Method for performing parallel quantum computations in a quantum register comprising qubits with an all-to-all interaction and quantum computer

Abstract

A method for performing parallel quantum computations in a quantum register comprising qubits (qb1, qb2, qb3, qb4) with an all-to-all interaction is specified herein, comprising the steps of: - applying a first periodic pulse sequence (1) to qubits (qb1, qb3) in a first sub-register (31), - applying a second periodic pulse sequence (2) to qubits (qb2, qb4) in a second sub-register (32), wherein - the first periodic pulse sequence (1) and the second periodic pulse sequence (2) are adjusted such that a time averaged pairwise interaction between any qubit (qb1, qb3) in the first sub-register (31) and any qubit (qb2, qb4) in the second sub-register (32) vanishes. Further, a quantum computer (10) is specified herein.

Description

[0001] P2023,1425 WO N November 13, 2024- 1 -Description METHOD FOR PERFORMING PARALLEL QUANTUM COMPUTATIONS IN A QUANTUM REGISTER COMPRISING QUBITS WITH AN ALL-TO-ALL INTERACTION AND QUANTUM COMPUTER A method for performing parallel quantum computations in a quantum register comprising qubits with an all-to-all interaction and a quantum computer are specified herein. At least one object of certain embodiments is to provide amethod for splitting a quantum register comprising qubitswith an all-to-all interaction into independent, virtual sub- registers. This object 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 performing parallel quantum computations in a quantum registercomprising qubits with an all-to-all interaction comprises astep of applying a first periodic pulse sequence to qubits ina first sub-register. In particular, each qubit is a quantummechanical two-state system comprising two linearly independent basis states, denoted by |0> and |1> in the following. For example, the quantum register comprises at least twoqubits or a plurality of qubits. In particular, the quantumcomputer performs quantum computations, such as gate operations, by manipulating states of the qubits in thequantum register. The quantum computer can comprise one, twoP2023,1425 WO N November 13, 2024- 2 -or more quantum registers that are physically separated by hardware elements. For example, the basis states of one qubitcan be represented by two hyperfine states of a trapped ionand multiple ions trapped in one ion trap form the quantumregister. For example, the basis states of one qubit can be represented by different states of a superconducting qubit and superconducting qubits connected to one quantum bus form the quantum register. For example, qubits in the same quantumregister interact with each other. For example, the all-to-all interaction refers to interactions between qubits, where each qubit in the quantum register interacts with every other qubit in the quantum register via a pairwise interaction. For example, the first sub-register is a part of the quantum register that comprises two or more qubits from the quantum register. For example, the first sub-register is a virtualsub-register. In particular, there are no hardware elementsthat physically separate the qubits in the first sub-registerfrom the remaining qubits in the quantum register. For example, quantum computations can be performed on qubits in the first sub-register independently from the remaining qubits in the quantum register. In particular, the first periodic pulse sequence is appliedto all qubits in the first sub-register. For example, thefirst periodic pulse sequence comprises or consists of single qubit gates acting on the qubits in the first sub-register. 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 ofeach qubit in the first sub-register by a given angle aroundP2023,1425 WO N November 13, 2024- 3 -a given axis of the Bloch sphere. For example, each pulse of the first periodic pulse sequence flips the state of thequbits in the first sub-register such that the basis 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 pulse sequence comprises one pulse and the pulse is applied to thequbits in the first sub-register repeatedly at equidistanttime-steps. According to a further embodiment, the method comprises astep of applying a second periodic pulse sequence to qubitsin a second sub-register. All features disclosed for thefirst sub-register and the first periodic pulse sequence can also apply to the second sub-register and the second periodic pulse sequence, and vice versa. For example, the second sub-register is a part of the quantum register that comprises two or more qubits from the quantumregister that are not part of the first sub-register. Inparticular, the affiliation of a qubit to the first sub- register or the second sub-register is mutually exclusive.For example, the second sub-register is a virtual sub-register. For example, quantum computations can be performedon qubits in the second sub-register independently from thequbits in the first sub-register. For example, the second periodic pulse sequence comprises or consists of single qubit gates acting on qubits in the second-sub register. 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 periodicP2023,1425 WO N November 13, 2024- 4 -pulse sequence rotates a state of each qubit in the secondsub-register by a given angle around a given axis of theBloch sphere. For example, each pulse of the second periodic pulse sequence flips the state of the qubits in the secondsub-register such that the basis state |0> is changed to |1>and vice versa. 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 thequbits in the second sub-register repeatedly at equidistanttime-steps. For example, the period of the second periodicpulse sequence is different from the period of the first periodic pulse sequence.According to a further embodiment of the method, the firstperiodic pulse sequence and the second periodic pulsesequence are adjusted such that a time averaged pairwiseinteraction between any qubit in the first sub-register andany qubit in the second sub-register vanishes. For example, time averaged pairwise interaction is zero or approximately zero. For example, the time averaged interaction strength is smaller than one tenth of an instantaneous or maximal interaction strength. For example, the period of the first periodic pulse sequence and the period of the second periodic pulse sequence are chosen such that the time averaged pairwise interaction between any qubit in the first sub- register and any qubit in the second sub-register is zero. In particular, the pairwise interaction is time averaged over the longer of the two periods. For example, the pairwise interaction is time averaged over two of the longer periods.P2023,1425 WO N November 13, 2024- 5 -According to a preferred embodiment, the method forperforming parallel quantum computations in a quantum register comprising qubits with an all-to-all interaction,comprises the steps of:- applying the first periodic pulse sequence to qubits in thefirst sub-register,- applying the second periodic pulse sequence to qubits inthe second sub-register, wherein- the first periodic pulse sequence and the second periodicpulse sequence are adjusted such that a time averagedpairwise interaction between any qubit in the first sub- register and any qubit in the second sub-register vanishes. The method disclosed herein is based on the idea to split a quantum register comprising qubits with an all-to-all interaction into at least two virtual sub-registers, such that that different multi-qubit quantum gates can be applied in parallel to the qubits in the separate sub-registers. In particular, this is achieved by tuning an interaction strength between qubits in the first sub-register and qubits in the second sub-register at least approximately to zero by applying suitably chosen periodic pulse sequences to the qubits. Advantageously, the method disclosed herein allows for aparallelization of multi-qubit quantum gates during a quantumcomputation with qubits in a quantum register that interactvia an all-to-all interaction. In particular, qubits in thequantum register that do not take part in a first multi-qubitquantum gate do not have to wait in an idle state during thetime evolution of the first multi-qubit quantum gate. Rather,a different, second multi-qubit quantum gate can be appliedsimultaneously to the qubits that do not take part in theP2023,1425 WO N November 13, 2024- 6 -first multi-qubit quantum gate. Accordingly, the methoddescribed herein allows to accelerate quantum computations, for example.According to a further embodiment of the method, the firstperiodic pulse sequence and the second periodic pulse sequence are adjusted such that a ratio between a period ofthe second periodic pulse sequence and a period of the firstperiodic pulse sequence is equal to a power of two. Inparticular, the period of the second periodic pulse sequence is larger than the period of the first periodic pulse sequence by a factor of two, four, eight, sixteen or thirty- two, for example. Accordingly, the states of the qubits in the first sub-register flip back and forth two times, four times, eight times, sixteen times, or thirty-two times more often than the states of the qubits in the second sub- register, such that the pairwise interaction between qubits in the first sub-register and qubits in the second sub- register time averaged over two periods of the secondperiodic pulse sequence at least approximately cancels out.According to a further embodiment of the method, the pairwiseinteraction between the qubits is an Ising or a XY interaction. For example, all qubits in the quantum 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 register interact via Ising or XY interactions. In equation (1) above Hintrefers to an interaction term in a Hamiltonian, theinteger indices i and j enumerate the N qubits in the quantumP2023,1425 WO N November 13, 2024- 7 -register, 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 to be understood as tensor products. Further, Jijhas units of energy and parametrizes the interaction strength betweenqubits i and j. For example Jij = J is independent of i andj.For α = β the interaction specified in equation (1) isdenoted as “Ising interaction” in the following, whereas forα β the interaction is denoted as “XY interaction” in thefollowing. For the case of Ising interactions with α = β, the basis 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 qubits arise for various physical qubit realizations, such as superconducting qubits or trapped ion qubits. For example, the pairwise Ising or XY interaction between the qubits can be used to implement two-qubit gates and / or multi-qubit gates, in particular entangling gates, between the respective qubits. 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).According to a further embodiment of the method, the firstperiodic pulse sequence comprises pi-pulses, wherein each pi-P2023,1425 WO N November 13, 2024- 8 -pulse flips the state of each qubit in the first sub-register, and / or the second periodic pulse sequence comprisespi-pulses, wherein each pi-pulse flips the state of eachqubit in the second sub-register. For example, the pi-pulseis a single qubit-rotation that rotates the state of the corresponding qubit to which the pi-pulse is applied by anangle of π radians about an axis of the Bloch sphere. Inparticular, the pi-pulse changes the basis state |0> of the corresponding qubit to the basis state |1> of the corresponding qubit and vice versa. 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 corresponding qubit. For example, the Rabi frequency is much larger, e.g. by at least a factor of ten larger, than the frequency |J| / h associated with the interaction strength J between the pairs of qubits in the quantum register. Accordingly, the frequency of the first periodic pulse sequence and / or the frequency of the second periodic pulse sequence, i.e. the inverse of the corresponding periods ofthe first and / or second periodic pulse sequences, can bechosen proportional to the Rabi frequency and thus is much larger than the frequency |J12| / h associated with the interaction strength.According to a further embodiment, the method comprises afurther step of applying an entangling quantum gate to qubits in the first sub-register and applying a further entangling quantum gate to qubits in the second sub-register in parallelat the same time. In particular, the entangling quantum gateis a multi-qubit quantum gate that acts on at least two qubits.P2023,1425 WO N November 13, 2024- 9 -According to a further embodiment of the method, the firstperiodic pulse sequence is a dynamical decoupling pulse sequence configured for protecting the qubits in the first sub-register from decoherence, and / or the second periodic pulse sequence is a dynamical decoupling pulse sequence configured for protecting the qubits in the second sub-register from decoherence. In particular, decoherence refersto 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 than the time interval between pulses. Advantageously, the periodic pulse sequences described herein have a dual purpose as dynamical decoupling pulse sequences for protecting the states of the qubits from decoherence, as well as for cancelling the interaction between qubits in different sub-registers.According to a further embodiment of the method, a frequencyof the first periodic pulse sequence and / or a frequency ofP2023,1425 WO N November 13, 2024- 10 -the second periodic pulse sequence is matched to anenvironmental noise spectrum. In particular, the frequency ismatched such that a coherence time of qubits in the first sub-register and / or in the second sub-register in the presence of environmental noise is increased compared to a situation, where no first and second periodic pulse sequences are applied to the qubits in the first and second sub- registers, 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 configuredto subdivide the quantum register into N independent sub- registers, such that each sub-register comprises an arbitrarynumber of qubits, and the qubits in each sub-register onlyinteract with qubits from the same sub-register. According to a further embodiment, the method comprises astep of applying a first to a N-th periodic pulse sequence toqubits in the first to N-th sub-register, respectively,wherein a period of the m-th periodic pulse sequence is equalto 2m-1times the period of the first periodic pulse sequence,wherein the integer m with 1 ≤ m ≤ N enumerates the N sub-registers. Accordingly, pairwise interactions between qubitsin different sub-registers at least approximately average out to zero.According to a further embodiment of the method, the sub-registers are virtual sub-registers, such that at least one qubit can be dynamically assigned to different sub-registers at subsequent time intervals.P2023,1425 WO N November 13, 2024- 11 -According to a further embodiment of the method, aninteraction strength of the pairwise interaction between a first qubit and a second qubit within any sub-register can be independently tuned by adjusting a time delay between pulses of the periodic pulse sequence applied to first qubit and pulses of the periodic pulse sequence applied to the second qubit. In particular, a time averaged interaction between the first qubit and the second qubit in the same sub-register is tuned by adjusting the time delay between pulses. For example, the time scale of the periodic pulse sequence applied to the first and second qubits is much faster than the interaction time scale between the first and the second qubits. Accordingly, the first and second qubits effectivelyinteract via the time averaged interaction that is averagedover one period of the periodic pulse sequence.For example, if the pulses of the m-th periodic pulse sequence are applied to the first qubit in the m-th sub- register at discrete equidistant time-steps tm, the pulses of the m-th periodic pulse sequence are applied to the second qubit in the m-th sub-register 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 m-th periodic pulsesequence. For example, the first and second periodic pulse sequences flip the measurement basis states of the first and second qubit, respectively. Accordingly, the sign of the interaction strength J12in equation (1) effectively changes from + to – and vice versa, if the first and second qubits are not flipped simultaneously. For example, the time averaged interaction strength between the first qubit and the second qubit takes the valueP2023,1425 WO N November 13, 2024- 12 - where Δt with T is the time delay and T is theperiod of the m-th periodic pulse sequence. Accordingly, the time averaged interaction strength can be continuously tuned between +J12and –J12by adjusting the time delay Δt accordingly.According to a further embodiment of the method, basis statesof each qubit correspond to different hyperfine states of atrapped ion. In particular, each qubit is encoded in twodifferent 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 qubitsin the quantum register interact via a magnetic gradientinduced coupling. For example, a magnetic field gradientalong a chain of trapped ions induces the pairwise Ising interaction between all pairs of qubits in the ion trap. For example, this Ising interaction in the presence of the magnetic field gradient is mediated by a common vibrationalmotion of the ions in the ion trap.Further a quantum computer is specified herein. In particular, the quantum computer implements the method for performing parallel quantum computations in a quantum register comprising qubits with an all-to-all interaction described above. All features of the method are also disclosed for the quantum computer and vice versa.According to an embodiment, the quantum computer has aquantum register comprising qubits interacting via an all-to-P2023,1425 WO N November 13, 2024- 13 -all interaction, and the quantum computer performs parallelquantum computations 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 pulse sequence, a second periodic pulse sequence, states of a qubit in a first sub-register and of a qubit in a second sub- register, as well as an interaction strength between the qubit in the first sub-register and the qubit in the second sub-register as a function of time according to an exemplary embodiment of the method for performing parallel quantum computations in a quantum register comprising qubits with an all-to-all interaction. Figure 2 shows a schematic graph of a time shifted periodic pulse sequences applied to a first qubit and a second qubit in a sub-register, states of the first qubit and a second qubit, as well as an interaction strength between the first qubit and the second qubit as a function of time according to an exemplary embodiment of the method for performing parallel quantum computations in a quantum register comprising qubits with an all-to-all interaction. 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 theP2023,1425 WO N November 13, 2024- 14 -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 qubits qb1, qb3 in a first sub-register31, and a second periodic pulse sequence 2 comprising pulses 21 applied to a qubits qb2, qb4 in a second sub-register 32of a quantum register of a quantum computer 10 according to an exemplary embodiment of the method. The period T2of the second periodic pulse sequence 2 is twice the period T1of the first periodic pulse sequence 1, i.e. T2 = 2·T1.All the qubits qb1, qb2, qb3, qb4 in the quantum register interact via a pairwise Ising interaction. In particular, a qubit qb1 in the first sub-register 31 and a qubit qb2 in the second sub-register 32 interact via an Ising interaction denotes the z-Pauli matrix acting on the qubit qb1 in the first sub-register 31, ^^^ = 0 −1^ denotes the z-Pauli matrix acting on the qubit qb2 in the second-sub-register 32, and J is the interaction strength.The pulses 11 of the first periodic pulse sequence 1 are pi- pulses that flip the basis states |0> and |1> of the qubits qb1, qb3 in the first sub-register 31 into each other. Inother words, after applying one pi-pulse 11 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. Similarly, the pulses 21 of the second periodicpulse sequence 2 are pi-pulses that flip the basis states |0>and |1> of the qubits qb2, qb4 in the second sub-register 32into each other. In other words, after applying one pi-pulseP2023,1425 WO N November 13, 2024- 15 -21 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 11 of the first periodic pulse sequence 1 aretwice as frequent as the pulses 21 of the second periodicpulse sequence 2. Accordingly, during the time intervalT2 = 2·T1 where the qubits qb1, qb3 in the first sub-register31 are flipped twice, the qubits qb2, qb4 in the second sub-register 32 are only flipped once by each respective pi-pulse11, 21. The second and third panels of Figure 2 show the expectation value〈^^^ 〉 for a qubit qb1 in the first sub-register 31 andthe expectation value〈^^^〉 for a qubit qb2 in the second sub-register 32, respectively, as a function of time t. In thisexample, the qubit qb1 in the first sub-register 31 and the qubit qb2 in the second sub-register 32 are both initialized in the basis state |1> at time t=0. Accordingly, both expectation values 〈^^^ 〉 and 〈^^^〉 are equal to one at time t=0.After applying the first pi-pulse 11 to the qubit qb1 in thefirst-sub-register 31, the state of the qubit qb1 changes to|0> and the expectation value changes accordingly to〈^^^ 〉 =−1. The second pi-pulse 11 flips the state of the qubit qb1 in the first sub-register 31 back to |1> and the expectation value〈^^^ 〉 changes accordingly. The same is true for thequbit qb2 in the second sub-register 32, albeit the stateflips only half as many times.The interaction energy ^ = 〈^ ^^^^^^〉 between the qubit qb1 inthe first sub-register 31 and the qubit qb2 in the second sub-register 32 as a function of time t is depicted in the lowest panel of Figure 1. The interaction energy is +J, ifP2023,1425 WO N November 13, 2024- 16 -both qubits qb1, qb2 are in the same state, e.g. both qubits qb1, qb2 are in the state |0> or both qubits qb1, qb2 are in the state |1>. The interaction energy is -J, if the qubits qb1, qb2 are in different states, e.g. the qubit qb1 in thefirst sub-register 31 is in state |0> whereas the qubit qb2in the second sub-register 32 is in state |1>, or vice versa. Accordingly, the time average of the interaction energy E over two periods 2·T2of the second periodic pulse sequence 2 vanishes or approximately vanishes, irrespective of a phase shift or time delay between pulses 11 of the first periodic pulse sequence 1 and pulses 21 of the second periodic pulsesequence 2. Time intervals between which the interactionenergy E mutually averages to zero are highlighted with the same shading in the lowest panel of Figure 1. In particular, the time dependence of the interaction energy E between times t = T2and t = 2·T2has the same functional form as the time dependence of the interaction energy E between times t = 0 and t = T2, albeit with opposite sign, such that these two contributions cancel in the time average. As the time average of the interaction energy E vanishes, thequbits qb1, qb3 in the first sub-register 31 effectively donot interact with the qubits qb2, qb4 in the second sub-register 32. By contrast, interactions between qubits qb1,qb3 or qb2, qb4 within each sub-register 31, 32 remain unaffected, if identical periodic pulse sequences 1, 2 areapplied to every qubit qb1, qb3 or qb2, qb4 in each sub-register 31, 32. Accordingly, the method described hereinallows to separate the qubits qb1, qb2, qb3, qb4 of a quantumregister with all-to-all interactions between the qubits qb1,qb2, qb3, qb4 into independent sub-registers 31, 32, andP2023,1425 WO N November 13, 2024- 17 -independent multi-qubit gates can be applied simultaneouslyto qubits qb1, qb3, or qb2, qb4 in each sub-register 31, 32.Figure 2 shows a first periodic pulse sequence 1 comprising pulses 11 applied to a first qubit qb1, and a time shifted first periodic pulse sequence 1 comprising pulses 11 appliedto a second qubit qb3 in the same sub-register 31 of aquantum computer 10 according to an exemplary embodiment of the method. The first periodic pulse sequence 1 and the timeshifted first periodic pulse sequence 1 have the same periodT as function of time t.The first qubits qb1 and the second qubit qb3 interact via anIsing interaction denotes the z- Pauli matrix acting on the first qubit qb1, ^^^ = denotes the z-Pauli matrix acting on the second qubit qb3,and J is the interaction strength.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 ^^^with eigenvalues -1 and +1, respectively. Similarly, the pulses 11of the time-shifted periodic pulse sequence 1 are pi-pulsesthat flip the basis states |0> and |1> of the second qubitqb3 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.P2023,1425 WO N November 13, 2024- 18 -The pulses 11 of the time-shifted first periodic pulsesequence 1 have a time delay Δt with respect to pulses 11 ofthe first periodic pulse sequence 1. Accordingly, the secondqubit qb3 is flipped at a different time t than the firstqubit qb1 by each respective pi-pulse.The second and third panels of Figure 2 show the expectationvalue〈^^^ 〉 for the first qubit qb1 and the expectation value the second qubit qb3, respectively, as a function oftime t. In this example, the first qubit qb1 and the secondqubit qb3 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 first qubit qb1 changes to |0> and the expectation value changes accordingly −1. The second pi-pulse flips the state of the firstqubit qb1 back to |1> and the expectation value〈^^^ 〉 changesaccordingly. The same is true for the second qubit qb3, albeit at later times delayed by the time delay Δt.The interaction energy ^ = between the first qubit qb1and the second qubit qb3 as a function of time t is depictedin the lowest panel of Figure 1. The interaction energy is +J, if both qubits qb1, qb3 are in the same state, e.g. both qubits are in the state |0> or both qubits are in the state |1>. The interaction energy is -J, if the first qubit qb1 andthe second qubit qb3 are in different states, e.g. the firstqubit qb1 is in state |0> whereas the second qubit qb3 is instate |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 between qubits qb1, qb3P2023,1425 WO N November 13, 2024- 19 -within one sub-register 31 can be tuned. In particular, thefrequency 1 / T of the first periodic pulse sequence 1 is muchlarger, 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 qb1 and the second qubit qb3 effectively interact via a time averaged Ising interaction with an effective interaction strength J that is a time average of the interaction energy Eover one period T of the periodic pulse sequence 1.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 qubits qb1,qb2, qb3, qb4 are divided into two virtual sub-registers 31,32 using the method described above, namely a first sub-register 31 comprising qubits qb1 and qb3, as well as asecond sub-register 32 comprising qubits qb2 and qb4.Accordingly, the interaction between qubits qb1, qb3 in thefirst sub-register 31 and qubits qb2, qb4 in the second sub- register 32 is averaged out to zero, while the qubits qb1,qb3 or qb2, qb4 within each of the two sub-registers 31, 32remain interacting. 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 inP2023,1425 WO N November 13, 2024- 20 -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.

[0002] P2023,1425 WO N November 13, 2024- 21 -References1 first periodic pulse sequence11 pulse2 second periodic pulse sequence21 pulse31 first sub-register32 second sub-register10 quantum computerqb1…4 qubitT1 period of the first periodic pulse sequenceT2 period of the second periodic pulse sequenceΔt time delayt timeJ interaction strengthE interaction energy

Claims

P2023,1425 WO N November 13, 2024- 22 -Claims 1. A method for performing parallel quantum computations in a quantum register comprising qubits (qb1, qb2, qb3, qb4) with an all-to-all interaction, comprising the steps of:- applying a first periodic pulse sequence (1) to qubits(qb1, qb3) in a first sub-register (31),- applying a second periodic pulse sequence (2) to qubits(qb2, qb4) in a second sub-register (32), wherein- the first periodic pulse sequence (1) and the secondperiodic pulse sequence (2) are adjusted such that a timeaveraged pairwise interaction between any qubit (qb1, qb3) inthe first sub-register (31) and any qubit (qb2, qb4) in the second sub-register (32) vanishes.

2. The method according to the previous claim, wherein thefirst periodic pulse sequence (1) and the second periodic pulse sequence (2) are adjusted such that a ratio between aperiod (T2) of the second periodic pulse sequence (2) and aperiod (T1) of the first periodic pulse sequence (1) is equalto a power of two.

3. The method according to any of the previous claims, wherein the pairwise interaction between the qubits (qb1, qb2, qb3, qb4) is an Ising or a XY interaction.

4. The method according to any of the previous claims, wherein- the first periodic pulse sequence (1) comprises pi-pulses(11), wherein each pi-pulse (11) flips the state of eachqubit (qb1, qb3) in the first sub-register (31), andP2023,1425 WO N November 13, 2024- 23 -- the second periodic pulse sequence (2) comprises pi-pulses(21), wherein each pi-pulse (21) flips the state of eachqubit (qb2, qb4) in the second sub-register (32).

5. The method according to any of the previous claims,comprising a further step of applying an entangling quantumgate to qubits (qb1, qb3) in the first sub-register (31) andapplying a further entangling quantum gate to qubits (qb2, qb4) in the second sub-register (32) in parallel at the same time.

6. 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 qubits (qb1, qb3) in the first sub-register (31) from decoherence, and / or- the second periodic pulse sequence (2) is a dynamicaldecoupling pulse sequence configured for protecting the qubits (qb2, qb4) in the second sub-register (32) from decoherence.

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

8. The method according to any of the previous claims, wherein- the method is configured to subdivide the quantum registerinto N independent sub-registers (31, 32), such that eachsub-register (31, 32) comprises an arbitrary number of qubits (qb1, qb2, qb3, qb4),P2023,1425 WO N November 13, 2024- 24 -- the qubits (qb1, qb2, qb3, qb4) in each sub-register (31,32) only interact with qubits (qb1, qb2, qb3, qb4) from thesame sub-register (31, 32), and- the method comprises the step of applying a first to a N-thperiodic pulse sequence (1, 2) to qubits (qb1, qb2, qb3, qb4)in the first to N-th sub-registers, respectively, wherein- a period (T2) of the m-th periodic pulse sequence is equalto 2m-1times the period (T1) of the first periodic pulsesequence (1), wherein the integer m with 1 ≤ m ≤ N enumeratesthe N sub-registers (31, 32).

9. The method according to any of the previous claims, wherein the sub-registers (31, 32) are virtual sub-registers,such that at least one qubit (qb1, qb2, qb3, qb4) can bedynamically assigned to different sub-registers (31, 32) at subsequent time intervals.

10. The method according to any of the previous claims, wherein an interaction strength (J) of the pairwise interaction between a first qubit (qb1) and a second qubit (qb3) within any sub-register (31) can be independently tunedby adjusting a time delay (Δt) between pulses (11) of theperiodic pulse sequence (1) applied to first qubit (qb1) andpulses (11) of the periodic pulse sequence (1) applied to thesecond qubit (qb3).

11. The method according to any of the previous claims, wherein basis states of each qubit (qb1, qb2, qb3, ab4) correspond to different hyperfine states of a trapped ion.

12. The method according to the previous claim, wherein the qubits (qb1, qb2, qb3, ab4) in the quantum register interact via a magnetic gradient induced coupling.P2023,1425 WO N November 13, 2024- 25 -13. A quantum computer (10) with a quantum register comprising qubits (qb1, qb2, qb3, ab4) interacting via an all-to-all interaction, wherein the quantum computer (10)performs parallel quantum computations using the methodaccording to any of claims 1 to 12.

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