A qubit processing method

Abstract

A method is disclosed for performing operations in a silicon chip qubit processor. The method involves providing a qubit at a first quantum dot, having a first resonant spin frequency and transferring the qubit to a second quantum dot, having a second resonant spin frequency. The method involves applying an AC electromagnetic field to the first and second quantum dots in order to manipulate the qubit spin state, wherein the applied AC electric field has a frequency which is calculated from a weighted average, derived from the first and second resonant spin frequencies. In another arrangement the method involves providing an enhanced tuneable range of g-factors by leveraging inter-dot tunnelling rather than Stark shift tuning. A variant of the method applies two-qubit operations on pairs of qubits that are shuttled along an array.

Description

[0001] A QUBIT PROCESSING METHOD

[0002] FIELD OF THE INVENTION

[0003] The present invention relates to qubit processing methods.

[0004] BACKGROUND TO THE INVENTION

[0005] Quantum computation involves the manipulation and processing of qubits. A qubit, or a quantum bit, is the quantum parallel to the classical “bit” used in classical computing, and contains information. There are a number of possible quantum computing schemes that can be used to process qubits.

[0006] Qubits are typically arranged in an array in a qubit processor. Examples of qubits include silicon spin qubits and trapped-ions qubits. Silicon spin qubits are considered to be one of the best candidates for building large-scale quantum computers because of their small footprint, long coherence times and fast operations. Because of their small footprint it is believed that millions of qubits could be placed on a single chip. In silicon and similar solid-state quantum computers, the qubits are associated with the states of electrons which are trapped within regions called quantum dots. In general the term “dot” can be used as shorthand for “quantum dot” as the physical site or structure associated with a qubit. One example of an implementation of a qubit processor is described in applicant’s co-pending application WO 2024 / 078949.

[0007] To perform a single qubit gate the spin of an electron can be controlled by providing an AC electromagnetic pulse at a frequency that is in resonance with the qubit spin frequency. This gives rise to Rabi oscillations for the two-level spin state of the qubit, and a particular rotation of the spin state can be achieved by setting the amplitude and the phase of the drive.

[0008] A number of different forms of two-qubit gate are possible in silicon; these include the exchange interaction which can give rise to root-SWAP and full-SWAP operations, the controlled-rotation which can give rise to CNOT operations, and the controlled-phase operation CZ. All these variants involve adjusting the so- called J coupling between the two qubits, typically by changing the electrical voltage on one or more of the electrodes that define the dot’s structures.

[0009] Successfully performing a one-qubit or two-qubit gate requires the proper choice of certain control parameters; such parameters include the frequency of any applied AC electromagnetic pulse or the duration and magnitude of any electrode voltage shift. The proper choice of parameters depends on the characteristics of the dots where the operation is to occur. In particular, the so-called g-factor associated with each dot, together with the precise magnetic field strength at that location, to determine the spin frequency of a qubit in that dot. This spin frequency in turn determines the proper control parameters for any operation involving that qubit.

[0010] Due to material variability within the processor the qubit spin frequencies and g-factors at individual dots are typically distinct from one another. This means that individual dots often need to be controlled with individual AC electromagnetic pulses at frequencies that are individually tuned. As the number of qubits on the chip increases this approach can become challenging in terms of providing individual wiring for AC control to individual gates. In addition, the use of many different frequencies can lead to undesirable cross-talk effects between the qubits in the device. There have been a number of attempts to mitigate this problem by attempting to provide global frequency control. These are attempts at providing a single AC electromagnetic pulse that can address multiple qubits simultaneously. Some frequency engineering techniques have been developed to control the resonant frequencies at individual quantum dots. These techniques often involve tuning the g-factors of individual dots using the Stark shift. Unfortunately the Stark shift only permits tuneability over a limited range of frequencies. This means that it may not always be possible to place all qubits in a desired narrow range of frequencies, also referred to as a frequency bin. In addition, in some scenarios it may be desirable to place different groups of qubits into different frequency bins so that they can be separately addressed. Stark shift tuning means that it is often possible only to achieve a small separation in frequency between qubits in different frequency bins. This can lead to cross-talk effects where the driving pulse intended to address qubits in one frequency bin undesirably also affects qubits in another frequency bin.

[0011] It is desirable to create improved qubit processing methods that can address some of these challenges.

[0012] SUMMARY OF THE INVENTION

[0013] According to an aspect of the present invention there is provided a method for performing operations in a silicon chip qubit processor, comprising the steps of: providing a first qubit at a first quantum dot, having a first g-factor, gi; transferring the first qubit to a second quantum dot, having a second g-factor, g2; and applying a control signal to the location of the first and second quantum dots to perform a gate operation, wherein the properties of the control signal are dependent on a weighted average of the first and second g-factors.

[0014] In this way, it is possible to apply a control signal that performs a gate operation to a charge carrier that has passed through multiple quantum dots, even though each quantum dot has a different g-factor. This is possible because of a realisation that a qubit moving between a cluster of dots acquires an effective g-factor that is equal to a weighted average of the g-factors of cluster. The inventors have recognised a practical application of this physical phenomenon in reducing the complexity in control signals in gate operations for spin qubits in silicon. Specifically, it is possible to implement single-qubit gates for qubits that are moved between two or more quantum dots with a single drive frequency, which is based on a weighted average of the g-factors of the quantum dots. Moreover, the same principle can apply to two-qubit gates.

[0015] In one example, the step of applying a control signal involves applying an AC electromagnetic field to the first and second quantum dots in order to manipulate the first qubit spin state. The applied AC electric field preferably has a frequency which is related to the weighted average of the first and second g-factors.

[0016] The frequency acquired by the first qubit is preferably dependent on the resonant frequencies of individual dots (i.e. their g-factors) and the amount of time that the qubit spends at each dot. In a simple case, a qubit spends an equal amount of time at each dot. In this situation, the weighted average may be the mean of the first and second resonant spin frequencies, or the mean of all of the resonant spin frequencies in a cluster of dots. In other scenarios the calculation of the weighted average may be more complex; for example, if a qubit spends a first time, T, at a first quantum dot and a second time, 2T, at a second quantum dot then the weighted average may be calculated as twice the resonant frequency of the second quantum dot plus the resonant frequency of the first quantum dot, divided by three. Extending this principle, it is possible to adjust the effective frequency, i.e. the weighted average, to any value between the frequency of the first dot and the frequency of the second. The same logic applies to the effective g-factor experienced by the first qubit.

[0017] There are several possible mechanisms that can be used for transferring the first qubit from the first quantum dot to the second quantum dot. In one method the confining electric field defining two adjacent dots, which is generated by proximal electrodes, can be adjusted so as to change the probability distribution of the confined electron bearing the qubit. If electrode adjustment, which is sometimes referred to in the art as ‘detuning’, results in a localisation that favours a dot with a given resonant frequency then that frequency will be dominant in the effective frequency of the qubit. Conversely if the electrode adjustment is such that the electron is delocalised uniformly over the double dot then the effective frequency will correspond to the average of the resonant frequencies of the two dots. In this mechanism the qubit remains stationary once the adjustment is complete; in other words, its wavefunction is static.

[0018] Alternatively, the qubit may be transferred back and forth between dots, or between successive dots in a sequence, while a gate operation is performed. One way in which this dynamical process can be performed is swapping. Another method involves shuttling. Both swapping and shuttling are special cases that can be achieved through appropriate control of electric fields. The rate at which qubits move from one location to another may be referred to as a transfer frequency. In both processes the transfer rate at which a qubit moves from one dot to another is controlled so that it is fast in comparison to the duration of the gate operation that is effected by the control signal. Fora single qubit gate, the transfer frequency may be at least three times the frequency of the applied AC electromagnetic field, preferably at least ten times the frequency of the applied AC electromagnetic field.

[0019] Qubit swapping provides one technique for allowing a qubit to experience different g-factor environments. This can be achieved by turning on an exchange interaction between adjacent qubits. This results in an exchange interaction in which the qubits swap their states at an exchange frequency, which may also be described as a transfer frequency. There may be relatively fast oscillations at the transfer frequency and relatively slow oscillations at the frequency of the applied AC electromagnetic field. This can allow the applied AC electromagnetic field to manipulate the qubit spin state as the qubit moves between the first and second gates.

[0020] When implementing qubit swapping it is inevitable that qubit will spend an equal amount of time at the first and second quantum dots. Therefore, the effective resonant frequency of the qubit may be calculated simply as the mean of the resonant frequencies of the first and second quantum dots.

[0021] The qubit may be transferred to the second quantum dot by shuttling at a transfer frequency. Shuttling provides a technique for transferring qubits between dots without turning on exchange interactions. As will be appreciated by a person skilled in the art, shuttling involves a manipulation of the electrical potential well in which a qubit is confined in order to move it from dot to another, or alternatively by displacing the dots themselves.

[0022] Shuttling allows qubits to be moved between a large number of potential locations. In some arrangements qubits may be shuttled around a cluster of gates that may comprise a large number of locations.

[0023] The qubit may be transferred from the second quantum dot to a third quantum dot having a third resonant spin frequency by shuttling. The applied AC electromagnetic field may have a frequency having a component which is a weighted average, derived from the first, second and third resonant spin frequencies. Further shuttling steps may be performed for moving the qubit to fourth, fifth and subsequent (i-th) quantum dots.

[0024] The qubit processor may be a 2xN array. The qubits may be shuttled along a path between dots in the array. In one embodiment a first group of qubits may be shuttled synchronously with a second group of qubits and different Stark shift tuning may be applied to the two groups in order to create resonant frequencies for the two grounds that are shifted with respect to one another.

[0025] The transfer frequency may be at least three times the frequency of the applied AC field, preferably tens or hundreds of times the frequency of the applied AC electromagnetic field. In the case of swapping, the transfer frequency is preferably at least ten times the frequency of the applied AC electromagnetic field and in the case of shuttling, at least three times. This means that the gate operation is performed slowly relative to the rate of information transfer. In this way, a gate operation can be applied to a qubit that acquires an effective resonant frequency, based on the g-factors of quantum dots that it visits while a driving pulse is applied.

[0026] The first quantum dot has a first g-factor, gi, the second quantum dot has a second g-factor, g2, and the difference between gi and g2is Agi2. Due to the transfer methods by swapping or shuttling, the effective resonant frequency for a qubit moving between these dots may be adjustable over a tuneable range of Agi2between gi and g2.

[0027] In some arrangements, a micromagnet or another magnetic field influencer such as superconducting wires may be positioned in the vicinity of the second quantum dot. The micromagnet can affect the DC magnetic field gradient which, in turn, affects the g-factor and the resonant spin frequency of a qubit at the location of the second dot. This can be used to increase the g-factor dispersion for the qubit between the first and second gates. This can advantageously increase the tuneable range of the g-factor. The method may further comprise the steps of providing a second qubit at a third quantum dot, having a third g-factor, g3; transferring the second qubit to a fourth quantum dot, having a fourth g-factor, g4; and applying control signals to the locations of the first, second, third and fourth quantum dots to perform a two-qubit gate operation, wherein the properties of the control signals are dependent on a weighted average of the first and second g-factors and on a weighted average of the third and fourth g-factors. In this way, the method may implement a two-qubit gate where the control signal properties are dependent on the g-factors of the first, second, third and fourth quantum dots.

[0028] According to another aspect of the invention there is provided a method fortuning the resonant frequency of a spin qubit in a silicon chip qubit processor, comprising the steps of: providing a first qubit at a first quantum dot, having a first g-factor, gi and a second quantum dot, having a second g-factor, g2wherein the difference between gi and g2is Agi2; and applying an electrode adjustment to the first and second quantum dots in order to delocalise the first qubit between the first and second quantum dots, to tune its effective g-factor over a tuneable range of Agi2between gi and g2, thereby to tune the resonant frequency of the first qubit.

[0029] Stark shift tuning is a well-known technique for adjusting the tuneable range of a qubit. However, the Stark shift is a weak effect. In many situations it provides an inadequate tuneable range to separate resonant spin frequencies of chosen qubits and avoid undesirable cross-talk effects. The present method provides a technique for increasing the tuneable range of a qubit. This is achieved by moving the spin qubit between different quantum dots in a qubit processor and allowing the qubit to experience different g-factor environments. The present inventors have realised that this can be leveraged to provide a significantly larger g-factor tunability.

[0030] The first qubit at the first gate may have a Stark shift tuneable range of 5g, which is typically ten times smaller than Agi2. The average dispersion of g-factors in a silicon chip qubit processor using current technology is roughly one order of magnitude higher than the tuneable g-factor range of an individual qubit by Stark shift. The method may further comprise providing a second qubit at a third quantum dot, having a third g-factor, g3; transferring the second qubit to a fourth quantum dot, having a fourth g-factor, g4wherein the difference between g3and g4is Ag34; and applying a DC electric field to the third and fourth quantum dots in order to use the Stark shift to tune the effective g-factor of the second qubit over a tuneable range of Ag34between g3and g4, thereby to tune the resonant frequency of the second qubit so that it is different to the resonant frequency of the first qubit.

[0031] The resonant frequencies of the first and second charge carrier spin qubits may be tuned to frequencies that are separated by at least 5MHz, preferably at least 10MHz, and more preferably at least 20MHz. This technique can provide two qubits that are provided in different frequency bins. The frequency bins are preferably separated from one another by a large enough frequency gap to allow them to be individually addressed by AC electromagnetic fields at different frequencies without introducing cross-talk effects. In theory, a large number of qubits can be provided in respective frequency bins which can allow first and second groups of qubits that have resonant frequencies that are sufficiently separated from one another.

[0032] The step of applying an electrode adjustment may, in one example, comprise shuttling the first qubit from the first quantum dot to the second quantum dot. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Embodiments of the invention will now be described with reference to the accompanying drawings in which:

[0034] Figure 1 is a schematic example of the present invention based on shuttling qubits between dots;

[0035] Figure 2A is a schematic example of the present invention with a 2xN qubit processor architecture;

[0036] Figure 2B is a schematic example of the arrangement of Figure 2A when an adjustment is applied;

[0037] Figure 3 is a flow diagram illustrating a method of operation in an embodiment of the invention;

[0038] Figure 4 is example of the present invention based on swapping qubits between a pair of quantum dots;

[0039] Figure 5 is a flow diagram illustrating another method of operation in an embodiment of the invention;

[0040] Figure 6 is another schematic diagram showing an operation of the present invention that involves shuttling qubits around a cluster of quantum dots;

[0041] Figure 7 is a flow diagram illustrating yet another method of operation in an embodiment of the invention;

[0042] Figures 8A-8F are schematic diagrams showing an operation of the present invention that involves shuttling qubits in a 2x6 array over six time steps; and Figure 9 is a schematic diagram showing how operations can be performed for two-qubit gates while the qubits are shuttled along an array.

[0043] DETAILED DESCRIPTION

[0044] As explained in the background section, it is known to use the Stark shift in order to tune the resonant frequency of individual qubits. This can be achieved by applying an electric potential to a gate electrode to slightly shift the g-factors and therefore the frequencies of chosen qubits. One can use this effect to separate target qubits from non-target qubits in the frequency spectrum. One can then drive the target qubits by applying a global AC electromagnetic field pulse at a frequency that is in resonance with these qubits only. A resulting source of infidelity is the marginal driving of non-target qubits by off-resonant pulses. The Stark shift alone provides a low level of tuneability. The g-factor of an individual spin qubit in silicon can be tuned by 5g, and in some scenarios 5g = 0.01% of the qubits’ average g-factor.

[0045] One aspect of the present disclosure is a method for increasing the separation between qubits in the frequency space.

[0046] Figure 1 schematically illustrates a first quantum dot 2 and a second quantum dot 4 positioned respectively at first and second voltage gates in a silicon spin qubit processor. At time TO (not shown) a qubit is initialised. At time T1 the qubit is confined at the location of the first quantum dot 2. The second quantum dot 4 is empty at time T 1. The present method involves shuttling the qubit from the first quantum dot 2 to the second quantum dot 4. This is achieved by manipulating the electrical potentials that are used to confine the charge carrier using gate electrodes and barrier electrodes, in a manner that would be understood by a person skilled in the art. As an alternative, the electrodes defining the first and second quantum dots 2, 4 can be adjusted so that the qubit’s wavefunction is redistributed between the quantum dots. Thus, qubits can either be moved entirely from one quantum 2 to another quantum dot 4, or their wavefunctions can be re-distributed between the dots. As a preliminary step the g-factor of each of the quantum dots 2, 4 is determined. This is achieved by initialising a qubit in the |0> state, providing an AC electromagnetic field, scanning it across a range of frequencies and detecting a resonant response from the relevant spin qubit by measuring its state, more specifically, this involves typically by measuring the probability that a qubit is transferred from the |0> to the |1> state. This reveals the resonant frequency of the qubit at each quantum dot and the associated g-factor. This information is stored in a classical data storage unit for each of the voltage gates in the processor.

[0047] In this schematic example, the g-factor of the first quantum dot 2 is gi and the g-factor of the second quantum dot 4 is g2. The normal tuneable range of each individual quantum dot is 5g, which may be only around 1MHz, or around 0.01% of the qubits’ average g-factor. The average dispersion of g-factors in the qubit processor is Ag, and in general Ag is significantly greater than 5g. In this case, for the sake of simplicity, gi -g2= Ag, which is the average dispersion of g-factors. The value gi-g2can be calculated trivially based on the relevant data entries in the classical data storage unit. In this case, gi -g2may be around 30MHz.

[0048] The present method can involve adjusting the electrodes defining the first quantum dot 2 and the second quantum dot 4. I n this way the qubit’s wavefunction can be redistributed between the first quantum dot 2 and the second quantum dot 4. The qubit can therefore become delocalised and experiences the g-factor at the location of both quantum dots. If it is equally delocalised between dots then it experiences an average g-factor which is the mean of the values gi and g2. A partial delocalisation that favours one of the dots, will correspondingly lead to an effective resonant frequency that favours the frequency of that dot. Thus the qubit also acquires a broader tuneable g-factor because the difference between gi and g2is roughly one order of magnitude greater than 5g which is the normal tuneable range of each individual quantum dot enabled by Stark shift.

[0049] The qubit, which is delocalised between the first and second quantum dots 2, 4, can therefore have its g-factor tuned by applying a DC electric field adjustment to the gate electrodes at the location of the dots. Unlike conventional Stark shift tuning, the tuning range is significantly enhanced. This is achieved because the method makes use of an inter-dot effect rather than the conventional intra-dot Stark shift. This effect can be utilised in a qubit processor to tune the frequencies of respective groups of qubits into different frequency bins. These frequency bins can be sufficiently spaced from one another in frequency that cross-talk effects are minimised. In other words, the driving pulse that is applied to one group of qubits can effectively manipulate the spin of those qubits without undesirably also manipulating the spin of a second group of qubits having a displaced resonant frequency.

[0050] Figure 2A is another schematic diagram showing an implementation of this technique that uses a 2xN qubit processor architecture. In this design, at least two quantum dots are used per electron. Each electron can be (partially) moved between a first row 6 and a second row 8 by applying an electrode adjustment. Figure 2B is another schematic diagram showing the effect of applying an example electrode adjustment. In this way, the g-factor of each qubit can vary by an average value of Ag instead of 5g. In this example, a 2x6 array is illustrated. A first group of qubits Q1 , Q4, Q5, Q6 are tuned to a first resonant frequency F1 by adjusting the electrodes to cause a suitable delocalisation for each qubit i.e. each pair of dots. A second group of qubits Q2, Q3 are tuned to a second resonant frequency F2 again by suitably adjusting the electric potential at the relevant gate electrodes. The broader tuneability of the frequencies allows the first and second resonant frequencies F1 , F2 to be significantly displaced from one another. This means that the first group of qubits can be manipulated using an AC electromagnetic pulse at frequency F1 without inadvertently manipulating the spin states of the second group of qubits.

[0051] In one alternative scenario the g-factors of the qubits of the first row 6 are all higher than the g-factors of the qubits of the second row 8. This special distribution can be engineered via the inclusion of a micromagnet next to the array or by using magnetic materials in the gate stack on one side of the array. This design is more technically complex but has advantages because all target qubits can be attributed the same frequency by placing them somewhere near the bottom row. This situation is thus equivalent to creating two large clusters of qubits, reducing the frequency spectrum to a single target frequency and a single non-target frequency. If the g-factor difference between the top and bottom row is sufficient, off-resonant effects are practically vanishing, and X gates can perfectly be implemented on all target qubits in parallel.

[0052] Figure 3 is a flow chart illustrating the steps that can be performed in implementing this method for tuning the resonant frequencies of silicon spin qubit in the 2xN qubit processor illustrated in Figure 2.

[0053] At step S102 six qubits Q1-Q6 are provided at respective quantum dots in the first row 6 having respective g-factors gi-6(which are measured in advance). Correspondingly they will have resonant frequencies fi-6. At steps S104 and S106 the qubits Q1-Q6 are delocalised between the first six quantum dots and quantum dots in the second row with respective g-factors g7-i4 (which are also measured in advance). Each pairing of quantum dots has a g-factor dispersion, Agi-6. In this example, at step S108, the first group of qubits Q1, Q4, Q5, Q6 are tuned to the first resonant frequency F1 and the second group of qubits Q2, Q3 are tuned to the second resonant frequency F2. F1 and F2 can be separated from one another by up to around 30MHz which means that each group can be driven by an AC electromagnetic pulse independently of the other.

[0054] Figure 4 is a schematic diagram illustrating two spin qubits 12, 14. In the laboratory frame of reference these two qubits 12, 14 precess with different frequencies fi2and fi4, set by their respective g-factors. In the example in Figure 4 the exchange interaction between the qubits 12, 14 is turned on by lowering the barrier between the dots and has a frequency, J, which is large in comparison with all other frequencies. This causes rapid swap of the qubits between the quantum dots. The inventors have recognised that the consequence of this effect is that the pair of qubits now rotate synchronously at a frequency, f=(fi2+fi4) / 2. Thus, when an AC electromagnetic field is applied at a frequency of f both qubits perform an X-rotation in the frame of reference, rotating at frequency f. In the present qubit-swapping technique the swapping frequency, J, is selected so that it is large in comparison with the frequency of the applied AC electromagnetic field, f, in one scenario J is preferably greater than around 10f.

[0055] The exchange interaction results in constantly entangling and swapping the qubits. Therefore, in order to implement an X gate on both qubits without any additional entanglement creation, J is tuned such that the qubits go back to their original positions in the time taken by the X gates. This condition reads:

[0056]

[0057] or equivalently

[0058] J = 2pfl

[0059] Where Q represents the frequency of the applied AC electromagnetic field and p is any natural number 1,2,... etc.

[0060] The present technique means that an AC electromagnetic field can be provided at a single frequency to drive two qubits simultaneously as they are swapped between the quantum dots 12, 14. This reduces the scale of the challenge in terms of frequency engineering qubits by a factor of two. Correspondingly, when the method is applied to bring N qubits into a common frequency, then the level of simplification to the engineering is N-fold which can therefore be very impactful. In addition, the tuneable range of the qubits is increased, as described above, because they experience different g-factor environments and the g-factor dispersion between the quantum dots 12, 14 is greater than the intra-dot g-factor dispersion which is used for Stark shift.

[0061] Figure 5 is a flow diagram illustrating a method of applying an AC electromagnetic field to a pair of qubits. At step S202 a qubit is provided at the first quantum dot 12, having a first resonant spin frequency and a qubit is provided at the second quantum dot 14, having a second resonant spin frequency. Optionally, at step 204 tuning by, for example, the Stark shift is applied as a preparatory step. At step 206 the exchange interaction for the pair of quantum dots is turned on so that exchange oscillations are initiated and the qubits are swapped between the quantum dots 12, 14 with frequency, J. At step 208 a drive frequency that has been pre-computed classically is applied as an AC electromagnetic pulse. The drive frequency in this case is calculated as the mean of the resonant frequencies of the qubit at the first quantum dot 12 and the second quantum dot 14. The effect is to apply a single-qubit gate simultaneously to the pair of qubits. Finally in step 210 the exchange interaction is switched off.

[0062] A further embodiment of the present disclosure involves shuttling qubits between quantum dots. In a simple example, qubits may be shuttled back and forth between from the first quantum dot 2 and the second quantum dot 4 in the example of Figure 1. In this way, the qubit can experience the g-factor at the location of both quantum dots. If the qubit spends an equal amount of time at both dots then the qubit acquires an effective resonant frequency according to the mean g-factor for the first and second quantum dots 2, 4. In the present method it is possible to drive the qubit at the effective resonant frequency by providing an off-chip microwave resonator that can apply an AC electromagnetic field to the first and second gates 2, 4 simultaneously.

[0063] Qubits can be shuttled between quantum dots at a shuttling speed of around 10ms'1. Atypical interdot spacing in a qubit processor is around 100nm. Thus, a shuttling or transfer frequency can be achieved of around 100MHz. In the present method the shuttling frequency is selected so that it is higher than the frequency of the AC electromagnetic field. In some arrangements the shuttling frequency is selected so that it is at least 3 times the frequency of the applied AC electromagnetic field. This means that the quantum gate is performed slowly relative to the rate at which charge carriers are shuttled between quantum dots. Therefore a quantum gate can be applied to a qubit that experience an average resonant frequency, based on the g-factors of gates that it visits.

[0064] Another example, which involves shuttling qubits, will now be described with reference to Figure 6 which illustrates two clusters each with four quantum dots. In this example four qubits Q1-Q4 are illustrated initially positioned at four respective gates in a first cluster and four qubits Q5-Q8 are initially positioned at respective gates in a second cluster. The qubits Q1-Q4 are shuttled around the first cluster of gates so that each qubit experiences the g-factor of each gate for an equal period of time. Simultaneously, the qubits Q5-Q8 are shuttled around the second cluster of gates in the same way. Thus, the qubits follow one another around the group of gates in the relevant cluster. Using a classical processor, the method involves pre-calculating an effective resonant frequency for each cluster of qubits. For the first cluster, this is achieved by determining the resonant frequency for each individual dot and calculating the mean resonant frequency. Similarly, a mean resonant frequency can be calculated for the second cluster. Using this method an AC electromagnetic drive pulse can be provided by an off-chip magnetic field resonator 22 at the respective calculated mean frequencies in order to perform a single-gate operation simultaneously for all of the qubits in the relevant cluster.

[0065] Stark shift tuning can be applied to each cluster in order to provide an additional frequency differential between the respective groups. In an alternative, optional, configuration a magnetic material 20 is provided in proximity with the gates in the second group. This provides a constant magnetic field gradient that modifies the g-factors. This can shift the mean g-factor of the second cluster relative to the mean g-factor of the first cluster. In this way, the separation between the average frequencies of the two clusters can be increased, compared to the sole use of Stark shift tuning. One advantage of the present technique is that shuttling can be applied to any number of qubits around a cluster. This offers an advantage in comparison with the qubit-swapping method which is limited to pairs of qubits.

[0066] Figure 7 is a flow chart illustrating a method of performing operations in a silicon chip qubit processor based on the schematic diagram in Figure 6. At step 302 qubits Q1-Q4 are provided at corresponding quantum dots in a first cluster and qubits Q5-Q8 are provided at corresponding dots in a second cluster. At step 304 optionally preparatory tuning, e.g. via the Stark shift, is applied. At step 306 the qubits Q1 -Q4 are shuttled from one gate to another so that they visit every location in the cluster; simultaneously the qubits Q5-Q8 in the second group are shuttled around the second cluster of gates so that they visit every location. At step 308, an average resonant frequency is applied for each group of qubits in the respective clusters. The required frequencies are pre-calculated using a classical processor and calculating the statistical mean of the resonant frequencies of the individual quantum dots. Because the two frequencies corresponding to the two groups are well-separated, the two pulses required to effect the gate operations can be applied globally across the entire device, not needing to be targeted physically to a specific group’s location. Similarly, the pulses can be applied at the same time.

[0067] In some configurations, the shuttling process may be varied in order to alter the effective frequencies away from the simple mean of the frequencies of the dots in the clusters. In one scenario the shuttling of the qubits may be performed unevenly so that qubits spend longer at some locations than at others. In one specific example, with reference to the first group of qubits, the qubits may spend twice as long at quantum dot QD2 than at QD1 , QD3 or QD4. In this example a weighted average is calculated for the resonant frequency with double the contribution from the resonant frequency at QD2.

[0068] The AC electromagnetic pulses are provided in step 308 may originate from an the off-chip resonator 22 in order to apply single-qubit gates simultaneously to all of the qubits in the relevant group.

[0069] In this configuration every qubit can be returned to its original position at the end of the single-qubit gate. This is expressed by the following equation:

[0070] pd n

[0071] v fT

[0072] or equivalently

[0073]

[0074] Where Q represents the frequency of the applied AC electromagnetic field, d is the distance between quantum dots, v is the shuttling speed and p is any natural number 1,2.. etc.

[0075] Figures 8A-8F illustrate another schematic example of a technique for implementing the present invention which involves shuttling. In this example, a 2xN qubit processor is provided, although only a limited 2x6 array is illustrated. Certain dots are initially empty 802 and do not contain a qubit. Other dots contain a qubit that belongs either to a first group 904 or a second group 906. There are six time steps as illustrated in Figures 8A-8F. At the first time step in Figure 8A six qubits Q1-Q6 are provided in the top row of quantum dots QD1-QD6. In this configuration the intention is to provide a first group of target qubits Q1 , Q3, Q4, Q6 and a second group of non-target qubits Q2, Q3. At the second time step in Figure 8B the second group of qubits Q2, Q3 are shuttled to the bottom row. At the third time step in Figure 8C a Stark shift +5g is applied to the first group of qubits Q1 , Q4, Q5, Q6 in the region 808. At the same time, an equal and opposite Stark shift -5g is applied to the second group of qubits Q2, Q3 in the region 810. Shuttling then proceeds as indicated by the arrows 812. At the third time step in Figure 8D the first and second groups of qubits are shuttled in a clockwise direction around the 2x6 array while continuing to apply a +5g Stark shift to the first group of qubits and a -5g Stark shift to the second group. This is achieved by applying a DC electric field to the gate electrodes corresponding to the positions of the qubits in the array. At the fifth time step in Figure 8E the qubits are further shuttled to the next set of quantum dots in a clockwise direction, and a further shuttling step is performed to move the qubits to the next position illustrated in Figure 8F. In this way the Shark shifts move around the structure synchronously with the corresponding groups of qubits. The motion continues until the quantum gate is implemented, during which process each qubit has experienced the g-factor at each position in the 2x6 array. In many instances, each qubit will have cycled many times through each position in the 2x6 array. The average g-factor for the quantum dots in the array is determined in advance to be g0. Therefore, while the first and second groups of qubits are being shuttled around the array a single driving frequency is provided by an AC electromagnetic field resonator that is off-chip. The frequency is selected to be in resonance for a g-factor of g0+bg in order to target the first group and be off-resonant with the second group thereby leaving it idle. In this way, a single-qubit gate can be provided simultaneously for all qubits in the first group while maintaining the qubits of the second group idle, and using only one driving frequency.

[0076] Figure 9 illustrates the application of the method to the case of a two-qubit gate process. The figure shows a 2xN architecture with a region having nine pairs of dots being depicted, and it is understood that this is a section of a longer array. Empty circles denote empty quantum dots 902 while circles with containing a filled circle 904 denote a quantum dot containing a qubit. Where a dot contains a qubit, there is also a partner qubit in the other row. Each pair of qubits is interacting to achieve a two-qubit gate, with the interaction denoted by the grey arrow 906. The general Hamiltonian describing the evolution of the i-th pair of qubits within the basis |H>,|TsL>,|sLT>,|TT>, is given by:

[0077] <>

[0078]

[0079] where g / 0and g2(l)are the g-factors of the i-th pair of qubits, J is the inter-dot coupling 906, B is the global DC magnetic applied to the system, and Q is a global AC electromagnetic driving field. CNOT and CZ gates are implemented in the low-interaction regime where J is small (with and without drive respectively), and SWAP gates are implemented in the high-interaction regime where J is large. In each two-qubit gate, the g-factors of the two dots involved, and indeed any other factor that differentiates them such as magnetic field strength variations, will affect the behaviour of the two-qubit gate. In conventional approaches the characteristics of the control pulse required to effect a given two-qubit gate (e.g. the choices of the magnitude and frequency of J and Q) must be uniquely adjusted to accommodate such details, so that a single control pulse could not create high fidelity two-qubit gates at multiple locations simultaneously. The present method provides a technique to achieve this result. In Figure 9, the pairs of qubits are shuttled along the array while the two-qubit operation is taking place. Each qubit pair is shuttled through multiple dot locations in the time taken to perform the two-qubit gate. In this way, the effect of the shuttling is to homogenise the irregularities in the g-factors between different locations. Mathematically, the Hamiltonians Hi’s, defined by the sequences of g-factors gi(l)and g2(l), become homogenised to a single Hamiltonian H, parameterised by g-factors gi and g2that are the respective averages of the gi(l)’s and g2(l)’s. This means that all pairs of qubits now evolve according to the same Hamiltonian, allowing for the simultaneous and high-fidelity implementation of two-qubit gates on all of them.

[0080] In a specific embodiment, the two-qubit gate is a CNOT gate. In this case, a global AC electromagnetic field Q is applied, with amplitude QRand frequency co. A CNOT gate between the i-th pair of qubits is implemented by turning on a small exchange coupling J compared to the difference in g-factors between the dots of the pair, and by setting .R= J / V15 and

[0081]

[0082] The gate is applied in a time TT / QR. Therefore, without shuttling, the drive frequency must be made location-dependent, meaning that device irregularities prevent one from implementing a CNOT gate simultaneously on all pairs of qubits with a unique global pulse. With shuttling however, the g-factors of all qubits in the first row are averaged to gi and the g-factors of all qubits in the second row are averaged to g2, A CNOT gate can therefore be simultaneously implemented on all pairs of qubits by setting

[0083]

[0084] While this figure illustrates the principle with a 2xN array, other topologies are possible including a one-dimensional array within which qubit pairs would exist in consecutive dots along that array. The skilled person could adapt the configuration in Figure 9 to implement any of a number of different forms of two-qubit gate that are possible in silicon. Examples of two-qubit gates include root-SWAP, full-SWAP, CNOT operations, and controlled-phase CZ operations. In each case, the operation is performed by applying a control signal that is dependent on the properties of the g-factors for each of the locations experienced by the constituent qubits. In the example of Figure 9, regardless of the gate operation applied, the control signal has properties that are dependent on the mean g-factor for the first row of qubits and the mean g-factorfor the second row of qubits. Thus, control signals can be applied to the first and second row with properties that are dependent on the mean g-factor in the first row and the mean g-factor in the second row.

Claims

CLAIMS1. A method for performing operations in a silicon chip qubit processor, comprising the steps of:providing a first qubit at a first quantum dot, having a first g-factor, gi;transferring the first qubit to a second quantum dot, having a second g-factor, g2; andapplying a control signal to the location of the first and second quantum dots to perform a gate operation, wherein the properties of the control signal are dependent on a weighted average of the first and second g-factors.

2. The method of claim 1, wherein the step of applying a control signal involves applying an AC electromagnetic field to the first and second quantum dots in order to manipulate the first qubit spin state, wherein the applied AC electric field has a frequency which is related to the weighted average of the first and second g-factors.

3. The method of any of the preceding claims, wherein the weighted average is the mean of the first and second g-factors.

4. The method of any of the preceding claims, wherein the first qubit is transferred to the second quantum dot by qubit swapping, wherein qubit swapping is performed between the first and second quantum dots at a transfer frequency.

5. The method of any of claims 1 to 3, wherein the first qubit is transferred to the second quantum dot by shuttling at a transfer frequency.

6. The method of claim 5, wherein the first qubit is transferred from the second quantum dot to an i-th quantum dot having an i-th g-factor, gi, by shuttling, and wherein the applied control signal has properties which are dependent on a weighted average of the first, second and i-th g-factors.

7. The method of claim 5 or claim 6, when dependent on claim 2, wherein the transfer frequency is at least three times the frequency of the applied ACelectromagnetic field, preferably at least ten times the frequency of the applied AC electromagnetic field.

8. The method of any of the preceding claims, wherein a micromagnet is positioned in the vicinity of the second quantum dot.

9. The method of claim 1 , further comprising the steps of:providing a second qubit at a third quantum dot, having a third g-factor, gs;transferring the second qubit to a fourth quantum dot, having a fourth g-factor, g4; andapplying control signals to the locations of the first, second, third and fourth quantum dots to perform a two-qubit gate operation, wherein the properties of the control signals are dependent on a weighted average of the first and second g-factors and on a weighted average of the third and fourth g-factors.

10. A method for tuning the resonant frequency of a spin qubit in a silicon chip qubit processor, comprising the steps of:providing a first qubit at a first quantum dot, having a first g-factor, gi and a second quantum dot, having a second g-factor, g2wherein the difference between gi and g2is Agi2; andapplying an electrode adjustment to the first and second quantum dots in order to delocalise the first qubit between the first and second quantum dots, to tune its effective g-factor over a tuneable range of Agi2between gi and g2, thereby to tune the resonant frequency of the first qubit.

11. The method of claim 10, wherein Agi2is at least two times, preferably at least five times, and more preferably at least ten times larger than the achievable Stark shift range, 5g, for the first quantum dot or the second quantum dot.

12. The method of claim 10 or claim 11, wherein the step of applying an electrode adjustment comprises shuttling the first qubit from the first quantum dot to the second quantum dot.

13. The method of any of claims 10 to 12, comprising:providing a second qubit at a third quantum dot, having a third g-factor, g3and a fourth quantum dot, having a fourth g-factor, g4wherein the difference between g3and g4is Ag34;applying an electrode adjustment to the third and fourth quantum dots in order to delocalise the second charge carrier spin qubit between the two dots, to tune its effective g-factor over a tuneable range of Ag34between g3and g4, thereby to tune the resonant frequency of the second qubit so that it is different to the resonant frequency of the first qubit.

14. The method of claim 13, wherein the resonant frequencies of the first and second qubits are tuned to frequencies that are separated by at least 5MHz, preferably at least 10MHz, and more preferably at least 20MHz.

Images