comprising adjusting the mechanical resonance frequency of the nanostructure by adjusting a magnitude of the DC component of the control signal.

6. The method of any preceding claim, wherein the pump component of the control signal has a dominant oscillatory component at a frequency that is substantially equal totwice a mechanical resonance frequency of the nanostructure.

7. The method of any preceding claim, wherein the input signal is of the form Vincos( jL>mt), where Vinis the amplitude of the input signal, a>mis the frequency of the input signal, and t is time.

8. The method of claim 7, wherein the output signal is of the formoutcos (rnmt), where Voutis the amplitude of the output signal.

9. The method of claim 7 or 8, wherein the pump component is of the form l^umpcos(2rnmt + ), where Vpumpis the amplitude of the pump component and is a phase difference between the pump component and the input signal.

10. The method of any preceding claim, where the forming of the single-electron transistor comprises cooling the nanostructure to a temperature below IK.

11. The method of any preceding claim, wherein a variation as a function of the control signal of an electrical conductance through the nanostructure comprises a Coulomb peak.

12. The method of claim 11, wherein the DC component of the control signal is within two standard deviations of the maximum of the Coulomb peak.

13. The method of claim 11 or 12, wherein the DC component of the control signal and a magnitude of the oscillatory pump component of the control signal are such that the control signal remains exclusively on one side of the maximum of the Coulomb peak.

14. The method of any of claims 11 to 13, wherein the DC component of the control signal and a magnitude of the oscillatory pump component of the control signal are such that the oscillations of the oscillatory pump component pass through, and are optionally substantially centred on, a steepest portion of the Coulomb peak.

15. A method of operating a qubit, comprising: controlling the qubit using a magnetic field; and using the method of any preceding claim to operate a parametric amplifier, wherein a qubit output from the qubit is provided as the input signal to the parametric amplifier and the parametric amplifier provides an amplified version of the qubit output as the output signal from the parametric amplifier.

16. The method of claim 15, wherein the qubit is a spin qubit.

17. A qubit system, comprising: a qubit configured to generate a qubit output; a magnetic field controller configured to control the qubit using a magnetic field; and a parametric amplifier configured to amplify the qubit output, the parametric amplifier comprising: a nanostructure; an input electrode; an output electrode; and a control electrode, wherein: the nanostructure is suspended between the input electrode and the output electrode; the nanostructure is spaced apart from the control electrode and configured such that a vibration of the nanostructure varies a distance between at least a portion of the nanostructure and the control electrode; and the parametric amplifier is configured to: apply the qubit output as an oscillatory input signal to the input electrode; and apply an oscillatory control signal to the control electrode, the control signal comprising a DC component to define a potential of aconducting island of a single-electron transistor formed by the nanostructure and an electrostatic tensioning of the nanostructure and an oscillatory pump component to provide an amplified version of the qubit output in an output signal provided at the output electrode.

18. The qubit system of claim 17, wherein the qubit is a spin qubit.

19. The qubit system of claim 17 or 18, wherein the island comprises a quantum dot.

20. The qubit system of any of claims 17 to 19, wherein the nanostructure is a carbon nanotube.

21. The qubit system of any of claims 17 to 20, wherein the input signal and pump component have dominant frequencies substantially equal to a mechanical resonance frequency of the nanostructure.

22. The qubit system of claim 21, wherein the parametric amplifier is configured to adjust the mechanical resonance frequency of the nanostructure by adjusting a magnitude of the DC component of the control signal.

23. The qubit system of any of claims 17 to 22, wherein the pump component of the control signal has a dominant oscillatory component at a frequency that is substantially equal to twice a mechanical resonance frequency of the nanostructure.

24. The qubit system of any of claims 17 to 23, wherein the input signal is of the form Vincos( jL>mt), where Vinis the amplitude of the input signal, a>mis the frequency of the input signal, and t is time.

25. The qubit system of claim 24, wherein the output signal is of the form Voutcos(rnmt), where Voutis the amplitude of the output signal.

26. The qubit system of claim 24 or 25, wherein the pump component is of the form l^umpcos(2rnmt + ), where Vpumpis the amplitude of the pump component and is a phase difference between the pump component and the input signal.

27. The qubit system of any of claims 17 to 26, wherein a variation as a function of the control signal of an electrical conductance through the nanostructure comprises a Coulomb peak.

28. The qubit system of claim 27, wherein the DC component of the control signal is within two standard deviations of the maximum of the Coulomb peak.

29. The qubit system of claim 27 or 28, wherein the DC component of the control signal and a magnitude of the oscillatory pump component of the control signal are such that the control signal remains exclusively on one side of the maximum of the Coulomb peak.

30. The qubit system of any of claims 27 to 29, wherein the DC component of the control signal and a magnitude of the oscillatory pump component of the control signal are such that the oscillations of the oscillatory pump component pass through, and are optionally substantially centred on, a steepest portion of the Coulomb peak.

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