Method for tuning an interaction strength between at least two qubits of a quantum computer and quantum computer

Abstract

A method for tuning an interaction strength (J) between at least two qubits (qb1, qb2) of a quantum computer (10) is specified herein, the method comprising the steps of: - applying a first periodic pulse sequence (1) to a first qubit (qb1), - applying a second periodic pulse sequence (2) to a second qubit (qb2), wherein - a period (T) of the first periodic pulse sequence (1) is equal to a period (T) of the second periodic pulse sequence (2), - pulses (21) of the second periodic pulse sequence (2) are time delayed with respect to pulses (11) of the first periodic pulse sequence (1), and - the interaction strength (J) is tuned by controlling said time delay (Δt). Further, a quantum computer (10) is specified herein.

Description

[0001]P2023,1428 WO N November 14, 2024- 1 -Description Method for tuning an interaction strength between at least two qubits of a quantum computer and quantum computer A method for tuning an interaction strength between at least two qubits of a quantum computer and a quantum computer are specified herein. At least one object of certain embodiments is to provide a method for tuning an interaction strength between at leasttwo qubits in a quantum register of a quantum computer. Thisobject is achieved by the method with the features according to the independent claim. Advantageous embodiments and further developments of the method are specified in the dependent claims. According to an embodiment, the method for tuning an interaction strength between at least two qubits of a quantumcomputer comprises a step of applying a first periodic pulsesequence to a first qubit. In particular, each qubit is aquantum mechanical two-state system comprising two linearly independent basis states, denoted by |0> and |1> in the following. For example, the at least two qubits are part of the same quantum register of the quantum computer. In particular, the quantum register comprises multiple qubits and the quantum computer performs quantum computations, such as gate operations, by manipulating states of the qubits in thequantum register. For example, qubits in the same quantumregister interact with each other.P2023,1428 WO N November 14, 2024- 2 -For example, the first periodic pulse sequence comprises or consists of single qubit gates acting on the first qubit. In other words, each pulse of the first periodic pulse sequence corresponds to one or more single qubit gates. For example, the single qubit gates are rotation operators about the x- axis, the y-axis, and / or the z-axis. In particular, one pulse of the first periodic pulse sequence rotates a state of the first qubit by a given angle around a given axis of the Bloch sphere. For example, each pulse of the first periodic pulsesequence flips the state of the first qubit such that thebasis state |0> is changed to |1> and vice versa. In particular, the first periodic pulse sequence is periodic in time. For example, one period of the first periodic pulsesequence comprises one pulse and the pulse is applied to thefirst qubit repeatedly at equidistant time-steps.According to a further embodiment, the method comprises astep of applying a second periodic pulse sequence to a secondqubit. All features disclosed for the first periodic pulsesequence can also apply to the second periodic pulse sequence, and vice versa. For example, the second periodic pulse sequence comprises or consists of single qubit gates acting on the second qubit. In other words, each pulse of the second periodic pulse sequence corresponds to one or more single qubit gates. For example, one pulse of the second periodic pulse sequence rotates a state of the second qubit by a given angle around a given axis of the Bloch sphere. Forexample, each pulse of the second periodic pulse sequenceflips the state of the second qubit such that the basis state |0> is changed to |1> and vice versa.P2023,1428 WO N November 14, 2024- 3 -In particular, the second periodic pulse sequence is periodic in time. For example, one period of the second periodic pulse sequence comprises one pulse and the pulse is applied to the second qubit repeatedly at equidistant time-steps.According to a further embodiment of the method, a period ofthe first periodic pulse sequence is equal to a period of thesecond periodic pulse sequence. In other words, timeintervals at which the first periodic pulse sequence and the second periodic pulse sequence repeat itself are identical for the first periodic pulse sequence and the second periodic pulse sequence.According to a further embodiment of the method, pulses ofthe second periodic pulse sequence are time delayed withrespect to pulses of the first periodic pulse sequence. Forexample, if the pulses of the first periodic pulse sequence are applied to the first qubit at discrete equidistant time- steps tm, the pulses of the second periodic pulse sequence are applied to the second qubit at discrete equidistant time-steps tm + Δt, where Δt is the time delay. In particular, 0 ≤Δt ≤ T, where T is the period of the first and secondperiodic pulse sequences. For example, the second periodic pulse sequence is identical to the first periodic pulse sequence with the only exception that there is a time delayor time shift Δt between the first periodic pulse sequenceand the second periodic pulse sequence. According to a further embodiment of the method, the interaction strength is tuned by controlling said time delay between pulses of the second periodic pulse sequence and pulses of the first periodic pulse sequence. For example, the interaction strength between the first qubit and the secondP2023,1428 WO N November 14, 2024- 4 -qubit is modulated as a function of time while the first periodic pulse sequence and the time-shifted second periodicpulse sequence is applied to the first qubit and the secondqubit, respectively. For example, the period of the first and second periodic pulse sequences is much shorter than a time scale associatedwith the interaction strength between the first qubit and thesecond qubit. In other words, an energy scale associated with pulses of the first and second periodic pulse sequences, such as Planck’s constant multiplied by a Rabi frequency of a single-qubit rotation operation applied to the first qubit and / or the second qubit, is much larger than an energy scale of the interaction strength between the first qubit and the second qubit. For example, the energy scale of pulses of the first and second periodic pulse sequence is at least by a factor of ten larger than the interaction strength.In particular, if the period of the first and second periodicpulse sequence is much shorter, e.g. at most one tenth, thanthe timescale h / |J|, where h is Planck’s constant and J isthe interaction strength that has physical units of energy, the first qubit and the second qubit effectively interact viaa time-averaged interaction strength while the first andsecond periodic pulse sequences are applied. Here, the time average can be performed over one period of the first and second periodic pulse sequence, for example. For example, the time averaged interaction strength is a function of the time delay between pulses of the second periodic pulse sequence and pulses of the first periodic pulse sequence. For example, the absolute value of the interaction strength remains constant during one period ofP2023,1428 WO N November 14, 2024- 5 -the first and second periodic pulse sequences, but the interaction flips its sign during the time interval between a pulse of the first periodic pulse sequence and the time delayed pulse of the second periodic pulse sequence. Accordingly, the absolute value of the time averagedinteraction strength changes with increasing time delay, forexample. For example, the time averaged interaction strength is zero if the time delay equals on half of the period of the first and second periodic pulse sequence. According to a preferred embodiment, the method for tuning an interaction strength between at least two qubits of a quantum computer comprises the steps of:- applying the first periodic pulse sequence to the firstqubit,- applying the second periodic pulse sequence to the secondqubit, wherein- the period of the first periodic pulse sequence is equal tothe period of the second periodic pulse sequence,- pulses of the second periodic pulse sequence are timedelayed with respect to pulses of the first periodic pulsesequence, and- the interaction strength is tuned by controlling said timedelay. The method disclosed herein is based on the idea to tune an interaction strength between qubits in a quantum register by applying time-shifted periodic pulse sequences to the respective qubits. For example, the periodic pulse sequences flip the qubit’s states at a frequency much higher than a frequency h / |J| associated with the interaction energy scale J. Accordingly, the interaction between the qubits is effectively time averaged over a period of the periodic pulseP2023,1428 WO N November 14, 2024- 6 -sequences and the strength of the time averaged interaction can be adjusted by tuning the time-delay of pulses between different periodic pulse sequences. Advantageously, these periodic pulse sequences have a dual purpose as dynamicaldecoupling pulse sequences for protecting the states of thequbits from decoherence. According to a further embodiment of the method, the interaction between the at least two qubits is a pairwiseIsing or XY interaction. For example, all qubits in thequantum register interact via an all-to-all pairwise Ising or XY interaction. For example, in a quantum register comprising N qubits the interaction takes the form In particular, all pairs of qubits in the quantum registerinteract via Ising or XY interactions. In equation (1) aboveHint refers to an interaction term in a Hamiltonian, theinteger indices i and j enumerate the N qubits in the quantumregister, and with ^, ^ ∈ {^, ^, ^} denotes one of the three2×2 Pauli matrices acting on the basis states of qubit i with ^ ∈ {1, 2, … , ^}. Here,products of Pauli matrices acting on different qubits are tobe understood as tensor products. Further, Jij has units ofenergy and parametrizes the interaction strength betweenqubits i and j. For example Jij = J is independent of i andj.For α = the interaction specified in equation (1) isdenoted as “Ising interaction” in the following, whereas forP2023,1428 WO N November 14, 2024- 7 -α β the interaction is denoted as “XY interaction” in thefollowing. For the case of Ising interactions with α = β, thebasis states |0> and |1> of the qubits are preferably eigenstates of the corresponding Pauli matrix and are denoted as measurement basis states in the following. For example, pairwise Ising or XY interactions between qubitsarise for various physical qubit realizations, such assuperconducting qubits or trapped ion qubits. For example, the pairwise Ising or XY interaction between the qubits canbe used to implement two-qubit gates and / or multi-qubitgates, in particular entangling gates, between the respectivequbits. For example, a global entangling gate, such as ageneralized Mølmer-Sørensen gate or a magnetic gradientinduced coupling gate, can be implemented by time evolving the qubits in the quantum register while the qubits interact via the Hamiltonian specified in Equation (1). Individually and simultaneously controlling the interaction strength between pairs of qubits using the method disclosed herein thus advantageously allows to implement specific quantumcircuits, where entangling gates act in a predeterminedmanner on selected qubits, for example.According to a further embodiment of the method, the firstperiodic pulse sequence comprises pi-pulses, wherein each pi-pulse flips the state of the first qubit. For example, thepi-pulse is a single qubit-rotation that rotates the state ofthe first qubit by an angle of π radians about an axis of theBloch sphere. In particular, the pi-pulse changes the basisstate |0> of the first qubit to the basis state |1> of the first qubit and vice versa.P2023,1428 WO N November 14, 2024- 8 -In particular, the pi-pulse acts on a time scale that is an inverse of a Rabi frequency of an applied electromagnetic field that couples the two basis states |0> and |1> of the first qubit. For example, the Rabi frequency is much larger,e.g. by at least a factor of ten larger, than the frequency|J12| / h associated with the interaction strength J12 betweenthe first and second qubits. Accordingly, the frequency of the first periodic pulse sequence can be much larger than thefrequency |J12| / h associated with the interaction strength.According to a further embodiment of the method, the secondperiodic pulse sequence comprises pi-pulses, wherein each pi-pulse flips the state of the second qubit. All featuresdisclosed for the pi-pulses of the first periodic pulse sequence can also apply to the pi-pulses of the second periodic pulse sequence, and vice versa. In particular, the pi-pulse changes the basis state |0> of the second qubit to the basis state |1> of the second qubit and vice versa.According to a further embodiment of the method, a timeaveraged interaction strength between the at least two qubits averaged over the period of the first periodic pulse sequence and the second periodic pulse sequence is a linear functionof the time delay. For example, the first and second periodicpulse sequences flip the measurement basis states of the first and second qubit, respectively. Accordingly, the sign of the interaction strength J12in equation (1) effectivelychanges from + to – and vice versa, if the first and secondqubits are not flipped simultaneously. For example, the timeaveraged interaction strength between the first qubit and thesecond qubit takes the value P2023,1428 WO N November 14, 2024- 9 -where Δt with 0 ≤ Δt ≤ T is the time delay and T is theperiod of the first and second periodic pulse sequence. Accordingly, the time averaged interaction strength can becontinuously tuned between +J12 and –J12 by adjusting the timedelay Δt accordingly.According to a further embodiment of the method, the firstperiodic pulse sequence is a dynamical decoupling pulse sequence configured for protecting the first qubit from decoherence, and / or the second periodic pulse sequence is a dynamical decoupling pulse sequence configured for protectingthe second qubit from decoherence. In particular, decoherencerefers to the loss of quantum coherence of a qubit due to an unwanted coupling of the qubit to the environment. For example, the dynamical decoupling pulse sequence suppresses decoherence of the qubit by at least approximately averaging the unwanted coupling between the qubit and the environment to zero. For example, the dynamical decoupling pulse sequence is based on a periodically iterated Hahn spin-echo or Carr-Purcell scheme for reducing the dephasing of qubits due to their coupling to environmental noise. In particular, the time interval between pulses of the dynamical decoupling pulse sequence is much shorter than a time scale on which the unwanted coupling between the qubit and the environment acts, e.g. the latter is at least by a factor of ten larger thanthe time interval between pulses.According to the method described above, dynamical decoupling pulse sequences for qubits can be further used to additionally tune the interaction strength between the qubitsP2023,1428 WO N November 14, 2024- 10 -by time-shifting the dynamical decoupling pulse sequences applied to different qubits with respect to each other.According to a further embodiment of the method, a frequencyof the first periodic pulse sequence and the second periodic pulse sequence is matched to an environmental noise spectrum. In particular, the frequency is matched such that a coherence time of the first qubit and the second qubit in the presence of environmental noise is increased compared to a situation, where no first and second periodic pulse sequences are applied to the first and second qubits, respectively. For example, the frequency of the first and second periodic pulse sequence is chosen such that it does not coincide with a resonance in the noise spectrum. According to a further embodiment, the method is configured to independently tune the interaction strength betweenmultiple pairs of qubits in the quantum register using thesteps of:- applying different periodic pulse sequences to differentqubits in the quantum register, wherein all periodic pulsesequences have the same period, and- adjusting the time delay of pulses between differentperiodic pulse sequences in order to independently tune theinteraction strength between multiple pairs of qubits. Inparticular, all features of the first periodic pulse sequence and / or the second periodic pulse sequence disclosed above can also apply to all other periodic pulse sequences applied to different qubits in the quantum register.According to a further embodiment of the method, basis statesof each qubit correspond to different hyperfine states of acorresponding trapped ion. In particular, each qubit isP2023,1428 WO N November 14, 2024- 11 -encoded in two different hyperfine states of a corresponding trapped ion. A system of two or more ions trapped in the same ion trap forms the quantum register, for example. According to a further embodiment of the method, the at least two qubits interact via a magnetic gradient induced coupling. For example, a magnetic field gradient along a chain of trapped ions induces the pairwise Ising interaction betweenall pairs of qubits in the ion trap. For example, this Isinginteraction in the presence of the magnetic field gradient is mediated by a common vibrational motion of the ions in the ion trap. Further a quantum computer is specified herein. In particular, the quantum computer implements the method for tuning an interaction strength between at least two qubits described above. All features of the method are also disclosed for the quantum computer and vice versa.According to an embodiment of the quantum computer, aninteraction strength between at least two qubits is tuned using the method described above.Further advantageous embodiments and further embodiments ofthe method and the quantum computer may become apparent fromthe following exemplary embodiments described in connectionwith the figures.Figure 1 shows a schematic graph of a first periodic pulsesequence, a second periodic pulse sequence, states of a first qubit and a second qubit, as well as an interaction strength between the first qubit and the second qubit as a function oftime according to an exemplary embodiment of the method forP2023,1428 WO N November 14, 2024- 12 -tuning an interaction strength between at least two qubits of a quantum computer. Figure 2 shows a schematic graph of periodic pulse sequencesapplied to four qubits in a quantum register according to anexemplary embodiment of the method for tuning an interaction strength between at least two qubits of a quantum computer. Figure 3 shows a schematic illustration of a quantum computer according to an exemplary embodiment. Elements that are identical, similar or have the same effect, are denoted by the same reference signs in the figures. The figures and the proportions of the elements shown in the figures are not to be regarded as true to scale. Rather, individual elements may be shown exaggeratedly large for better representability and / or better understanding. Figure 1 shows a first periodic pulse sequence 1 comprising pulses 11 applied to a first qubit qb1, and a second periodic pulse sequence 2 comprising pulses 21 applied to a second qubit qb2 in a quantum register of a quantum computeraccording to an exemplary embodiment of the method. The firstperiodic pulse sequence 1 and the second periodic pulse sequence 2 have the same period T as function of time t. The first qubits qb1 and the second qubit qb2 interact via anIsing interaction ^ ^^^^^^, where ^^^ = denotes the z- Pauli matrix acting on the first qubit qb1, ^^^ = denotes the z-Pauli matrix acting on the second qubit qb2,and J is the interaction strength.P2023,1428 WO N November 14, 2024- 13 -The pulses 11 of the first periodic pulse sequence 1 are pi- pulses that flip the basis states |0> and |1> of the first qubit qb1 into each other. In other words, after applying one pi-pulse the state |0> is changed to the state |1> and vice versa. Here, |0> and |1> are eigenstates of ^^^witheigenvalues -1 and +1, respectively. Similarly, the pulses 12of the second periodic pulse sequence 2 are pi-pulses that flip the basis states |0> and |1> of the second qubit qb2 into each other. In other words, after applying one pi-pulse the state |0> is changed to the state |1> and vice versa. Here, |0> and |1> are eigenstates of ^^^with eigenvalues -1 and +1, respectively. The pulses 21 of the second periodic pulse sequence 2 are time-shifted and thus have a time delay Δt with respect to pulses 11 of the first periodic pulse sequence 1. Accordingly, the second qubit qb2 is flipped at a differenttime t than the first qubit qb1 by each respective pi-pulse.The second and third panels of Figure 1 show the expectation value〈^^^ 〉 for the first qubit qb1 and the expectation value the second qubit qb2, respectively, as a function oftime t. In this example, the first qubit qb1 and the secondqubit qb2 are both initialized in the basis state |1> at timet=0, accordingly, both expectation values〈^^^ 〉 and 〈^^^〉 areequal to one at time t=0. After applying the first pi-pulse to the first qubit qb1, the state of the qubit changes to |0> and the expectation value changes accordingly −1.The second pi-pulse flips the state of the first qubit qb1 back to |1> and the expectation value 〈^^^ 〉 changesaccordingly. The same is true for the second qubit qb2, albeit at later times delayed by the time delay Δt.P2023,1428 WO N November 14, 2024- 14 -The interaction energy ^ = 〈^ ^^^^^^〉 between the first qubit qb1and the second qubit qb2 as a function of time t is depicted in the lowest panel of Figure 1. The interaction energy is +J, if both qubits are in the same state, e.g. both qubits are in the state |0> or both qubits are in the state |1>. Theinteraction energy is -J, if the first qubit qb1 and thesecond qubit qb2 are in different states, e.g. the first qubit qb1 is in state |0> whereas the second qubit qb2 is in state |1>, or vice versa.Accordingly, by tuning the time delay Δt the time average ofthe interaction energy E and thus the time average of the effective interaction strength J can be tuned. In particular, the frequency 1 / T of the first and second periodic pulse sequences 1, 2 is much larger, e.g. by a factor of at least ten, than the frequency scale J / h associated with the interaction strength J (here h denotes Planck’s constant). Accordingly, the first qubit and the second qubit effectively interact via a time averaged interaction with an effectiveinteraction strength J that is a time average of theinteraction energy E over one period T of the periodic pulse sequences 1, 2.Figure 2 shows four different periodic pulse sequences 1, 2,3, 4 applied to four different qubits qb1, qb2, qb3, qb4 according to an exemplary embodiment of the method, in order to tune the interaction strength Jijbetween each pair of the four qubits qb1, qb2, qb3, qb4. Apart from the different time delays Δt, the pulse sequences are identical to the pulse sequences described in connection with Figure 1. In particular, the effective interaction strength J12betweenqubits qb1 and qb2 depends on the time delay Δt = τbk2 – τbk1,whereas the effective interaction strength J23between qubitsP2023,1428 WO N November 14, 2024- 15 -qb2 and qb3 depends on the time delay Δt = τbk2 – τbk3, forexample. In general, the effective interaction strength Jijbetween qubits qbi and qbj, with ^, ^ ∈ {1, 2, 3, 4} depends on thetime delay Δt = τbki – τbkj. By choosing the timing of thepulses in each periodic pulse sequence 1, 2, 3, 4, the interaction strength between each pair of qubits can thus be independently tuned. The quantum computer 10 according to the exemplary embodiment in Figure 3 comprises a quantum register with four qubits qb1, qb2, qb3, qb4. The two basis states |0> and |1> of each qubit qb1, qb2, qb3, qb4 are two hyperfine states of a corresponding trapped ion. All ions belonging to the quantum register are trapped in the same ion trap, such as a Paul trap or a Penning trap. A magnetic gradient along the chain of trapped ions induces a pairwise Ising interaction between all pairs of qubits in the quantum register. The invention is not restricted to the exemplary embodiments by the description on the basis of said exemplary embodiments. Rather, the invention encompasses any new feature and also any combination of features, which in particular comprises any combination of features in the patent claims and any combination of features in the exemplary embodiments, even if this feature or this combination itself is not explicitly specified in the patent claims or exemplary embodiments.P2023,1428 WO N November 14, 2024- 16 -References1 first periodic pulse sequence11 pulse2 second periodic pulse sequence21 pulse3, 4 periodic pulse sequences10 quantum computerqb1…4 qubitT periodΔt time delayt timeJ interaction strengthE interaction energy

Claims

P2023,1428 WO N November 14, 2024- 17 -Claims 1. A method for tuning an interaction strength (J) between atleast two qubits (qb1, qb2) of a quantum computer (10),comprising the steps of:- applying a first periodic pulse sequence (1) to a firstqubit (qb1),- applying a second periodic pulse sequence (2) to a secondqubit (qb2), wherein- a period (T) of the first periodic pulse sequence (1) isequal to a period (T) of the second periodic pulse sequence (2),- pulses (21) of the second periodic pulse sequence (2) aretime delayed with respect to pulses (11) of the firstperiodic pulse sequence (1), and- the interaction strength (J) is tuned by controlling saidtime delay (Δt).

2. The method according to the previous claim, wherein the interaction between the at least two qubits (qb1, qb2) is a pairwise Ising or XY interaction.

3. The method according to any of the previous claims, wherein- the first periodic pulse sequence (1) comprises pi-pulses,wherein each pi-pulse flips the state of the first qubit (qb1), and- the second periodic pulse sequence (2) comprises pi-pulses,wherein each pi-pulse flips the state of the second qubit (qb2).

4. The method according to any of the previous claims, wherein a time averaged interaction strength (J) between theP2023,1428 WO N November 14, 2024- 18 -at least two qubits (qb1, qb2) averaged over the period (T) of the first periodic pulse sequence (1) and the second periodic pulse sequence (2) is a linear function of the timedelay (Δt).

5. The method according to any of the previous claims, wherein- the first periodic pulse sequence (1) is a dynamicaldecoupling pulse sequence configured for protecting the first qubit (qb1) from decoherence, and / or- the second periodic pulse sequence (2) is a dynamicaldecoupling pulse sequence configured for protecting the second qubit (qb2) from decoherence.

6. The method according to any of the previous claims, wherein a frequency of the first periodic pulse sequence (1)and the second periodic pulse sequence (2) is matched to anenvironmental noise spectrum.

7. The method according to any of the previous claims, wherein the method is configured to independently tune theinteraction strength (J) between multiple pairs of qubits(qb1, qb2, qb3, qb4) in a quantum register using the steps of:- applying different periodic pulse sequences (1, 2, 3, 4) todifferent qubits (qb1, qb2, qb3, qb4) in the quantumregister, wherein all periodic pulse sequences (1, 2, 3, 4)have the same period (T),- adjusting the time delay (Δt) of pulses between differentperiodic pulse sequences (1, 2, 3, 4) in order to independently tune the interaction strength (J) betweenmultiple pairs of qubits (qb1, qb2, qb3, qb4).P2023,1428 WO N November 14, 2024- 19 -8. The method according to any of the previous claims, wherein basis states of each qubit (qb1, qb2, qb3, qb4) correspond to different hyperfine states of a corresponding trapped ion.

9. The method according to the previous claim, wherein the at least two qubits (qb1, qb2) interact via a magnetic gradient induced coupling.

10. A quantum computer (10) wherein an interaction strength(J) between at least two qubits (qb1, qb2) is tuned using the method according to any of claims 1 to 9.

Images