Quantum computational method and apparatus for implementing a quantum operation

Abstract

The invention relates to a quantum computational method for implementing a quantum operation modifying a logical input state to obtain a logical output state, comprising the following steps: providing at least one input particle (2), wherein the logical input state is encoded into an internal state of the at least one input particle (2), wherein the internal state of the at least one input particle (2) is a superposition of at least two basis states; providing at least two component particles (3), wherein distinct component states chosen from a set of at least two component states are encoded into the respective internal states of each of the at least two component particles (3), wherein the set of component states is linearly independent, wherein the at least two component particles (3) are indistinguishable particles; performing a rearrangement operation controlled by the internal state of the at least one input particle (2), wherein, conditioned on each of the at least two basis states of the internal state of the at least one input particle (2), the rearrangement operation populates an output site (4) with a respective component particle (3) of the at least two component particles (3); outputting at least the internal state of the population (6) at the output site (4) as logical output state.

Description

[0001] QUANTUM COMPUTATIONAL METHOD AND APPARATUS FOR IMPLEMENTING A QUANTUM OPERATION FIELD OF THE INVENTION

[0002] Embodiments described herein relate to a quantum computational method and an apparatus for implementing a quantum operation modifying a logical input state to obtain a logical output state.

[0003] BACKGROUND OF THE INVENTION

[0004] Quantum computations make use of quantum mechanical effects to solve computational problems. Quantum computations are carried out on an apparatus comprising a quantum processing unit. The quantum processing unit may comprise particles, wherein logical quantum information may be encoded into the (physical) internal states of the particles. For example, a quantum processing unit may comprise electrostatically defined quantum dots, wherein the particles are spin particles, especially electrons or holes, and wherein the internal states of the particles are the up- and downspin states of the electrons or the holes. Another example of a quantum processing unit may comprise polarizable particles such as atoms or molecules trapped in optical potentials, for instance optical tweezers and / or optical lattices, wherein the internal states of the particles may be different electronic or magnetic states of the atoms or molecules.

[0005] Performing a quantum computation comprises implementing (typically several) quantum operations in the quantum processing unit. A quantum operation modifies a logical input state to obtain a logical output state. Especially, the quantum operation may be a unitary operation, such as an X gate or a CNOT gate or an operation creating entangled states. The implementation of a quantum operation on a quantum processing unit involves encoding the logical input state into the quantum processing unit (especially the internal states of the particles), applying physical operations on the particles of the quantum processing unit, and outputting a modified physical quantum state as logical output state.

[0006] To modify the internal state of a particle according to a unitary operation, in state-of-the-art implementation the internal state is rotated from the input internal state corresponding to the logical input state to the output internal state corresponding to the logical output state. “Rotating an internal state” means that the internal state is transformed from one internal state to another internal state such that a continuous or at least partially discontinuous trajectory of intermediate internal states is attained in course of performing the quantum operation. Which intermediate internal states are attained depends on the physical implementation of the quantum operation. When the (logical and internal)states are qubits, i.e., two-level quantum states, the rotation of the internal states corresponds to a rotation on the Bloch sphere.

[0007] For instance, performing an X gate on a particle in the |f ) state (defined to be on the north pole of the Bloch sphere) would typically involve a rotation from the |f) state (on the north pole of the Bloch sphere) to the |T) state (defined to be on the south pole of the Bloch sphere), wherein during performing the operation the (| f)+ 1 T)) / 2 state (on the equator of the Bloch sphere) or another state on the equator of the Bloch sphere is attained as intermediate internal state.

[0008] A disadvantage of implementing a quantum operation by rotating the internal state through intermediate states is that the intermediate states might have undesirable properties such as higher error rates.

[0009] Especially, this is relevant for biased-noise systems. Using biased-noise systems for fault-tolerant quantum computation can save resources needed for quantum error correction. In quantum error correction, the quantum information is redundantly encoded according to a quantum error correction code, which leads to an overhead of particles. For qubits, two types of errors need to be corrected: bit-flips and phase-flips. In a biased-noise system, bit-flip error rates may be small and phase-flip error rates may be large (or vice versa). The quantum error correction code can be adapted to the noise bias, having strong error correcting capabilities for phase-flip errors and weaker or no error correcting capabilities for bit-flip errors. By this adaptation, resources, that is, several particles, can be saved.

[0010] An example of a biased-noise system are spin particles in electrostatically defined quantum dots in semiconductors. The spin particles, e.g., electrons or holes, comprising spin-up and spin-down state as internal states (also referred to as spin qubits) experience a strong noise bias towards dephasing. That means that the error rate of an erroneous transition from spin-up to spin-down states (bit-flips) is much smaller than the error rate of an erroneous transition from the even superposition of spin-up and spin-down states to the odd superposition of spin-up and spin-down states (phase-flips). A further example of biased-noise systems are some internal states in atoms, molecules or ions.

[0011] To make use of the resource-saving adaptation of error correction codes to the noise bias of the internal states of the particles, one needs to be able to implement noise-bias-preserving quantum operations, since otherwise the noise bias may be destroyed by applying a quantum operation and the quantum error correction code is maladapted. For qubits, a noise-bias-preserving quantum operation does not translate bit-flips into phase-flips, or vice versa, at least approximately.In noise-biased systems, an intermediate internal state attained during an implementation of a quantum operation may be subject to a stronger noise and a higher error rate than the input internal state and the output internal state, which are subject to weaker noise and thus a lower error rate. Since an error flipping the intermediate internal state may translate into an error in the output internal state, the output state becomes subject to the stronger noise and the higher error rate of the intermediate internal states. Applying a quantum operation by rotating the internal state is thus not always a noise-bias-preserving quantum operation.

[0012] A further disadvantage of the implementation of a quantum operation by the rotation of the internal state is that rotations of the internal state are - in some physical systems - slow or technically not or only hardly feasible.

[0013] OBJECT OF THE INVENTION

[0014] An object of the invention is to provide a quantum computational method and an apparatus for implementing quantum operations avoiding rotations of internal states.

[0015] SUMMARY OF THE INVENTION

[0016] The object is solved by the quantum computational method according to claim 1 and by the apparatus according to claim 19 or 20.

[0017] According to the invention, a quantum computational method for implementing a quantum operation modifying a logical input state to obtain a logical output state comprises the following steps:

[0018] • providing at least one input particle, wherein the logical input state is encoded into an internal state of the at least one input particle, wherein the internal state of the at least one input particle is a superposition of at least two basis states;

[0019] • providing at least two component particles, wherein distinct component states chosen from a set of at least two component states are encoded into the respective internal quantum states of each of the at least two component particles, wherein the set of component states is linearly independent, wherein the at least two component particles are indistinguishable particles; • performing a rearrangement operation controlled by the internal state of the at least one input particle, wherein, conditioned on each of the at least two basis states of the internal quantumstate of the at least one input particle, the rearrangement operation populates an output site with a respective component particle of the at least two component particles;

[0020] • outputting at least the internal state of the population at the output site as logical output state.

[0021] Thereby, a quantum operation modifying a logical input state to obtain a logical output state can be implemented without rotating the internal state of the at least one input particle. Instead of rotating the internal state of the at least one input particle and outputting the rotated internal state of the at least one input particle, the internal state of the at least one input particle controls a rearrangement operation. The rearrangement operation, in turn, populates an output site with a respective component particle of at least two component particles. Thereby, a component particle may be transferred to the output site or kept at the output site, i.e., an external degree of freedom is changed, but their internal states do not need to be rotated. Since populating the output site with the at least two component particles is controlled by the basis states of the internal state of the at least one input particle, the population at the output site contains information about the internal state of the at least one input particle, that is, about the logical input state. Since distinct component states chosen from a set of at least two component states are encoded into the respective internal states of each of the at least two component particles before populating the output site with the at least two component particles, a new internal state carried by the population at the output site can be composed. Due the indistinguishability of the at least two component particles the population at the output site corresponds to one “composed” particle (where “composed” refers to a superposition from two origins). The population at the output site (i.e., the “composed” particle), and optionally the population on further sites, can be used as output of the quantum operation, especially its internal state represents the logical output state of the quantum operation.

[0022] In short, the concept according to the invention can be used to implement a quantum operation modifying a logical input state to a logical output state by a controlled population of an output site with indistinguishable particles, without the necessity of rotating the internal state which encodes the logical quantum information.

[0023] In the following the underlying physical principles of the invention are explained in more detail.

[0024] According to the invention, the at least two component particles are indistinguishable particles. When at least two indistinguishable particles populate the (same) output site, i.e., are in the same external quantum state, such that the total probability of having a particle at the output site is one, the population at the output site corresponds to one “composed” particle at the output site. Spin particles,e.g., electrons or holes, in electrostatically defined quantum dots may be indistinguishable particles. The output site may correspond to a quantum dot, which may be realized by an electrostatic potential well in a (semiconductor) heterostructure defined by metallic gates. When the same electrostatic potential well is populated by at least two spin particles, the at least two spin particles may attain the same external quantum state and form a population of the electrostatic potential well. Polarizable particles, e.g., atoms or molecules, trapped in optical potential wells (such as optical tweezers or optical lattices) may be indistinguishable particles. The output site may be defined by an optical potential well created by an optical tweezer or an optical lattice. When the same optical potential well is populated by at least two polarizable particles, the at least two polarizable particles may attain the same external quantum state and form a population of the optical potential well. Preferably, only the orbital ground state of said quantum dots or optical potential wells is populated. The at least one input particle may be indistinguishable from the at least two component particles in some embodiments but may be distinguishable from the at least two component particles (for instance, it might be a different species or an artificial particle) in other embodiments.

[0025] The internal state of the population (the “composed” particle) at the output site may be determined by a superposition of the respective internal states of the at least two component particles used for populating the output site (i.e., of the particles the “composed” particle is composed of). Since, according to the invention, distinct component states chosen from a set of at least two component states are encoded into the respective internal states of each of the at least two component particles, the internal state of the population at the output site is determined by a superposition of the at least two component states. Further since, according to the invention, the set of component states is linearly independent, any internal state of the population at the output site can be attained by superposing the at least two component states. For example, for qubits (two-state Hilbert space), there may be two (orthogonal) component states |f) and |T) which create a linearly independent set {| f), |T)}. Any qubit state can be created by a superposition a |f) + <z2|h) with complex amplitudes a and a2and | < Zi |2+ \a2\2= 1. The state |T) may be a spin-up state of a spin-1 / 2 particle, wherein the state |f) may be a spin-down state of a spin-1 / 2 particle. In another system, states |T) and |f) may be two orthogonal internal states or two sets of internal states of a more complex level structure of a polarizable particle such as an atom, for instance.

[0026] The rearrangement operation creates the desired population at the output site. The final state of the quantum operation depends on the logical input state (since the quantum operation modifies the logical input state), which is encoded into the internal state of the at least one input particle. For this reason, according to the invention, the rearrangement operation is controlled by the internal state ofthe at least one input particle. That means that, conditioned on the basis states of the internal state of the at least one input particle, the rearrangement operation populates the output site with a respective component particle of the at least two component particles. Thereby, the rearrangement operation may create entanglement between the basis states of the internal state of the at least one input particle and a superposition of arrangements of the at least two component particles, wherein in each arrangement associated with a respective basis state, a respective component particle populates the output site (while the other component particles do not populate the output site). Hence by the rearrangement operation, the probability amplitudes of the at least two basis states of the at least one input particle are translated to probability amplitudes of arrangements of the at least two component particles (potentially up to correctable phase factors). Since within each arrangement exactly one of the at least two component particles may populate the output site and since each arrangement has a probability amplitude corresponding to one of the basis states of the internal state of the input particle, a respective component particle populates the output site with probability amplitudes corresponding to those of the basis states of the internal state of the input particle. The absolute squares of the probability amplitudes (i.e., the probabilities) sum up to one, so the total probability of having a particle at the output site is one. Hence, there may be a population of one particle (one “composed” particle) at the output site after applying the rearrangement operation.

[0027] The rearrangement operation populates the output site with a respective component particle of the at least two component particles in such a way that, given the respective component states and the quantum operation to be implemented, the population at the output site after performing the rearrangement operation carries at least a part of the logical input state modified by the quantum operation (i.e., the desired logical output state), at least approximately. The exact operations defining how the output site is populated with the at least two component particles depend on the quantum operation to be implemented, the choice of the component states, and the type of the quantum processing unit (spin particles or polarizable particles, for instance). Different detailed implementations of the rearrangement operation are given in the embodiments of the invention.

[0028] An important technical feature of the quantum mechanical method for implementing a quantum operation according to the invention is that the rearrangement operation can be performed essentially without rotating the internal states, in which the logical states are encoded, of neither the at least one input particle (used for control in the rearrangement operation) nor the at least two component particles (with which the output site is populated by the rearrangement operation). This, for example, allows for implementing a noise-bias-preserving quantum operation for biased-noise quantumsystems, since noise channels are not mixed by rotating internal states. In some cases, a rearrangement operation may be implementable faster than rotations of the internal states.

[0029] In the context of the current application, “outputting at least the internal state of the population at the output site as logical output state” means that the logical output state is at least partially encoded in the internal state of the population at the output site. The invention also comprises the case where a many-particle state, especially a two-particle state, comprising the internal state of the population at the output site is outputted as logical output state - but at least the internal state of the population at the output site must be part of the logical output state. In other words, the logical output state may be the combined state of the population at the output site and at least one further site, for instance a site involving the at least one input particle.

[0030] “Outputting” does not necessarily mean that the state is measured immediately after applying the operation, for instance to use information about the logical output state in a (classical) logical unit. Additionally or alternatively, “outputting” may mean that the population at the output site is used in a subsequent quantum operation. For instance, the subsequent quantum operation may be applied according to a quantum circuit specifying the quantum computation.

[0031] Preferably, a particle populating a site may mean that the particle attains a ground state and / or any other state in a potential well in position space, such as an optical potential well or a quantum dot (an electrostatic potential well, known as electrostatic confinement potential), so a site may be identified with a certain potential well in position space. Alternatively, a particle populating a site may mean that the particle attains a ground state and / or any other state in a potential well in momentum space. The potential well may be physically realized by a site of a momentum space lattice created by counterpropagating lasers, for instance. More generally, a particle populating a site may mean that the particle attains a specific external state or specific subset of external states. Different sites are distinguishable (while different component particles and, optionally, input particles and further particles are indistinguishable).

[0032] “Rearrangement” means a rearrangement of particles between different sites, i.e., involves a change of the external states of at least some of the particles. It may mean spatial rearrangement, but in other embodiments it may mean a more general rearrangement among different external states, for instance a rearrangement of particles among different momentum states. Likewise, an “arrangement” may be an arrangement of particles between different sites, e.g., in position space or momentum space.The quantum operation may be given as a mapping between basis states and component states. In other words, a mapping which maps each basis state to one of the component particles, wherein the mapping is based on the quantum operation to be implemented, may be provided. The mapping may be stored in a (classical) logical unit and trigger a sequence of control signals of a (classical) control unit which control the quantum processing unit to perform the quantum operation. Then, conditioned on a respective basis state, the rearrangement operation populates the output site with the respective component particle to which the respective basis state is mapped by the mapping.

[0033] For populating the output site, the rearrangement operation may, conditioned on each of the basis states of the internal state of the at least one input particle,

[0034] • transfer the respective component particle of the at least two component particles to the output site (when the respective component particle was initially not at the output site), or

[0035] • keep the respective component particle at the output site (when the respective component particle was initially at the output site), and

[0036] • optionally transfer at least one further component particle of the at least two component particles out of the output site (when the further component particle was initially at the output site).

[0037] There may be one input particle, wherein the at least two basis states are single-particle states. Alternatively, there may be at least two input particles, wherein the at least two basis states are composite basis states of the respective single-particle basis states of the individual input particles.

[0038] An apparatus for implementing a quantum operation modifying a logical input state to obtain a logical output state according to the invention comprises:

[0039] • a quantum processing unit comprising a plurality of sites,

[0040] • a control unit for providing and rearranging the population in the plurality of sites,

[0041] • a logical unit connected to the control unit.

[0042] The logical unit is configured to instruct the control unit to implement the quantum operation modifying the logical input state to obtain the logical output state. The logical unit may comprise a mapping which maps each basis state to one of the component particles, wherein the mapping is based on the quantum operation to be implemented.

[0043] Implementing the quantum operation comprises:• providing, using the control unit, the at least one input particle in at least one input site of the quantum processing unit, wherein the logical input state is encoded into an internal state of the at least one input particle, wherein the internal state of the at least one input particle is a superposition of at least two basis states;

[0044] • providing, using the control unit, at least two component particles in at least one component site of the quantum processing unit, wherein distinct component states chosen from a set of at least two component states are encoded into the respective internal states of each of the at least two component particles, wherein the set of component states is linearly independent, wherein the at least two component particles are indistinguishable particles;

[0045] • performing, using the control unit, a rearrangement operation controlled by the internal state of the at least one input particle, wherein, conditioned on each of the at least two basis states of the internal state of the at least one input particle, the rearrangement operation populates an output site with a respective component particle of the at least two component particles;

[0046] • outputting at least the internal state of the population at the output site as logical output state.

[0047] A register of sites may comprise the plurality of sites used to implement the quantum operation. The apparatus may comprise several registers.

[0048] The sites of the quantum processing unit may be electrostatically defined quantum dots and the at least one input particle and the at least two component particles may be spin particles, preferably electrons or holes, confined in the electrostatically defined quantum dots.

[0049] The sites of the quantum processing unit may be optical potential wells and the at least one input particle and the at least two component particles may be polarizable particles, preferably atoms, trapped in the optical potential wells.

[0050] BRIEF DESCRIPTION OF THE DRAWINGS

[0051] A full and enabling disclosure to one of ordinary skill in the art is set forth more particularly in the remainder of the specification including reference to the accompanying drawings, wherein:

[0052] Fig. 1 shows a schematic representation of the arrangement of an input particle and two component particles before and after performing the rearrangement operation for implementing an X-gate applied to a qubit.Fig. 2 shows a schematic representation of the arrangement of an input particle and three component particles before and after performing the rearrangement operation for implementing an exemplary qutrit gate.

[0053] Fig. 3 shows a schematic representation of the arrangement of two input particles and two component particles before and after performing the rearrangement operation for implementing a CNOT gate.

[0054] Figs. 4a, 4b and 4c show schematic representations of the arrangement of one input particle and two component particles before and after performing different embodiments of the rearrangement operation for implementing an X-gate by populating the output site with component particles using direct intemal-state-dependent interaction between input particle and component particles.

[0055] Figs. 5a and 5b show schematic representations of the arrangement of one input particle and two component particles before and after performing different embodiments of the rearrangement operation for implementing an X-gate, by translating the internal state of the input particle into an external state by using an internal-state-dependent transfer and populating the output site with component particles depending on the external state of the input particle.

[0056] Figs. 6a, 6b and 6c show schematic representations of the arrangement of one input particle and two component particles before and after performing different embodiments of the rearrangement operation for implementing an X-gate, by translating the internal state of the input particle into an external state by using the Pauli spin blockade for the intemal-state-dependent transfer and populating the output site with component particles depending on the external state of the input particle.

[0057] Fig. 7 shows individual steps of an embodiment of the rearrangement operation corresponding to Fig.

[0058] 6a implementing the X-gate.

[0059] Fig. 8 shows individual steps of an embodiment of the rearrangement operation corresponding Fig.

[0060] 6b implementing the X-gate.

[0061] Fig. 9 shows individual steps of an embodiment of the rearrangement operation corresponding to Fig.

[0062] 6c implementing the X-gate.Fig. 10 shows an embodiment of the rearrangement operation for implementing a CNOT-gate using the Pauli spin blockade as internal-state-dependent transfer, comprising eight sites, preferably for spin particles in electrostatically defined quantum dots.

[0063] Fig. 11 shows the individual steps of the operation on the right four-site register of the embodiment ofFig. 10.

[0064] Fig. 12 shows an alternative embodiment of the rearrangement operation for implementing a CNOT-gate using the Pauli spin blockade as intemal-state-dependent transfer, comprising nine sites, preferably for spin particles in electrostatically defined quantum dots.

[0065] Fig. 13 shows the individual steps of the operation on the right five-site register of the embodiment ofFig. 12.

[0066] Fig. 14 shows an embodiment of the rearrangement operation for implementing a CNOT-gate using an alternative internal-state-dependent transfer, preferably via internal-state-dependent shuttling and van-der-Waals interaction for polarizable particles in optical potential wells.

[0067] Fig. 15a shows a schematic representation of spin particles in electrostatically defined quantum dots.

[0068] Fig. 15b shows the energy level scheme of a spin particle in two quantum dots.

[0069] Fig. 16 shows the energy level scheme of zero, one and two spin particles in two sites depending on the energy bias detuning between the sites.

[0070] Fig. 17 shows a simplified level scheme of the physical internal states of a polarizable particle which may be used for intemal-state-dependent shuttling and intemal-state-dependent van-der-Waals interactions.

[0071] Figs. 18a and 18b show a scheme for internal-state-dependent shuttling.

[0072] Fig. 19 shows a schematic depiction of an apparatus for implementing the quantum operation.

[0073] A, General descriptionA logical input state encoded into the internal state of the at least one input particle may be written as \Bi). The states {|B,)}“1are linearly independent basis states of the at least one input particle. For a single input particle, these are single-particle states and M=N for a Hilbert space with N states (for qubits N=2). For several input particles K, these are many -particle states and M = NK(for qubits and two input particles: M=4).

[0074] The quantum state of the register after applying the rearrangement operation may be given by lG)olei)r, wherein L is smaller than or equal to M. The states {|Q)0}f=iarethe linearly independent component states of the component particles, wherein the output site o was populated with these particles. The states | et )rare the state of the rest of the subsystem involved in the operation, i.e., outside of the output site (involving the sites of the at least one input particle and of the at least two component particles which are not at the output site). By the rearrangement operation the population of the output site and the rest of the system becomes entangled. It may be that a further site other than the output site (which is populated by the “composed” particle originating from component particles) is taken as logical output of the operation.

[0075] A quantum operation may be given by a mapping from the states {|B,)}“1to states {| )}{=1. The mapping for a specific quantum operation depends on the relation of the states.

[0076] In a preferred case, particularly where there is one single input particle, at least one, preferably each, of the basis states of the internal state of the at least one input particle is physically alike to a respective component state of one of the at least two component particles. That is, |B£) = |C£) for each i. Additionally assuming a qubit as input particle, one may have {IQ), |B2)} = {IQ), |Q)} = {|f), |T)} (here M=L=2). For instance, one may use spin-up states and spin-down states of spin particles for input particle as well as for the component particles.

[0077] When there are several input particles, the internal states of the individual input particles may be superpositions of at least two single-particle basis states, and at least one, preferably each, of the single-particle basis states of the internal states of the individual input particles may be physically alike to a respective component state of one of the at least two component particles. For two qubits as input particles, one may have the (composite) basis states

[0078]

[0079] |B2), |B3), |B4)} = {|ff), |fT), |T f), |ff)J, wherein the individual input particles have the single-particle basis states {|f), |T)}. The component states may be physically alike, i.e., {I ), | C2)} = {|f), |T)} (here M=4 and L=2). For instance, one may use spin-up states and spin-down states of spin particles for input particle as well as for the component particles.We will restrict the discussion in the following mainly on the cases described in the previous two paragraphs where the basis states or single-particle basis states are physically alike to the component states without limiting the scope of the application.

[0080] We define the logical meaning with respect to the standard notation in quantum informatics as |0) = |f) and |1) = |T) for all embodiments involving spin qubits. The gates (X gate, CNOT gate and so on) are defined in the standard way in the following.

[0081] A,1 Example for a single-qubit unitary: X gate

[0082] Fig. 1 shows a schematic representation of the arrangement of an input particle 2 and two component particles 3 before (initial arrangement 8) and after (superposition of modified arrangements 9) performing the rearrangement operation for implementing an X-gate applied to a qubit. That is, a physical implementation of the X-gate (a quantum operation) modifying the logical input state | / ) = \q) = a |f) + <z2|T) to the logical output state |O) = \Xq) = <z2|f) +ai ) avoiding rotations of internal states is shown.

[0083] The top part of Fig. 1 shows an overview formula of the effect of the quantum operation. The logical input state is encoded into the input particle 2 forming the population at the input site 5 denoted by the index i: \q)t= oq |f ); + a2|T)i. The logical output state is, after the application of the rearrangement operation, given by the internal state of the population 6 at the output site 4 denoted by the index o: |Xq)0= <z2|f)0+ffi l^o-

[0084] The bottom part of Fig. 1 shows an example of a physical implementation from an initial arrangement 8 of a register 1 of a quantum processing unit 20 to a superposition of modified arrangements 9 of the register 1 due to the application of a rearrangement operation. By the rearrangement operation the quantum operation, here the X gate, is implemented. The probability amplitudes alta2of the modified arrangements 9 in the superposition are shown above the arrows from the initial arrangement 8 to the modified arrangements 9. The output site 4 is depicted by a dashed square in all arrangements 8, 9.

[0085] The arrangements 8, 9 may be spatial arrangements between different spatial sites, for instance different potential wells. Alternatively, the arrangements 8, 9 may be more general arrangements ofparticles populating external states, wherein the sites may be different potential wells of a momentum space lattice, for instance.

[0086] Two distinct component states |Q) = |f) and |Q) = |T) are encoded into the respective internal states of each of the two component particles 3. The set of component states {|Q)> |Q)} = {|f)> |T)} is linearly independent, especially the component states are orthogonal. In the example of Fig. 1, the two component particles 3 populate a respective component site 7.

[0087] The rearrangement operation is controlled by the internal state of the input particle 2. Conditioned on each of the two basis states |Q) = |4) and |B2) = |T) of the internal state of the input particle 2, the rearrangement operation populates an output site with a respective component particle 3. In this example, the output site 4 is populated by transferring the component particles 3 to the output site 4, which is depicted by arrows in Fig. 1. The transfer of the component particles 3 may be conditioned on the basis states of the internal state of the input particle 2 by an interaction between the input particle 2 at the input site 5 and the component particles 3 at the component sites 7, which is depicted by zig-zag lines in Fig. 1. The interaction may be a intemal-state-dependent van-der-Waals interaction between polarizable particles in this example, but it can in general be any suitable physical interaction. Technical details on the implementation will be explained at a later section in this application.

[0088] Summarizing, a first component particle 3 is prepared in the first basis state |Q) = |4) (i.e., the component state of the first component particle 3 is |Q) = |4) ) and a second component particle 3 is prepared in the second basis state |B2) = |T) (i.e., the component state of the second component particle 3 is |Q) = |T)), wherein performing the rearrangement operation comprises: conditioned on the first basis state |Q ) of the internal state of the input particle 2, populating the output site 4 with the second component particle 3 (carrying states |B2) = |Q)), and / or conditioned on the second basis state |B2) of the internal state of the input particle 2, populating the output site 4 with the first component particle 3 (carrying states |Q) = IQ)).

[0089] The conditional transfer of the component particles 3 causes a superposition of a first and a second modified arrangement 9, wherein the first (upper) modified arrangement 9 is conditioned on the basis state |Q) = |4) and a second (bottom) modified arrangement 9 is conditioned on the basis state |B2) = |T) of the internal state of the input particle 2. The first modified arrangement 9 was created by transferring the component particle 3 with the component state |Q) = |T) to the output site 4, thereby populating the output site 4. The second modified arrangement 9 was created bytransferring the component particle 3 with the component state I ) = | X) to the output site 4, thereby populating the output site 4. This logic has been chosen according to a mapping for the X-gate, the quantum operation to be implemented. The mapping maps each basis state |Q) = |4) and |B2) = |f) to one of the component particles 3 with respective component states |Q) = |f) and IQ)=| ). Conditioned on a respective basis state, the rearrangement operation populates the output site 4 with the respective component particle 3 to which the respective basis state is mapped by the mapping. For the X-gate, the mapping is given by |Q) = |4) -> |Q) = |f) and |B2) = |T) -> IQ) = 14). By choosing other states for basis states and / or component states, other gates may be implemented by using the method according to the invention.

[0090] The resulting superposition of modified arrangements 9 comprises the desired logical output state of the X gate as the internal state of the population 6 at the output site 4 (within dashed square in Fig. 1) |Xq)0= a214)0+ |T)0- The final step is thus outputting the internal state of the population 6 at the output site 4 as logical output state.

[0091] Note that the internal state of the population 6 at the output site 4 is still entangled with the population at the other sites, i.e., the full physical state is «2|T)(|X)CI |T)c2|f)0+ 11Q| 4)cl\x)c21 T)o, wherein cl and c2 denote the component site 7 which was initialized with the component particle 3 carrying component state |Q) = |4) (lower in Fig. 1) and |Q) = |T) (upper in Fig. 1), respectively. |x) denotes an empty site. For further computations the populations at the other site should be protected against noise or discarded by performing a discarding operation.

[0092] Optionally, a discarding operation can be performed to the modified arrangements 9 to disentangle the internal state of the population 6 at the output site 4 from the population in the rest of the register 1 of the quantum processing unit, i.e., the population at the input site 5 and the component sites 7. Alternatively, performing the discarding operation may involve a measurement of the internal state of the population 6 at the output site 4, if a measurement is anyways intended after performing the X gate according to a quantum circuit to be implemented. Especially, a Z measurement of the internal state of the population 6 at the output site 4 may lead to a collapse of the superposition statea2 )i lx)ci )c2 l^)o +ail' )il' )cilx)c2 l1')o to one of its constituents, rendering a further disentangling operation obsolete.

[0093] Note that the rearrangement operation described above may additionally add known phases to the individual modified arrangements 9 due to the nature of the physical implementation. These phasesmay be compensated for as long as they are known. An explicit example for phases is given below for the example of spin particles.

[0094] A.2 Example for a single-qutrit unitary

[0095] Fig. 2 shows a schematic representation of the arrangement of an input particle 2 and three component particles 3 before (initial arrangement 8) and after (modified arrangement 9) performing the rearrangement operation for implementing an exemplary qutrit gate denoted by G. A qutrit comprises three levels. That is, a physical implementation of said qutrit gate (a quantum operation) modifying the logical input state | / ) = |q3) = a010) + a 11) + a2{2) to the logical output state |O) = | Gq3) = <z2|0) + cr0|l) +ai|2) avoiding rotations of internal states is shown. Hereby the states |0), |1), |2) form a linearly independent set.

[0096] The presentation of Fig. 2 resembles the scheme of Fig. 1. The logical input state is encoded into the input particle 3 which is situated at the input site 5, which has the initial state \q3)t = a0|0)j + «i|l>i + a2|2>f.

[0097] The top part of Fig. 2 shows an overview formula of the effect of the quantum operation. The logical input state is encoded into the input particle 2 forming the population at the input site 5 denoted by the index i: |q3)j = a 10)^ + a211); + a312)^. The logical output state is, after the application of the rearrangement operation, given by the internal state of the population 6 at the output site 4 denoted by the index o: | Gq3)0= a310)o+ < Zi|l)0+ a2{2)0.

[0098] The bottom part of Fig. 2 shows an example of a physical implementation from an initial arrangement 8 of a register 1 of a quantum processing unit 20 to a superposition of modified arrangements 9 of the register 1 due to the application of a rearrangement operation. By the rearrangement operation the quantum operation, here the exemplary qutrit gate, is implemented. The probability amplitudes a1, a2, a3of the modified arrangements 9 in the superposition are shown above the arrows from the initial arrangement 8 to the modified arrangements 9. The output site 4 is depicted by a dashed square in all arrangements 8, 9.

[0099] The arrangements 8, 9 may be spatial arrangements between different spatial sites, for instance different potential wells. Alternatively, the arrangements 8, 9 may be more general arrangements of particles populating different external states, wherein the sites may be different potential wells of a momentum space lattice, for instance.Three distinct component states |CX) = |0), |C2) = |1) and |C3) = |2) are encoded into the respective internal states of each of the three component particles 3. The set of component states {I ), |C2), IQ)} = {|0>, |1), 12)} is linearly independent, especially the component states may be orthogonal. In the example of Fig. 2, the three component particles 3 populate a respective component site 7.

[0100] The rearrangement operation is controlled by the internal state of the input particle 2. Conditioned on each of the three basis states |Q ) = |0), |B2) = |1) and |B3) = |2) of the internal state of the input particle 2, the rearrangement operation populates an output site with a respective component particle 3. In this example, the output site 4 is populated by transferring the component particles 3 to the output site 4, which is depicted by arrows in Fig. 2. The transfer of the component particles 3 may be conditioned on the basis states of the internal state of the input particle 2 by an interaction between the input particle 2 at the input site 5 and the component particles 3 at the component sites 7, which is depicted by zig-zag lines in Fig. 2.

[0101] Summarizing, a first component particle 3 is prepared in the first basis state |Q ) = |0) (i.e., the component state of the first component particle 3 is |Q) = |0)), a second component particle 3 is prepared in the second basis state |B2) = |1) (i.e., the component state of the second component particle 3 is |C2) = |1)) and a third component particle 3 is prepared in the third basis state |B3) = |2) (i.e., the component state of the third component particle 3 is |C3) = |2)), wherein performing the rearrangement operation comprises: conditioned on the first basis state |BX) of the internal state of the input particle 2, populating the output site 4 with the second component particle 3 (prepared in states |B2) = |C2)), and / or conditioned on the second basis state |B2) of the internal state of the input particle 2, populating the output site 4 with the third component particle 3 (prepared in states |B3) = |C3)), and / or conditioned on the third basis state |B3) of the internal state of the input particle 2, populating the output site 4 with the first component particle 3 (prepared in states |Q ) = |Q )).

[0102] The conditional transfer of the component particles 3 causes a superposition of a first, a second and a third modified arrangement 9, wherein the first (upper) modified arrangement 9 is conditioned on the basis state |BX) = |0), the second (central) modified arrangement 9 is conditioned on the basis state |B2) = |1) and the third (bottom) modified arrangement 9 is conditioned on the basis state |B3) = 12) of the internal state of the input particle 2. The first modified arrangement 9 was created by transferring the component particle 3 with the component state |C2) = 11) to the output site 4, thereby populating the output site 4. The second modified arrangement 9 was created bytransferring the component particle 3 with the component state |C3) = 12) to the output site 4, thereby populating the output site 4. The third modified arrangement 9 was created by transferring the component particle 3 with the component state |CX) = |0) to the output site 4, thereby populating the output site 4. This logic has been chosen according to the mapping for the exemplary qutrit gate, the quantum operation to be implemented. The mapping maps each basis state |BX) = |0>, |B2) = |1) and |B3) = |2) to one of the component particles 3 with respective component states i) = |0), |C2) = |1) and |C3) = |2). Conditioned on a respective basis state, the rearrangement operation populates the output site 4 with the respective component particle 3 to which the respective basis state is mapped by the mapping. For the exemplary qutrit gate, the mapping is given by |BX) = |0) |C2) = |1), |B2) = |1) |C3) = |2>, and |B3) = |2) |CX) = |0). By choosing other states for basis states and / or component states, other gates may be implemented by using the method according to the invention.

[0103] The resulting superposition of modified arrangements 9 comprises the desired logical output state of the exemplary qutrit gate as the internal state of the population 6 at the output site 4 (within dashed square in Fig. 2) |Gq3)0= <z3|0)o+ cr l^ + a2l2)0. The final step is thus outputting the internal state of the population 6 at the output site 4 as logical output state.

[0104] Note that the internal state of the population 6 at the output site 4 is still entangled with the population at the other sites, i.e., the full physical state is tr3|2) x)cl|l)c2|2)c3|0)o+ai|0)i|0)ci|x)c2|2)c3|l)o+ tr2|l)i|0)cl|l)c2|x)c3|2)o, wherein cl, c2 and c3 denote the component site 7 which was initialized with the component particle 3 carrying component state |CX) = |0), |C2) = |1) and |C3) = |2), respectively. |x) denotes an empty site. For further calculations the populations at the other site should be protected against noise or discarded by performing a discarding operation.

[0105] Optionally, a discarding operation can be performed to the modified arrangements 9 to disentangle the internal state of the population 6 at the output site 4 from the population in the rest of the register 1 of the quantum processing unit, i.e., the population at the input site 5 and the component sites 7. Alternatively, performing the discarding operation may involve a measurement of the internal state of the population 6 at the output site 4, if a measurement is anyways intended after performing the exemplary qutrit gate according to a quantum circuit to be implemented. Especially, a projective measurement onto the states |0), 11) and |2) of the internal state of the population 6 at the output site 4 may lead to a collapse of the superposition state to one of its constituents, rendering a further disentangling operation obsolete.Note that the rearrangement operation described above may additionally add known phases to the individual modified arrangements 9 due to the nature of the physical implementation. These phases may be compensated for as long as they are known.

[0106] Note that the method according to the invention can also be used for quantum operations acting on qudits, i.e., arbitrary d-level states with d = 2, 3, 4, 5, 6, and so on. For a general single-qudit operation one would need d component particles 3, each being prepared in a distinct component state, wherein the component states form a linearly independent set. In special cases fewer component particles may suffice.

[0107] A.3 Example for a two-qubit unitary: CNOT gate

[0108] Fig. 3 shows a schematic representation of the arrangement of two input particles 2 and two component particles 3 before (initial arrangement 8) and after (superposition of modified arrangements 9) performing the rearrangement operation for implementing a CNOT gate applied to a control qubit ql and a target qubit q2. That is, a physical implementation of CNOT gate (a quantum operation) modifying the logical input state | / ) = |QI, Q2)=«i|H) + a2|1 T) + <z3|T4) + <z4|TT) to the logical output state |O) = |CNOT(q1, q2)) = cti|W) + cr2l' ) + <z4|Tf) + cr3|TT) avoiding rotations of internal states is shown.

[0109] The top part of Fig. 3 shows an overview formula of the effect of the quantum operation. The logical input state is encoded into the two input particles 2 forming the population at the two input sites 5 denoted by the indices il and i2: |Qi, Q2>i1,t2

[0110]

[0111] =ai + <

[0112]

[0113] z4|TT)i1;i2. The logical output state is, after the application of the rearrangement operation, given by the internal state of the population 6 at the output site 4 denoted by the index o and a further site, namely the first input site 5 denoted by index il:

[0114]

[0115] |CNOT(q1, q2))i1,0= cti|W)£10+ a2|4T)£1 0+ c

[0116]

[0117] r4|Tf)£ij0+ a3|TT>£10. This is an example where the logical output state is not solely given by the internal state of the population 6 at the output site 4 but comprises the internal state of the population 6 at the output site 4.

[0118] The bottom part of Fig. 3 shows an example of a physical implementation from an initial arrangement 8 of a register 1 of a quantum processing unit 20 to a superposition of modified arrangements 9 of the register 1 due to the application of a rearrangement operation. By the rearrangement operation the quantum operation, here the CNOT gate, is implemented. Theprobability amplitudes alta2, a3, <z4of the modified arrangements 9 in the superposition are shown above the arrows from the initial arrangement 8 to the modified arrangements 9. The output site 4 is depicted by a dashed square in all arrangements 8, 9. The input sites 5 are schematically depicted to be in a solid rectangle in the initial arrangement 8 since the interaction depends on both states.

[0119] The arrangements 8, 9 may be spatial arrangements between different spatial sites, for instance different potential wells. Alternatively, the arrangements 8, 9 may be more general arrangements of particles populating different external states, wherein the sites may be different potential wells of a momentum space lattice, for instance.

[0120] Two distinct component states |Q) = |f) and |C2) = |T) are encoded into the respective internal states of each of the two component particles 3. The set of component states {| ), |C2)} = {|f)> |T)} is linearly independent, especially the component states are orthogonal. In the example of Fig. 3, the two component particles 3 populate a respective component site 7.

[0121] The rearrangement operation is controlled by the internal state of the two input particles 2.

[0122] Conditioned on each of the four basis states |Q) = |J ), |B2) = |4T), |B3) = |T4) and |B4) = |TT) of the internal state of the two input particles 2, the rearrangement operation populates an output site 4 with a respective component particle 3. In this example, the output site 4 is populated by transferring the component particles 3 to the output site 4, wherein the output site 4 is initially empty. The transfer is depicted by arrows in Fig. 3. The transfer of the component particles 3 may be conditioned on the basis states of the internal state of the two input particles 2 by an interaction between the two input particles 2 at the input sites 5 and the component particles 3 at the component sites 7, which is depicted by zig-zag lines in Fig. 3. The interaction may be an intemal-state-dependent van-der-Waals interaction between polarizable particles in this example, but it can in general be any suitable physical interaction. Technical details on the implementation will be explained in section D of this application.

[0123] The four basis states are a composite basis made up from single-particle basis states of the individual input particles 2. For the first basis state |Q) = |J ), both the first input particle 2 and the second input particle 2 are in a first single-particle basis state \bt) = |4). For the second basis state |B2) = |4T), the first input particle 2 is in the first single-particle basis state \bt) = |4) and the second input particle 2 is in a second single-particle basis state |b2) = |T). For the third basis state |B3) = |T4), the first input particle 2 is in the second single-particle basis state |b2) = |T) and the second input particle 2 is in the first single-particle basis state \bt) = |4). For the fourth basis state |B4) = |TT), both thefirst input particle 2 and the second input particle 2 are in the second single-particle basis state |b2) = in.

[0124] A first component particle 3 of the at least two component particles 3 is prepared in the first singleparticle state \bt) = |f) (i.e., |Q) ■= |Q)) and a second component particle 3 of the at least two component particles 3 is prepared in the second single-particle state |b2) = |T) (i.e., |Q):=IQ)). The rearrangement operation comprises the steps:

[0125] • Conditioned on the first basis state |Q) = |ff ) and / or the fourth basis state |B4) = | TT) of the internal state of the two input particles 2, populating the output site 4 with the first component particle 3 (carrying state |Q) = IQ)). These are the upper and the bottom modified arrangements 9 in Fig. 3.

[0126] • Conditioned on the second basis state |B2) = |4T) and / or the third basis state |B3) = |T4) of the internal state of the two input particles 2, populating the output site 4 with the second component particle 3 (carrying state |b2) = |Q)). These are the two central modified arrangements 9 in Fig. 3.

[0127] The conditioned transfer of the component particles 3 causes a superposition of all four modified arrangements 9. The mapping is given by |Q) = |J4) -> |Q) = |4), |B2) = |4T) -> |Q) = |T), |B3) = |T4) -> |Q) = |T) and |B3) = |TT) -> |Q) = |4). By choosing other states for basis states and / or component states, other two-body gates may be implemented by using the method according to the invention. For other two-body gates, one might need four different component states.

[0128] The resulting superposition of modified arrangements 9 comprises the desired logical output state of the CNOT gate as the internal state of the population 6 at the output site 4 (within dashed square in Fig. 3) and the internal state of the first input particle 2 at the first input site 5 |CNOT(q1, q2))£i 0=

[0129]

[0130] < Zi | J4)i1;0+ a^l'Ll')^ + <z4|T4)£1 o+ a3|TT)£10. The final step is thus outputting the internal state of the population 6 at the output site 4 and the internal state of the population at the first input site 5 (i.e. of the first input particle 2) as logical output state.

[0131] Note that the internal state of the population 6 at the output site 4 and the internal state of the population at the first input site 5 is still entangled with the population at the other sites, i.e., the full physical state is tfil^iil^Ql^l^cil^o + «2l'l')iil )i2l^)c2l'l')cil )04-< X3|T)£i|X)£2l^)<2l'l')<7il )0+ «4|T)n |T)£2)C2 l )ci l^)0, wherein cl and c2 denote the component site 7 which was initialized with the component particle 3 carrying component state |Q) = |X) and |Q) = |T), respectively. For further calculations the populations atall involved sites should be protected against noise or sites other than the output site 4 and the first input site 2 may be discarded by performing a discarding operation.

[0132] Optionally, a discarding operation can be performed to the modified arrangements 9 to disentangle the internal state of the population 6 at the output site 4 and the population at the first input site 2 from the population in the rest of the register 1 of the quantum processing unit 20, i.e., the population at the second input site 5 (denoted by i2) and the component sites 7. Alternatively, performing the discarding operation may involve a measurement of the internal state of the population 6 at the output site 4 and / or the population 6 at the first input site 5, if a measurement is anyways intended after performing the CNOT gate according to a quantum circuit to be implemented. Especially, a Z measurement of the internal state of the population 6 at the output site 4 and / or the internal state of the population at the first input site 2 may lead to a collapse of the superposition state to one of its constituents, rendering a further disentangling operation obsolete.

[0133] Note that the rearrangement operation described above may additionally add known phases to the individual modified arrangements 9 due to the nature of the physical implementation. These phases may be compensated for as long as they are known.

[0134] A.4 Various embodiments for the X gate suitable for different internal-state-dependent transfer operations

[0135] Figs. 4a, 4b, 4c, 5a, 5b, 6a, 6b and 6c show alternative embodiments for implementing an X gate by performing a rearrangement operation, which differ in the way the output site 4 is populated with component particles 3. As Fig. 1, said figures depict a schematic representation of the arrangement of an input particle 2 and two component particles 3 before (initial arrangement 8) and after (superposition of modified arrangements 9) performing the rearrangement operation for implementing an X-gate applied to a qubit.

[0136] Figs. 4a, 4b and 4c show variations of the embodiment of Fig. 1, wherein Fig. 4a essentially reprints Fig. 1 for comparison of all three embodiments on one page. The intemal-state-dependent transfer of the component particles is realized by a (direct) intemal-state-dependent interaction between input particle and component particles. In the embodiments of said figures populating the output site 4 with the respective component particle 3 conditioned on the at least two basis states |BX) = |4) and |B2) = |T) of the internal state of the at least one input particle 2 involves using an intemal-state-dependent transfer directly transferring at least one of the two component particles 3. In other words, conditionedon the internal state of the input particle 2, at least one of the component particles 3 is transferred. The input particle 2 itself does not move in these embodiments. Such an internal-state-dependent interaction between input particle 2 and component particles 3 (depicted by zig-zag line) may be realized by an internal-state-dependent van-der-Waals interaction of polarizable particles in Rydberg states, for example. In these embodiments, the two component particles 3 are indistinguishable particles, while the input particle 2 may be distinguishable or indistinguishable. That is, the input particle 2 may be a different species than the two component particles 3. Furthermore, the input particle 2 may be an artificial, non-moveable particle such as a superconducting qubit.

[0137] In the embodiment of Figs. 1 and 4a, the output site 4 is unpopulated in the initial arrangement 8. Populating the output site 4 with the respective component particle 3 conditioned on each of the at least two basis states |BX) = |4) and B2) = |T) of the internal state of the at least one input particle 2, comprises transferring a first component particle 3 to the output site 4 from a first component site 7 (denoted by cl) and keeping a second component particle 3 at the second component site 7 (denoted by c2) (leading to lower modified arrangement 9), and / or transferring a second component particle 3 to the output site 4 from a second component site 7 and keeping the first component particle 3 at the first component site 7 (leading to upper modified arrangement 9). Arrows indicate the transfer. The transfer of the component particles may be accomplished by an exchange of the T or X populations at respective component sites 7 and output site 5 conditioned on the internal state of the input particle 2 (these operations may be denoted

[0138]

[0139] by and C / 'T^o). These operations may be realized by internal-state-dependent shuttling and van-der-Waals interaction.

[0140] In the embodiment of Fig. 4b, the output site 4 is populated by a first component particle 3 in the initial arrangement 8. Populating the output site 4 with the respective component particle 3 conditioned on each of the at least two basis states |BX) = |4) and |B2) = |T) of the internal state of the at least one input particle 2, comprises keeping the first component particle 3 at the output site 4 and keeping the second component particle 3 at the second component site 7 (leading to the lower arrangement 9), and / or transferring the first component particle 3 out of the output site 4 (to a first component site 7) and transferring the second component particle 3 to the output site 4 from the second component site 7 (leading to upper modified arrangement 9). Arrows indicate the transfer.

[0141] In the embodiment of Fig. 4c, the output site 4 is populated by a first component particle 3 in the initial arrangement 8. Populating the output site 4 with the respective component particle 3 conditioned on each of the at least two basis states |BX) = |4) and |B2) = |T) of the internal state of the at least one input particle 2, comprises keeping the first component particle 3 at the output site 4and keeping the second component particle 3 at a component site 7 (leading to the lower arrangement 9), and / or transferring the first component particle 3 out of the output site 4 (to the component site 7) and transferring the second component particle 3 to the output site 4 from the component site 7 (leading to upper modified arrangement 9). Arrows indicate the transfer.

[0142] Figs. 5a, 5b, 6a, 6b and 6c show embodiments, wherein populating the output site 4 with the respective component particle 3 conditioned on the at least two basis states Bl, B2 of the internal state of the at least one input particle 2 involves two steps:

[0143] • first performing an internal-state-dependent transfer of the at least one input particle 2, thereby creating a modified arrangement of the at least one input particle 2, and

[0144] • second performing a population-dependent transfer of at least one of the component particles 3 depending on the modified arrangement of the at least one input particle 2.

[0145] In some systems this two-step process may be easier to implement than the embodiments of Figs. 1, 4a, 4b and 4c, since it avoids a direct intemal-state-dependent interaction of the component particles 3 depending on the internal state of the input particle 2 (i.e., of two different particles). Here the internal-state-dependent transfer depends on the internal state of the same particle (input particle 2) which is to be transferred. In these embodiments, the input particle 2 and the two component particles 3 are indistinguishable particles.

[0146] In Figs. 5a and 5b the internal-state-dependent transfer of the input particle 2 may be implemented via state-dependent shuttling of polarizable particles, for instance. Instead, in Figs. 6a, 6b and 6c the Pauli spin blockade for fermionic particles is used.

[0147] Fig. 5a shows an embodiment comprising four sites.

[0148] In a first step of Fig. 5a, an intemal-state-dependent transfer of the input particle 2 is performed, i.e., the input particle 2 is transferred from the input site 5 to a target site 10 depending on the internal state of the input particle 2. Especially, the input particle 2 is transferred conditioned on the second basis state |B2) = |T) and not transferred conditioned on the first basis state |BX) = |4). Thereby a modified arrangement of the input particle 2 at the input site 5 and the target site 10 is created. In other words, the input particle 2 is in a superposition between the input site 5 and the target site 10, which is entangled with its internal state. With the probability amplitude a the input particle 2 is not transferred and stays at the input site 5 and with the probability amplitude a2the input particle 2 is transferred to the target site 10 leaving the input site 5 empty.The intemal-state-dependent transfer may be realized by exchanging the T-populations between input site 5 and target site 10 (i.e., sites with indices zero and one) depending on the state at the input site 5 (i.e. site with index one) being T. This operation may be denoted by

[0149]

[0150] and may be realized via state-dependent shuttling of polarizable particles.

[0151] In a second step of Fig. 5a, a population-dependent transfer of a component particle 3 depending on the modified arrangement of the input particle 2 is performed. Depending on if the input site 5 is populated, the second component particle 3 with component state |C2) = |T) (initially populating the output site 4 indexed by two) is transferred to the input site 5 (indexed by one) or not (first populationdependent transfer). After that, depending on if the output site 4 is populated, the first component particle 3 with component state | ) = |4) (initially populating a component site 7 indexed by three) is transferred to the output site 4 or not (second population-dependent transfer). By this, the output site 4 is populated depending on the modified arrangement of the input particle 2 and thus depending on the internal state of the input particle 2.

[0152] The population-dependent transfer may be realized by configuring the system such that a double occupation of a site is energetically suppressed, for example due to a repulsive Coulomb interaction or collisional interaction. Alternatively, it may be realized by state-dependent shuttling of polarizable particles.

[0153] The internal state of the population 6 at the output site 4 after applying the rearrangement operation may be used as the logical output state of the X gate operation. Optionally, the population at the sites apart from the output site 4 may be discarded by performing a discarding operation.

[0154] Fig. 5b shows an alternative embodiment with five sites in a different arrangement. The input particle 2 is here arranged on a site via which the component particles 3 are to be transferred in order to reach the output site 4.

[0155] In a first step of Fig. 5b, an intemal-state-dependent transfer of the input particle 2 is performed, i.e., the input particle 2 is transferred from the input site 5 to a target site 10 depending on the internal state of the input particle 2. Especially, the input particle 2 is transferred conditioned on the second basis state |BX) = |4) and not transferred conditioned on the first basis state |B2) = |T). Thereby a modified arrangement of the input particle 2 at the input site 5 and the target site 10 is created. In other words, the input particle 2 is in a superposition between the input site 5 and the target site 10, which is entangled with its internal state. With the probability amplitude a2the input particle 2 is nottransferred and stays at the input site 5 and with the probability amplitude a the input particle 2 is transferred to the target site 10 leaving the input site 5 empty. The intemal-state-dependent transfer may be realized by internal-state-dependent shuttling of polarizable particles, for instance.

[0156] In a second step of Fig. 5b, a population-dependent transfer of a component particle 3 depending on the modified arrangement of the input particle 2 is performed. Depending on if the input site 5 is populated, the second component particle 3 with component state |C2) = |f) (initially populating a second component site 7 indexed by zero) is transferred to the input site 5 and then to the output site 4, or not. Depending on if the target site 10 is populated, the first component particle 3 with component state |CX) = |4) (initially populating a first component site 7 indexed by two) is transferred to the target site 10, and then to the output site 4, or not. By this, the output site 4 is populated depending on the modified arrangement of the input particle 2 and thus depending on the internal state of the input particle 2. The population-dependent transfer may be realized by configuring the system such that a double occupation of a site energetically suppressed, for example due to a repulsive Coulomb interaction or collisional interaction.

[0157] The internal state of the population 6 at the output site 4 after applying the rearrangement operation may be used as the logical output state of the X gate operation. Optionally, the population at the sites apart from the output site 4 may be discarded by performing a discarding operation.

[0158] Figs. 5a and 5b comprise an intemal-state-dependent-transfer of the input particle 2 from the input site 5 to an initially empty target site 10, which can be realized by internal-state-dependent shuttling of polarizable particles, for instance.

[0159] Figs. 6a, 6b and 6c use the Pauli spin blockade of fermionic particles for the intemal-state-dependent-transfer of the input particle 2 instead. For this the embodiments of Figs. 6a, 6b and 6c comprise a reference particle 11 whose internal state is prepared in one of the basis states of the input particle 2 (the reference state | / ?) = |4) in the examples of the figures). An intemal-state-dependent transfer may be achieved by inducing to transfer the input particle 2 from the input site 5 to the target site 10, wherein by the Pauli spin blockade mechanism the input particle 2 transfers conditioned on its internal state being mapped to the two-particle singlet state |

[0160]

[0161] S) = (|4T) — |T4)) / V2 (denoted by S in the figures) of the internal states of the reference particle 11 and the input particle 2 by the dominant transition matrix element. The singlet state of two indistinguishable particles in the ground state of a doubly populated potential well (e.g., quantum dot) is antisymmetric under particle exchange (Pauli exclusion principle). Preferably, the quantum processing unit 20 comprises a longitudinal (magnetic)field gradient between the sites such that a Pauli spin blockade which is adiabatic with respect to the (magnetic) field gradient allows transitions between |S) in one potential well (e.g., quantum dot) and one of the states |f) |T) and |T) | f) with particles in two potential wells. I.e., when the reference state is | / ?) = |f) the input particle 2 transfers only when being in the state |B2) = |T).

[0162] Fig. 6a shows an embodiment with four sites. The structure of the embodiment of Fig. 6a resembles the one of Fig. 5a, but the target site 10 is initially populated with a reference particle 11. Individual steps 12, 13, 14 of the rearrangement operation and an optional discarding step 15 are depicted in Fig.

[0163] 7. The intemal-state-dependent transfer 12 is done via employing the Pauli spin blockade mechanism. The population-dependent transfer is similar to Fig. 5a, i.e. there is a first population-dependent transfer 13 of the second component particle 3 to the input site 4 (conditioned on the changed arrangement of the input particle 2) and a second population-dependent transfer 14 of the first component particle 3 to the output site 4 (conditioned on the changed arrangement of the second component particle 3 due to the first population-dependent transfer 13). Fig. 7 additionally depicts an optional discarding step 15, after which the population apart from the population 6 at the output site 4 is discarded.

[0164] Fig. 6b shows an embodiment with five sites. Fig. 8 shows the individual steps 12, 13, 14 of the rearrangement operation and an optional discarding step 15 for the embodiment of Fig. 6b.

[0165] In a first step of Figs. 6b and 8, an internal-state-dependent transfer 12 of the input particle 2 is performed, i.e., the input particle 2 is transferred from the input site 5 to a reference site 10 depending on the internal state of the input particle 2. Especially, the input particle 2 is transferred conditioned on the second basis state |B2) = |T) and not transferred conditioned on the first basis state |BX) = |4). This is realized by placing a reference particle in state | / ?) = | X) at the reference site 10. Thereby a modified arrangement of the input particle 2 at the input site 5 and the reference site 10 is created. With the probability amplitude a the input particle 2 is not transferred such that the target site 10 stays populated by a single particle (the reference particle) and with the probability amplitude a2the input particle 2 is transferred to the reference site 10 such that the reference site 10 is populated by two particles (the reference particle and the input particle) in the singlet state S.

[0166] In a second step of Figs. 6b and 8, a first population-dependent transfer 13 of a component particle 3 depending on the modified arrangement of the input particle 2 is performed. Depending on whether the reference site 10 is populated by two particles or just a single particle, the second component particle 3 with component state |C2) = |T) (initially populating the output site 4) is transferred to theempty second component site 7 or not. After that, depending on if the output site 4 is populated, the first component particle 3 with component state I ) = |f) (initially populating the first component site 7) is transferred to the output site 4 or not (second population-dependent transfer 14). By this, the output site 4 is populated depending on the modified arrangement of the input particle 2 and thus depending on the internal state of the input particle 2. The first population-dependent transfer 13 may be realized a repulsive interaction, preferably a Coulomb interaction, acting from the population at the reference site 10 to the population at the output site 4, which pushes the second component particle 3 at the output site 4 to the second component site 7. The second population-dependent 14 transfer may be realized by a repulsive interaction, preferably a Coulomb interaction, allowing the first component particle 3 to move to the output site 4 only when it is empty. Fig. 8 additionally depicts an optional discarding step 15, after which the population apart from the population 6 at the output site 4 is discarded.

[0167] An advantage of the embodiment of Figs. 6b and 8 compared to the embodiment of Figs. 6a and 7 is that it allows for realizing a cascaded transfer of the first and the second component particle 3, whereas in Figs. 6a and 7, the first and the second component particle 3 must be transferred individually.

[0168] Fig. 6c shows a further embodiment involving five sites, whereas Fig. 9 shows the individual steps 12, 13, 14 of the rearrangement operation and an optional discarding step 15. The output site 4 is initially unpopulated. The intemal-state-dependent transfer 12 works in the same as described for Fig.

[0169] 6b. The population-dependent transfer step again has two substeps 13, 14. First, the first component particle 7 is transferred from the first component site 7 (indexed by two) adjacent to the reference site 10 when the reference site 10 is occupied by two particles (reference particle 11 and input particle 2). Second, the second component particle 7 is transferred from the second component site 7 (indexed by zero) to the output site 4 only when the output site 4 is unpopulated. Fig. 9 additionally depicts an optional discarding step 15, after which the population apart from the population 6 at the output site 4 is discarded.

[0170] A, 5 CNOT gate implementations using the Pauli spin blockade

[0171] Fig. 3 showed an exemplary implementation of a CNOT gate. The embodiment of Fig. 3 necessitates an internal-state-dependent transfer of the component particles 3 depending on the internal state of the two input particles 2, i.e., direct intemal-state-dependent interaction between input particles and component particles. Such an operation may be implemented for instance for polarizable particleswith Rydberg states and other platforms. Some platforms, however, may not provide such a physical operation. The embodiment of the CNOT gate in Figs. 10 and 12 show alternative implementations of the CNOT gate which only involves an internal-state-dependent transfer of a first input particle 2 depending on its own internal state and an internal-state-dependent transfer of a second input particle 2 to an an cilia particle 19 depending on their composite state. This may be implementable on a quantum processing unit comprising spin particles in electrostatically defined quantum dots.

[0172] The register for implementing the CNOT gate in Fig. 10 comprises eight sites. Fig. 10 depicts a linear configuration comprising a left four-site register and a right four-site register, wherein the method involves long distance shuttling from the left four-site register to the right four-site register. Shuttling over long distances may be avoided by changing to a two-dimensional configuration with seven sites, wherein the two sites between which shuttling is envisioned are merged to one site.

[0173] The left four-site register comprises a first input particle 2 corresponding to a logical control qubit of the CNOT gate, which has the internal state ql and which populates a first input site 5, a reference particle 11 populating a target site 10 (i.e., the site the first input qubit 2 is transferred to) and prepared in the internal state | / ?) = |f), a first component particle 3 prepared in the internal state | ) = |f) and populating a component site 7 and a second component particle 3 prepared in the internal state IQ) = |f) and populating an intermediate output site 18. The right four-site register comprises a second input particle 2 corresponding to logical target qubit of the CNOT gate, which has the internal state q2 and which populates a second input site 2. Further, it comprises a reference site 10, which is initially empty and will then later be populated by the population at the intermediate output site 18 of the left four-site register by shuttling (ancilla particle 19). Also, the right four-site register includes a first component particle 3 prepared in the internal state |Q) = |f) and populating a component site 7 and a second component particle 3 prepared in the internal state |C2) = |f) and populating an output site 4.

[0174] The first part of the protocol modifies the left four-site register and resembles the method for the X gate from Figs. 6a and 7. An intemal-state-dependent transfer depending on the internal state of the first input qubit 2 (with the state ql) is realized via the Pauli spin blockade of the first input qubit 2 and a reference particle 11 which is prepared the state | / ?) = |4), i.e., depending on the internal state of the first input qubit 2 it is transferred from the first input site 5 to the target site 10 which is initially populated by the reference particle 11. In the following two steps including population-dependent transfer, the component particles 7 are transferred to the left or not depending on the arrangement of the first input qubit 2. By this, an intermediate output site 18 is populated by the component particles7 such that the population at the intermediate output site 18 is anti-correlated to the state ql of the first input particle 2. In other words, the first input particle 2 and the population at the intermediate output site 18 are in a maximally entangled state.

[0175] In a next step, the population at the intermediate output site 18 is shuttled to the reference site 10 of the right four-site register, i.e., the population at the intermediate output site 18 is used as an ancilla particle 19 mediating the internal state of the first input particle 2 (to which the internal state of the ancilla particle 19 is anti-correlated) to the right four-site register.

[0176] These two steps lead to a superposition of the intermediate arrangements 16 of the particles with amplitudes ava2(first intermediate arrangement 16) and a3, <z4(second intermediate arrangement 16).

[0177] In the following, a further method similar to the X gate from Figs. 6a and 7 with the ancilla particle 19 acting as reference particle is performed. A difference to the method for the X gate from Figs. 6a and 7 is that the ancilla particle 19 carries an arbitrary internal state while the reference particle 11 was prepared in a known state | / ?) = |f). The internal-state-dependent transfer of the second input particle 2 (carrying state q2) depends on the internal state of the ancilla particle 19 and the second input particle 2.

[0178] Fig. 11 shows exemplary detailed steps for implementing the further steps of this method (second part of Fig. 10). In a first internal-state-dependent transfer 32, the state | ff)4;5is transferred to |S, x)45(wherein x denotes the empty site and S the singlet state) by a employing the Pauli spin blockade (this affects the arrangement with amplitude a4). Then a population-dependent transfer 34 between sites five and six is performed (this affects the arrangement with amplitude <z4). In a second internal-state-dependent transfer 33, the state |TF)45is transferred to |S, x)45(this affects the arrangement with amplitude Then a further population-dependent transfer 34 between sites five and six is performed (this affects the arrangement with amplitude a ). Finally, a population-dependent transfer 34 between sites six and seven is performed (this affects the arrangements with amplitudes a and tr4).

[0179] The method ends up with a superposition of four modified arrangements 9 with respective amplitudes alta2, a3, <z4. As can be verified from Figs. 10 and 11, the internal state of the population at the input site 2 of the left four-site register (denoted by a dash-dotted square) and the internal state of the population 6 at the output site 4 of the right four-site register (denoted by a dashed square) give theoutput of the CNOT gate a |

[0180]

[0181] TT)X 6+ a2|4T)1;6+a4l^)i,6 +a3 |ff)i,6 (up to phases). That is, the internal state of the population 6 at the output site 4 (and a further site, namely the first input site 2) are outputted as logical output state. Optionally, the population at the other sites may be discarded.

[0182] Fig. 12 shows a further embodiment using the Pauli spin blockade.

[0183] The register for implementing the CNOT gate in Fig. 12 comprises nine sites. Fig. 12 depicts a linear configuration comprising a left four-site register and a right five-site register, wherein the method involves shuttling from the left four-site register to the right five-site register. Long distance shuttling may be avoided by changing to a two-dimensional configuration with eight sites, wherein the two sites between which shuttling is envisioned are merged to one site.

[0184] The operations on the left four-site register may be the same as described for the embodiment of Fig.

[0185] 10. Further, the shuttling operation may be the same as described for the embodiment of Fig. 10. The further operations on the right five-site register in Fig. 12 (and Fig. 13) differ from the operations on the right four site of Figs. 10 and 11.

[0186] Fig. 13 shows exemplary detailed steps for implementing the further steps of this method (second part of Fig. 12). First (after the shuttling), a Pauli spin blockade between sites five and four is applied (transferring the state |TT)45to |x, S)45), followed by a transfer between sites six and seven depending on the population of site five (intemal-state-dependent transfer and population-dependent transfer 35). Second, the Pauli spin blockade is undone 36. Third, the sites four and five are swapped 37. Fourth, a Pauli spin blockade between sites five and four is applied (transferring state | TT)45to |x, 5)45), followed by a transfer between sites six and seven depending on the population (as the first step 35). Fifth, a population-dependent transfer 34 between sites seven and eight is performed.

[0187] The method ends up with a superposition of four modified arrangements 9 with respective amplitudes alta2, a3, <z4. As can be verified from Figs. 12 and 13, the internal state of the population at the input site 2 of the left four-site register (denoted by a dash-dotted square) and the internal state of the population 6 at the output site 4 of the right five-site register (denoted by a dashed square) give the output of the CNOT gate < ZI|LL)I7+ a2|4T)1 7+ <z4|T4)1;7+ a3|TT)i7(up to phases). That is, the internal state of the population 6 at the output site 4 (and a further site, namely the first input site 2) are outputted as logical output state. Optionally, the population at the other sites may be discarded.

[0188] A.6 CNOT gate implementations using direct internal-state-controlled interactionThe embodiment of Fig. 3 may be used in quantum processing units 20 in which direct intemal-state-dependent interaction between input particles 2 and component particles 3 is possible. The input particles 2 at input sites 5 (indexed by il and i2) directly interact with the component particles 3 at component sites (indexed by cl and c2). The internal-state-dependent interaction may amount to applying the operations

[0189] ^ii,i2^ci,o^ which exchanges (e.g., by tunneling with angle pi) the f -components between sites cl and o conditioned on the internal state of site il being |f) and of site i2 being |f) (i.e., the composite state |J4)),

[0190] Ci^i2Tci,0, which exchanges the f -components between sites cl and o conditioned on the internal state of site il being |T) and of site i2 being |T) (i.e., the composite state | TT)), ^ii,i2^c2,o^ which exchanges the T-components between sites c2 and o conditioned on the internal state of site il being |f) and of site i2 being |T) (i.e., the composite state |TT)), and ^ii,i2^c2,o^ which exchanges the T-components between sites c2 and o conditioned on the internal state of site il being |T) and of site i2 being |f) (i.e., the composite state |Tf)).

[0191] These operations may be implementable on a quantum processing unit 20 using polarizable particles with internal-state-dependent shuttling and intemal-state-dependent van-der-Waals interactions.

[0192] The embodiment of Fig. 14 is a modification from the embodiment of Fig. 10 using direct intemal-state-contr oiled interaction instead of the Pauli spin blockade. It is suitable for bosonic as well as fermionic particles.

[0193] As the embodiment of Fig. 10, the register for implementing the CNOT gate in Fig. 14 comprises a left four-site register and a right four-site register. The left four-site register is modified according to the method for the X gate introduced in the embodiment of Fig. 5a. This amounts to applying the operation C^T^, i.e., depending on the internal state of the population of site 1 being T (denoted by (), an exchange (e.g., tunneling with angle pi) between the sites 0 and 1 is done for the component of the population being in state T (denoted by T ). Furthermore, a population-dependent transfer from site three to site two is performed.

[0194] In a next step, the population (ancilla particle 19) at the intermediate output site 18 is shuttled to the reference site 10 of the right four-site register. As described for Fig. 10, long distance shuttling may be avoided when using a 2D configuration, wherein the two sites indexed by two and four which are connected by shuttling are merged into one site.The left four-site register is not changed anymore after the shuttling, the following operations are all performed on the right four-site register.

[0195] First, the operation C4, 5^4,5 exchanges the T-components between sites indexed by four and five conditioned on the internal state of site four being |f) and of site five being |T) (i.e., the composite state | XT)). This causes a transfer of the population of site five to site four in the fourth (bottom) arrangement with amplitude a4, which creates a singlet state S at site four, leaving site five empty. Further, the operation £4,5 T45exchanges the f -components between sites indexed by four and five conditioned on the internal state of site four being |T) and of site five being |f) (i.e., the composite state |Tf )). This causes a transfer of the population of site five to site four in the first (top) arrangement with amplitude a4, which creates a singlet state S at site four, leaving site five empty. By these operations, an intemal-state-dependent transfer of the second input particle 2 conditioned on the internal states of the populations at site four and five being (i.e., ancilla particle 19 and second input particle 2) orthogonal is performed.

[0196] Second, the operation

[0197]

[0198] 6exchanges the T-components between sites indexed by five and six conditioned on the composite state of sites five and six not being |TT). Hence, the populations of sites five and six are exchanged in all arrangements but arrangement three (center bottom) with amplitude a3. This causes a transfer of the population from site six to site five in the first (top) and fourth (bottom) arrangements (amplitudes a4and a4). The exchange does not change the internal states on the second arrangement (top center), since the populations at sites five and six are in the same internal state. By this operation, a population-dependent transfer from site six to site five is realized.

[0199] Third, the operation

[0200]

[0201] exchanges the f -components between sites indexed by six and seven conditioned on the composite state of sites six and seven not being |Ti). Hence, the populations of sites six and seven are exchanged in all arrangements apart from arrangements two (center top) and three (center bottom). This causes a transfer of the population from site seven to site six in the first (top) and fourth (bottom) arrangements (amplitudes a4and a4). By this operation, a populationdependent transfer from site seven to six is realized.

[0202] In total this rearrangement operation leads to a superposition of four modified arrangements 9 with respective amplitudes a1, a2, a3, a4. As can be verified from Fig. 14, the internal state of the population at the input site 2 of the left four-site register (denoted by a dash-dotted square) and theinternal state of the population 6 at the output site 4 of the right four-site register (denoted by a dashed square) give the output of the CNOT gate. That is, the internal state of the population 6 at the output site 4 (and a further site, namely the first input site 2) are outputted as logical output state. Optionally, the population at the other sites may be discarded.

[0203] A.7 Creation of entangled states

[0204] Some of the embodiments implementing an X-gate above can be used to create entangled pairs of qubits. In the embodiments for the X-gate given in Figs. 4a, 4b, 4c, 5a and 6a, for the X-gate the population 6 at the output site 4 (indicated by the dashed square) is outputted as logical output state. For the creation of an entangled pair the population 6 at the output site 4 (indicated by the dashed square) and the population at a further site (indicated by the dash-dotted square) may be outputted as logical output state. In said embodiments the further site is always the input site 2, but this is not necessarily the case. The output state is 14T)£ 0+ a2|1'f)i,o (up to phases potentially), wherein i indicates the input site 5 and o indicates the output site 4. The population at all other sites may optionally be discarded or protected.

[0205] Furthermore, embodiments implementing the CNOT-gate above can be used to create entangled three-qubit states. For instance, in the embodiments for the CNOT-gate given in Figs. 10 and 14, for the CNOT-gate the population 6 at the output site 4 (indicated by the dashed square) and the population at the first input site 5 on the left four-site register (indicated by the dash-dotted square) is outputted as logical output state. For the creation of entangled three-qubit states the population 6 at the output site 4 (indicated by the dashed square), the population at the first input site 5 on the left four-site register (indicated by the dash-dotted square) and the population at the second input site 5 on the right four-site register (indicated by the dotted square) may be outputted as logical output state. The output state is

[0206]

[0207] + <*2|f1'1')i,5,6 +as l^)i,5,6 +a4l^Wi,5,6 (UP to phases). The population at all other sites may optionally be discarded. This method can be extended by shuttling the population at the output site to a third four-site register comprising a third input qubit and repeating the intemal-state-dependent transfer operation. Further, the method may be extended to an arbitrary numbers of input qubits.

[0208] To create highly entangled pairs, it may be simpler to use an intemal-state-dependent transfer which realizes the mapping | T ) to the singlet state but without realizing the mapping |T4) to the singlet state. Then no transfer will take place in the first (top) modified arrangement 9 and the entangled outputstate may end up in <

[0209]

[0210] ZI|WT)1 56+ cr2I^TT^i56+ ^311^)1, 5, 6 +a4l^Wi,5,6 (UPt0phases). Furthermore, the embodiment of Fig. 3 may be used for the creation of entangled three-qubit states.

[0211] B, Implementation with spin particles in electrostatically defined quantum dots

[0212] At least the embodiments for implementing an X-gate of Figs. 6a, 6b, 6c, 7, 8, 9 and for implementing a CNOT-gate of Figs. 10, 11, 12, 13 may be realized on a quantum processing unit 20 with spin particles 22, especially fermionic spin- 1 / 2 particles (e.g., electrons or holes), in electrostatically defined quantum dots 21. The quantum dots 21 are preferably semiconducting quantum dots 21. The quantum dots 21 each correspond to a site of the quantum processing unit 20. These devices may exhibit excellent coherence properties, fast operation, small size, and compatibility with established CMOS fabrication. The logical quantum states are stored in the two (physical) spin states | f) (spin down) and | T) (spin up) ofthe spin particles 22. Here, the definition |0) = | f) and |1) = | T) is used. In some materials, the valley pseudospin can be used to encode logical states.

[0213] The internal (spin) state of spin particles 22 in electrostatically defined quantum dots may experience a strong noise bias towards dephasing, the coherence time T2is orders of magnitude shorter than the relaxation time especially

[0214]

[0215] ~ 10-3. This noise bias is intrinsic and originates in the microscopic nature of the spin particle 22 and the mechanisms that exchange energy quanta between the spin particle 22 and its environment.

[0216] In Fig. 15a an exemplary quantum processing unit 20 with two electrostatically defined quantum dots 21 is illustrated. Spin particles 22 such as electrons or holes are confined to a plane by creating a two-dimensional spin particle gas (e.g, electron gas or hole gas), for example, in a semiconductor heterostructure such as Si / SiGe or Ge / SiGe. In the example of Fig. 15a the electron gas or hole gas is formed in the Si layer 30 which is located between two SiGe layers 29. Confinement within the plane is achieved by means of an electrostatic potential created by metallic gates 23, 24, 25, 26. By this, effectively zero-dimensional potential wells are formed (quantum dots 21, corresponding to sites). Plunger gates 23 are used to control the on-site energy of the particle and barrier gates 24 are used to lower or raise the tunneling barrier 28 between neighboring quantum dots 21. By this one may control the population of each quantum dot 21 down to the level of single spin particles 22 and control tunneling (that is, the transfer) of spin particles 22 between adjacent quantum dots 21. Source gate 25 and drain gate 26 allow for the formation of electron reservoirs. A micromagnet 27 is used to create an inhomogeneous magnetic field which is used to implement a synthetic spin-orbit coupling, i.e., a coupling between the internal and the external degrees of freedom of the spin particles 22.The overall state of the system is determined by the internal (spin) state of the spin particles 22 and the external (charge) state of the spin particles 22, i.e., the arrangement in the array of quantum dots 21. The quantum dots 21, especially their ground state, correspond to a site, i.e., an external state which can be populated by particles.

[0217] Fig. 15b illustrates the level diagram of two quantum dots 21 comprising a single spin particle 22. In each quantum dot 21 the two spin states | f) and |T) can be attained, wherein the spin states are split in energy by the Zeeman splitting. Spin-conserving tunneling through the barrier 28 with matrix element tcconnects the quantum dots 21. The micromagnet 27 splits the energy of the spin states with the magnetic fields BLand BRin the left (L) and right (R) quantum dot 21. Spin-flip tunneling with matrix element t is possible as well. A difference in the plunger gate 23 voltage leads to an energy detuning £ between the f energies (on-site energy bias). It is noted that tcand tf may have a different magnitude so that spin-conserving tunneling and spin-flip tunneling can be addressed separately, especially spin-flip tunneling can be minimized.

[0218] The confinement in a quantum dot 21 may be sufficiently strong such that the orbital states are separated by a large energy splitting. In this preferred case it is sufficient to consider only the orbital ground state (along with the spin of the particles). A quantum dot 21 thus corresponds to a single external state (a site). Due to the Pauli exclusion principle, a quantum dot 21 can be populated by zero, one or two fermionic spin particles 22 (and not by more spin particles 22).

[0219] For population of a quantum dot 21 with two spin particles 22, a charging energy due to the Coulomb repulsion between the charged spin particles 22 must be overcome and the Pauli principle requires that the two spins particles 22 must form a singlet state, |

[0220]

[0221] S) = (| Tf ) — | f T)) / V2. Atransfer of a spin particle 22 to an already occupied quantum dot 21 thus depends on the internal state of the involved particles and forms an internal-state-dependent transfer operation due to the Pauli spin blockade.

[0222] B,1 Transfer operations with spin particles in electrostatically defined quantum dots

[0223] Transfer (here tunneling) between two adjacent quantum dots 21 denoted by the indices L (left) and R (right) can be induced via at least two methods. These transfer methods may be used to implement the rearrangement operation.In a first tunneling method ("on-site bias energy sweep"), the on-site energy bias ELand ERis swept. Especially, the detunings are swept from (EL— ER / 2 = £ < 0 to (EL— ER / 2 = £ > 0. The sweep needs to be sufficiently slow, so that the evolution is adiabatic with respect to the tunnel coupling. Other than that, the exact timing is not important in this method. Single-site resolved control of the energy bias may be achieved via tuning the voltage at the plunger gates 24 of the respective quantum dot 21 while compensating for crosstalk due to cross-capacitances between the gates. This is the preferred method, which will be used for the explanation in the following.

[0224] In a second tunneling method (" Rabi tunneling") one may lower the tunneling barrier 28 and wait for a certain time (half a Rabi oscillation). This method is time sensitive and one needs to know the Rabi frequency precisely. Single-site resolved control of the tunneling barrier 28 may be achieved via tuning the voltage at the barrier gates 23 of the respective quantum dots 21 while compensating for crosstalk due to cross-capacitances between the gates.

[0225] These methods may be used for intemal-state-dependent transfer as well as for population-dependent transfer of spin particles 22 in electrostatically defined quantum dots 21.

[0226] Further, a substantial nearest-neighbor (Coulomb) repulsion U can be utilized to realize an automatic or cascaded tunneling. To achieve this, the on-site energies are tuned such that the change in energy due to a particle tunneling into or out of a neighboring site brings the system into an energetic configuration where another successive tunneling event is favorable. This may be used for automatically triggered population-dependent transfer in some embodiments.

[0227] B.1,1 Internal-state-dependent transfer with spin particles in electrostatically defined quantum dots

[0228] In embodiments of X-gate of Figs. 6a, 6b, 6c, 7, 8, 9 and of the CNOT gate of Figs. 10, 11, 12, and 13 performing the rearrangement operation involves performing an intemal-state-dependent transfer using the Pauli spin blockade.

[0229] Fig. 16 shows a level diagram of two sites occupied by zero (dotted), one (dashed) and two (solid) particles, as a function of the detuning of the on-site energies ELand ER. The notation (< JL, < JR) indicates that a particle with spin projection <L(R)is localized in the left L (right R) site, 0 indicates an empty site and S a singlet state. A sweep through zero detuning traverses the avoided level crossing.The solid lines of Fig. 16 with two particles are relevant for the internal-state-dependent transfer using the Pauli spin blockade. A general formulation of the Pauli principle is that the total fermionic wavefunction must be antisymmetric under the exchange of any two particles. If two particles occupy the same site, the wavefunction of their internal (spin) states must be antisymmetric under particle exchange. The basis for the total spin of two spin- 1 / 2 particles is given by the triplet states |T_) = |ff), \T+) = | TT), \T0) = (| Ti) + | fT)) / 2 and the singlet state |S) = (| Ti) — | 4T)) / 2. Among the basis states, only the singlet state is antisymmetric under particle exchange. Thus, when inducing to merge two spin particles 22 into the same site, only the singlet state allows tunneling to the doublepopulated configuration, while for all other states the tunneling is suppressed or “Pauli spin blockaded”. Due to the Pauli spin blockade, one may realize an intemal-state-dependent transfer.

[0230] A longitudinal magnetic field gradient (for instance due to the micromagnet 27), i.e., BLBR, which is strong enough to dominate the exchange coupling leads to non-degenerate eigenstates | Tf ) and | f T). Fig. 13 shows the situation for a first direction of the magnetic field gradient, for which an adiabatic sweep of the state | ft) (in the notation of the figure: (f, T)) leads to the state |S, x) or \x, S) (in the notation of the figure: (S,0) or (0, S)). Switching the direction of the magnetic field gradient (second direction) would couple state | Tf) to the state |S, x) or \x, S).

[0231] An example of a Pauli spin blockade for semiconductor quantum dots in prior art is disclosed in Petta et al., Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots, Science, Vol 309, Issue 5744, pp. 2180-2184 (2005).

[0232] B.1,2 Population-dependent transfer with spin particles in electrostatically defined quantum dots

[0233] In embodiments of X-gate of Figs. 6a, 6b, 6c, 7, 8, 9 and of the CNOT gate of Figs. 10, 11, 12, and 13 performing the rearrangement operation involves performing a population-dependent transfer using the Coulomb interaction.

[0234] In a first type of population-dependent transfer, a particle is not transferred to a site which is already populated by a particle and is transferred to a site which is empty. This can be achieved by inducing tunneling according to the first or second tunneling method introduced above, while keeping the tunneling amplitude small compared to the on-site Coulomb repulsion U2minus the nearest-neighbor Coulomb repulsion U. By this, the avoided level crossings in Fig. 16 are clearly resolved. In the case of tunneling induced by a sweep of the detuning (i.e., the difference of the energy bias of adjacent sites), the sweep should be adiabatic with respect to the tunnel coupling to avoid errors. If the systemsupports spin-flip tunneling, it may be necessary to avoid Landau-Zener transitions at the respective avoided level crossings, either by implementing a sufficiently strong spin splitting or by making the sweep adiabatic with respect to all tunnel couplings. If resonant Rabi tunneling is used, U2— U1should be large enough such that the Rabi oscillations are much faster than the timescale of non-resonant tunneling in the two-particle states and that the splitting is much larger than the energy -time uncertainty related to the change in the tunnel barrier or on-site energies. The timescale on which the barriers or the potentials are changed, should be much faster than the Rabi oscillations and slow enough such that the energy -time uncertainty of the pulse is significantly smaller than U2— U1. Furthermore, the spin-conserving tunneling must be much larger than all spin-flip tunneling terms.

[0235] In a second type of population-dependent transfer, a particle is pushed away from a site whose population increases by one particle but is not pushed away from a site with constant or decreasing population. This may be used for cascaded tunneling which automatically follows internal -state-dependent tunneling using the Pauli spin blockade without actively inducing the tunneling event.

[0236] B.2 X-gate with spin particles in electrostatically defined quantum dots

[0237] In the following the physical implementation of method for the X gate (Figs. 6a, 6b, 6c, 7, 8, 9) and of the left-four site register for the CNOT gate (Figs. 10, 11, 12, and 13; resembling the method for the X gate) with a quantum processing unit 20 comprising spin particles 22 in electrostatically defined quantum dots 22 is described.

[0238] Implementing a first embodiment of X gate (Figs. 6a and 7) and the left-four site register for the CNOT gate (Figs. 10, 11, 12, 13) with spin particles 22 in electrostatically defined quantum dots 21 is described in the following.

[0239] A pair of component particles 3 may be initialized by preparing a singlet state in a quantum dot 21 coupled to a reservoir and adiabatically separating the component particles 3 into a separable state with total spin 0 in the presence of a magnetic field gradient. This yields two component particles 3 placed in two separate quantum dots 21 (indexed by two and three). Further, the reference particle 11 is prepared in the f state in a further quantum dot 21 (indexed by zero). The input particle 2 is placed in a quantum dot 21 (indexed by one).

[0240] The register 1 of the quantum processing unit 20 which is used for performing the X gate comprises four electrostatically defined quantum dots 21 in this embodiment.To implement the first step (internal-state-dependent transfer 12), one may consider the sites zero and one. The state of the reference particle 11 is X (left site in the figures) and the state of the input particle 2 (or first input particle 2) is a general internal state with parts in 4 and parts in T (right site in the figures). According to the protocol, the part (X, T) should be translated to (S,0) and the part (4,4) should stay the same (see Fig. 7). From Fig. 16 one may see that when going from the center (detuning is zero) to the left, the state (X, X) stays the same (straight line) while the state (X, T) goes through an avoided crossing to the (S,0) state, for a magnetic field gradient in the first direction. With an induced tunneling according to the first (sweeping the energy bias detuning) or the second (Rabi oscillation) tunneling method one may realize the state-dependent transfer mapping | XT) to |S, x), while not changing the other states | TX), | XX) and | TT). Fast and precise control can be achieved with control pulses which are tailored to the targeted outcome.

[0241] This is a Pauli spin blockade or spin-to-charge conversion. In a real physical system, the resulting state may comprise a phase < >x, so that the state becomes (

[0242]

[0243] cri|X)0|X)1+ e a2|S)0|x)x) | T)21 X)3after the intemal-state-dependent transfer 12. The phase is known and may be compensated.

[0244] Note that the component particles 3 may also be initialized only after the internal-state-dependent transfer 12 and before the first population-dependent transfer 13.

[0245] In the second step (first population dependent transfer 13), tunneling is induced from site two to site one. A particle only tunnels when site one has been vacated due to the internal-state-dependent tunneling 12, otherwise the Coulomb repulsion hinders tunneling. In the third step (second population dependent transfer 14), tunneling is induced from site three to site two. Again, a particle only tunnels when site two has been vacated due to the first population dependent transfer 13, otherwise the Coulomb repulsion hinders tunneling. In other words: After the internal-state-dependent tunneling 12, the electrostatic potential of adjacent quantum dots 21 depends on the internal state of the input particle 2. This energy difference is used to control the movement of component particles 3 in the second and third step. The resulting state is |

[0246]

[0247] X)o| X)x| T)21 X)3+ e1^2a2|S)0|T)X|X)21 x)3, with an additional phase (p2depending on the physical implementation of the tunneling (e.g., the pulse shapes of voltage changes of the gates). The population-dependent transfers may be implemented by energy-selective bucket brigade shuttling or by cascaded displacement of the particles.

[0248] The population 6 at the output site 4 (indexed by two) is used as logical output qubit. The population in other quantum dots 21 may be discarded by a discarding operation 15 which disentangles themfrom the population 6 at the output site 4 without measuring the charge distribution. This results in the state a

[0249]

[0250] ±| x)01 | T)2Ms + el(t>a2|x)0|x)i |f)2|%)3, with a final phase (p.

[0251] For the CNOT gate on the left four-site register in Figs. 10, 11, 12, 13 only the population in quantum dots 21 indexed by zero and three are discarded. The populations at the quantum dots 21 indexed by one and two form an entangled pair a |x)0|4)i |T)2s

[0252]

[0253] + cr2|x)01 T)11 X)21, which may be used for the further CNOT gate protocol (shuttling the ancilla particle, and so on) or for another purpose.

[0254] When discarding the populations one must not measure the charge distribution, since this would correspond to a measurement of the internal state of the input particle. This can be achieved, for example, by measuring a suitable observable or by sequentially applying the protocols which discard any possible particle back into the reservoir.

[0255] In total, the method leaves the new particle in state | T) (| 4)) in the designated output position if the initial qubit state was | 4) (| T)) corresponds to the quantum gate Rz(p)X, an inversion of the Bloch sphere (X) and a phase shift ( / ?z). The angle (p depends on the chosen detuning pulses. The phase shift (p can be compensated by a physical Rz—(p) gate.

[0256] The methods for implementing the X-gate of Figs. 6b and 8 and of Figs. 6c and 9 may be implemented in a similar way. A difference is that cascaded tunneling may be employed for the populationdependent tunneling steps. In Figs. 6a and 7, the population-dependent tunneling was realized by discriminating between the charging energy in an occupied quantum dot 21 compared to an empty quantum dot 21. Here, the increased on-site energy due to double occupation (with a singlet state) affects the energy at an adjacent quantum dot 21. The tunneling in the first population-dependent transfer 13 may be triggered automatically by the internal-state dependent tunneling event 12. This technique may trigger a cascade of tunneling events and can therefore be utilized to modify the method such that more successive movements of additional particles triggered. In Figs. 6b and 8 five quantum dots 21 in two dimensions are used in the register 1 for implementing the X gate, in Figs.

[0257] 6c and 9 the register 1 is adapted to a linear array.

[0258] B.3 CNOT-gate with spin particles in electrostatically defined quantum dots

[0259] To implement the protocol on the right register for the CNOT gate (Figs. 10, 11, 12, 13), a more complex internal-state-dependent transfer than for the X gate and the left register for the CNOT gatemust be realized. This is because both the spin particle 22 on the left site (ancilla particle 19 indexed by four) is in a general superposition state (which is even entangled with the left four-site register) and the spin particle 22 of the right site (second input particle 2 indexed by five) is in a general superposition state. The aim is to coherently map the components of the combined two-site states as follows: (T, X) -> (S, 0), (X, T) -> (S, 0) (tunneling) and (T, T) -> (T, T), (4,4) -> (4,4) (no tunneling) (notation of Fig. 16 is used here). This can be done by two intemal-state-dependent sweeps with opposite magnetic field gradients and with an intermediate reverse sweep. First, with the magnetic gradient in a first direction as for the X gate (for (4, T); the situation is depicted in Fig. 16), and second with the magnetic gradient in a second direction (for (T, 4)). Inverting the magnetic field may be done by physically inverting the magnetic field gradient (if there is a tunable gradient, e.g., created by an electric wire). Other methods which effectively invert the magnetic gradient may be used, for instance physically swapping the populations at the two sites, shuttling all the particles on the opposite side of the micromagnet, or flipping both spins independently with the X-gate protocol introduced in this disclosure. When the magnetic field is effectively inverted, in both intemal-state-dependent transfers the state (4, T) may be mapped to the singlet state. In between the two internal-state-dependent sweeps and before the reverse sweep one may perform a population-dependent transfer to avoid that the reverse sweep reverts the effect of the first intemal-state-dependent sweep.

[0260] The steps in Fig. 11 (of the CNOT gate embodiment depicted in Figs. 10 and 11) can be implemented as follows: The first internal state-dependent-transfer 32 can be implemented by applying a magnetic field gradient in the first direction and doing an energy bias detuning sweep from site five to site four (i.e., from the center to the left in Fig. 16). Only the component 14T)4;5((4, T)) will be transferred to |S, )45((S> 0))- Inthe next step, a population-dependent transfer 34 from site six to site five is done. The Coulomb repulsion allows only for a tunneling in the arrangement with amplitude a4, i.e., the one where the transfer took place in the first internal-state-dependent transfer 32. Following the population-dependent transfer 34, the reverse sweep of the energy bias is performed (from the left to the center in Fig. 16). By the reverse sweep, no population will be transferred since all sites are blocked by Coulomb repulsion (due to the population-dependent transfer). In the following the magnetic gradient is inverted to a second direction, alternatively the magnetic gradient is effectively inverted. Then, the second intemal-state-dependent transfer 33 can be performed with the (physically or effectively) inverted magnetic gradient. Now, only the component |TT)45will be transferred to |S, x)4;5. The protocol is completed by further population-dependent transfers 34 from site six to five and from site seven to six.The resulting state i

[0261]

[0262] s <z1|ii)16et^1+ a2\lT)1 6e1^2+ <z4|H)16et^>3+ <z3|TT)16et^>4wherein the phases ( q, <p2, <p-3 and < >4, depend on the detailed implementation of the protocol and may be removed by single-particle Rzgates and the native CPhase gate. Both gates are known to be noisebias preserving operations for spin particles in electrostatically defined quantum dots 21.

[0263] The steps in Fig. 13 (of the CNOT gate embodiment depicted in Figs. 12 and 13) can be implemented as follows: The first internal-state dependent transfer 35 is done with the magnetic field in a first direction, such that only the component |4T)4;5will be transferred to \x, S)45(from the center to the right in Fig. 16). This affects the arrangement at <z4. The changed charge distribution triggers a cascaded population-dependent tunneling of the population at site six to site seven, i.e., the population at site six is pushed away by nearest-neighbor Coulomb repulsion. Following the cascaded population-dependent transfer, the reverse sweep of the energy bias is performed (from the right to the center in Fig. 16). By the reverse sweep, |x, S)45will be transferred back to | 4T)4;5. Note that this does not revert the whole arrangement since the population from six has already been transferred to site seven with the cascaded transfer (for the arrangement at <z4with the single state). In the following, in this example a swap operation 37 between the site four and five is performed. Again, only the state |ff)4;5 will be transferred to |x, S)45but due to the swap operation 37 this now affects the arrangement at a. The changed charge distribution triggers a cascaded population-dependent tunneling of the population at site six to site seven. A final population-dependent transfer based on Coulomb repulsion from site eight to seven finishes the protocol.

[0264] B.4 Noise-bias preserving features of spin particles in electrostatically defined quantum dots

[0265] For idling spin-1 / 2 particles a phase error is significantly more likely than a bit error. Thus, a noise-bias-preserving gate should suppress the translation of a phase error into a bit error. Phase errors arise from environmental magnetic or electric noise or imprecise control of the electric potential. Spurious spin-spin interaction (residual exchange) in the presence of a magnetic gradient can also lead to a phase error conditional on the spin projection of adjacent spin particles. In a shuttling-based architecture residual exchange can be suppressed by storing spin particles separated and bringing them close only if required.

[0266] The intemal-state-dependent tunneling preserves the noise bias for following reasons: A phase error can couple the spin singlet |

[0267]

[0268] S) = (| T4) — | 4T)) / V2 to the unpolarized spin triplet | To)=(| T4) + | 4T)) / V2 by changing the phase of the superposition between - and +. However, this issuppressed if the longitudinal magnetic gradient is large compared to the exchange coupling between the spin particles. In that case the states |S) and \T0) are no eigenstates and decompose into | FT) = (|T0) - |^» / 2 and | U) = (|T0) + |S» / 2, while leaving \T_) = | U) and \T+) = | TT) unaffected. For these states, a phase error on the spin particles results in a phase shift only.

[0269] During the tunneling of the auxiliary qubits the noise bias can be suppressed as well. In general, quantum interference between different non-adiabatic trajectories of the state may occur due to Landau-Zener transitions between different spin and charge states. A phase error in that step will affect the result of the interference and therefore may lead to a wrong spin projection or charge state. However, these Landau-Zener transitions and therefore the observation interference are suppressed if 2tcis sufficiently larger than the Zeeman splitting of the moving spin, where tcis the spin-conserving tunneling matrix element. A sufficiently adiabatic (i.e., slow) sweep of the energy bias detuning between the two quantum dots suppresses the transition probability further.

[0270] C. Implementation with fermionic polarizable particles trapped in optical potential wells

[0271] The results of spin particles 22 in quantum dots 21 may be translated to fermionic polarizable particles, for instance fermionic atoms or molecules, trapped in optical potentials wells (i.e., optical traps). The sites correspond to potential wells of the optical potential. For polarizable particles which are cold enough, only the ground state of the potential wells is occupied (a site is defined as the ground state). Single-site control may be necessary for tuning the energy bias or the tunneling barrier to induce tunneling. The internal states may be two suitable internal states or sets of internal states of the (more complex) level structure of the polarizable particles.

[0272] For the internal-state-dependent transfer, induced tunneling by single-site control of the optical potential and the Pauli spin blockade may be employed. For the population-dependent transfer, collisional repulsion (tunable by Feshbach resonances) may be employed (instead of the Coulomb repulsion).

[0273] D. Implementation with polarizable particles trapped in optical potential wells with intemal-state-dependent shuttling and van-der-Waals interaction

[0274] At least the embodiments for implementing an X-gate of Figs. 1, 4a, 4b, 4c, 5a, 5b and for implementing a CNOT-gate of Figs. 3, 14 may be realized on a quantum processing unit 20 with polarizable particles 44, especially atoms or molecules, trapped in optical potentials wells 38 of anoptical potential (for fermionic as well as bosonic particles). The optical potential wells 38 each correspond to a site of the quantum processing unit 20, more precisely the ground states of the potential wells 38 may define a site. The logical quantum states are stored in a first set 42 (corresponding to a logical state |4)) and a second set 43 (corresponding to a logical state |T)) of internal states of the polarizable particle 44, preferably wherein transitions within each set 42, 43 of internal states are performable but no transitions between different sets 42, 43 of internal states are performed.

[0275] The polarizable particle 44 may be an Alkali or Alkaline earth atom which can be used as " Rydberg atoms". An example may be the fermionic isotope of Strontium87Sr, but also bosonic particles are suitable for this embodiment (since the Pauli spin blockade is not used).

[0276] Fig. 17 shows a simplified level scheme of a polarizable particle 44 usable for internal-state-dependent shuttling and internal-state-dependent van-der-Waals interactions. The first set 42 of internal states may comprise a first state |0) of a ground manifold 39 (e.g. ground states such as the 'So hyperfine manifold of87Sr), a first state |0') of an excited manifold 40 (e.g. metastable excited states such as the3Po hyperfine manifold of87Sr) and a first state |0R) of a Rydberg manifold 41 (e.g. highly excited states of87Sr). The second set 43 of internal states may comprise a second state |1) of the ground manifold 39, a second state 11') of the excited manifold 40 and a second state |1R) of the Rydberg manifold 41. The logical states are defined as follows: |4): = {| 0), |0'), 10R)} and |T) ■= {| 1), |1'), |1R)}, meaning that a respective logical state may be encoded into either of the physical states in a set. The transitions between states {| 0), | O'), |0R)} of the first set 42 and the transitions between the states {| 1), 11'), |1R)} of the second set 43 are separably addressable. Cross-transitions should be avoided. The energy scale of the transition to the Rydberg manifold 41 is much larger than the energy scale of the transitions between the ground 39 and excited 40 state manifold.

[0277] The states of the ground manifold 39 may be used to encode a logical input state in an idling particle.

[0278] Partial excitation to the excited manifold 40 may be used for intemal-state-dependent shuttling. The ground manifold 39 may be trapped by laser light with a first frequency, while the excited manifold 40 may be trapped by laser light with a second frequency. Only the projection onto either ground manifold 39 or excited manifold 40 may be shuttled.

[0279] The states of the Rydberg manifold 41 may be used to realize intemal-state-dependent van-der-Waals interactions. The two Rydberg states may be chosen such that there is no exchange interaction (sameRydberg level for both) and such that the population only decays into the states within the respective set (first 42 or second 43 set) without any mixture.

[0280] D.l Transfer operations with polarizable particles trapped in optical potential wells

[0281] The intemal-state-dependent transfer and the population-dependent transfer may be realized using state-dependent shuttling and / or the van-der-Waals interaction (instead of the Pauli spin blockade). The transfer operations are also suitable for bosonic particles.

[0282] D.1.1 Intemal-state-dependent shuttling

[0283] Polarizable particles 44 can be trapped by laser light when the laser is tuned to a wavelength where a certain internal state of the atom has a non-zero polarizability. The state-dependent polarizability in turn depends on the polarizability vector of the trapping laser and its wavelength. Two different wavelengths may be used: the wavelength of a first trapping laser may be tuned such that the ground manifold 39 has a non-zero polarizability while the excited manifold 40 has a polarizability close to zero, and the wavelength of a second trapping laser may be tuned such that the excited manifold 40 has a non-zero polarizability while the ground manifold 39 has a polarizability close to zero. With the first trapping laser one may thus trap the projection of the polarizable particle 44 onto the ground manifold 39 and with the second trapping laser one may trap the projection of the polarizable particle 44 onto the excited manifold 40. Separating the two respective trapping positions (internal-state-dependent shuttling) may lead to a spatial superposition of the polarizable particle 44 depending on the internal state of the polarizable particle 44 (i.e., depending on the internal population of the ground manifold 39 and excited manifold 40).

[0284] The ground manifold 39 may be a hyperfine manifold. Especially, for87Sr the states may be part of the hyperfine manifold of thexSo level. For instance, for |0) the hyperfine state mF= —9 / 2 and for |1) the hyperfine state mp= —7 / 2 may be used. The trapping laser wavelength of the ground manifold 39 may be around 500 nm. The excited manifold 40 may be separated from the states of the ground manifold 39 levels by an optical frequency, while the levels |0) and |1) or | O') and 11') may be separated by a frequency much smaller than optical frequencies (e.g., microwave frequency). The excited manifold 40 may be the meta-stable excited hyperfine manifold of the3Po level of87Sr. The trapping laser wavelength of the excited manifold 40 may be essentially 689.2 nm.The electronic population may be transferred between ground manifold 39 and excited manifold 40, i.e., on the transitions |0) <-> | O') and / or 11) <-> 11'), by applying n pulses resonant to the transitions. Importantly, the transitions are individually addressable, e.g., have different frequency (see arrows in Fig. 17).

[0285] Figs. 18a and 18b show a schematic explaining an exemplary implementation of an internal-state-dependent shuttling operation, where the trapping lasers form optical tweezers 45, 46, 47, i.e., optical potential wells 38 with one trapping site. During the shuttling operation, a transport tweezer 47 is moved from a first (left) site at location L of a left static tweezer 45 to a second site (right) at location R of a right static tweezer 46. The static tweezers 45, 46 correspond to physical realizations of sites. The transport tweezer 47 is formed by a second trapping potential trapping the excited manifold 40 and the static tweezers 45, 46 are formed by the first trapping potential trapping the ground manifold 39. The static tweezers 45, 46 may as well be formed by different sites of an optical lattice. A similar shuttling operation was disclosed in D. Gonzalez-Cuadra et al., Fermionic quantum processing with programmable neutral atom arrays, PNAS, Vol. 120, No. 35, e2304294120 (2023) in a different context.

[0286] Initially, the polarizable particle 44 may be in an arbitrary superposition of the ground manifold 39 states |0) and |1), i.e., the logical state cr- i) + cr2|T) may be encoded into the physical state cr O) + a211) in the ground manifold 39. The polarizable particle 44 is trapped with the left static tweezer 45 at a location L.

[0287] In a first step ((1) in Fig. 18a), the transport tweezer 47 trapping the excited manifold 40 is switched on such that it overlaps the left static tweezer 45 at location L. The internal population in 11) (ground manifold 39) is excited to 11') (in the excited manifold 40) via a n pulse, while the internal population of |0) remains in the ground manifold 39. The operation Rx(j ) denotes the physical X rotation with angle n on transition 11) <-> 11'). The atom is now trapped by both left static tweezer 45 and transport tweezer 47.

[0288] In a second step ((2) in Fig. 18a), the transport tweezer 47 is spatially separated from the left static tweezer 45 and moved to overlap with the right static tweezer 46 at location R. This operation ML^Rbrings the polarizable particle 44 into a spatial superposition corresponding to the internal superposition state encoded into the polarizable particle 44 (internal-state-dependent shuttling).In a third step ((3) in Fig. 18a), an X rotation with angle (p is implemented on the transition |1) 11') with a (p pulse. This corresponds to the operation Rx.(p).

[0289] In a fourth step ((4) in Fig. 18b), the transport tweezer 47 is moved to the location L of the first static tweezer 45, which is denoted by the operation MR^L.

[0290] In a fifth step ((5) in Fig. 18b), the internal population is de-excited by performing a n rotation on the transition |1) <-> 11') with a n pulse. I.e., operation ( / ?^)+(TT) is applied.

[0291] Summarizing, the following operation was applied by the above-described procedure, wherein the index i denotes the left static tweezer 45 (at location L) and the index j denotes the right static tweezer 46 (at location R):

[0292] T

[0293]

[0294] “-(0) = ( / ?“)+(7T)M^R / ?“(0)MR^ / ?“(TT)

[0295] wherein a denotes the first 42 or second set 43 of internal states. The index a may be T when the internal population of |1) is excited to 11') (as in the sequence of steps given above). The index a may be 4 when the internal population of |0) is excited to | O').

[0296] The effect of this sequence using state-dependent shuttling is a unitary operation with an angle (p TfiW =e-4((c“)+5“+H c)

[0297]

[0298] Herein c“ is the (bosonic or fermionic) annihilation operator of a polarizable particle 44 with internal state a =4 or a =T at the site (static tweezer) denoted by index i. The angle (p of the rotation in fourth step can be adapted to apply the desired unitary operation.

[0299] D.1.2 Internal -state-dependent van-der-Waals interactions

[0300] Internal-state-dependent Van-der-Waals interactions may be used for implementing an internal-state-dependent entangling gate.

[0301] The dipolar moment of polarizable particles in Rydberg states is large so that neighboring polarizable particles 44 (or polarizable particles 44 up to a certain range, the Rydberg blockade radius) may interact via the van-der-Waals interaction. Two different Rydberg lasers may be used to excite the internal population to the Rydberg states of the Rydberg manifold 41. Especially, the transitions |0') <- |0R) and 11') <- |1R) may be selectively excited, i.e., the transition should be energeticallyseparated (alternatively the transitions |0) <-> |0R) and |1) <-> |1R) may be selectively excited). For a single transition two-photon processes may be used as well.

[0302] An entangling gate (or CZ gate) may be implemented by using the Rydberg blockade mechanism, for instance at two neighboring sites created by static tweezers 45, 46. Two polarizable particles 44 are positioned within the distance of the Rydberg blockade radius. The polarizable particles 44 are driven with two lasers with a specific pulse shape known from the state of the art (for instance, Evered et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268-272 (2023)). By this an interaction of the form e~l9nini can be implemented, wherein ntand rj is the internal population of the state from which the Rydberg level was excited of polarizable particles 44 denoted at sites i and j, respectively. Specifically, one may use 0 = n corresponding to the maximal entangling power of the gate (this is the case considered in the following). For the van-der-Waals interaction, the states |0R) and 11R) of the two atoms may be in the same level, for instance |nS), such that no exchange interaction occurs.

[0303] For the internal-state-dependent van-der-Waals interaction, the excitation to the Rydberg manifold 41 is made state-selective, i.e. depending on the internal state a of the first atom and on the internal state / 3 of the second atom. The intemal-state-dependent entangling gate is denoted by

[0304] -innfn?

[0305] E

[0306]

[0307] i,j =e 1 J- Here and n is the population of polarizable particles 44 at site i and j in states a and?, respectively, a and / 3 can be 4 or T, depending on if atoms are excited to |0R) or |1R).

[0308] For example, when the Rydberg lasers addressing polarizable particles 44 at sites i and j are resonant with the 11R) state, the operation takes the following form:

[0309] £-V _e-i7minj

[0310] For example, when the Rydberg laser addressing the polarizable particle 44 at site i is resonant with the |1R) state, and the Rydberg laser addressing the polarizable particle 44 at site j is resonant with the |0R) state, the operation takes the following form:

[0311] £. U _e-mn[ni-

[0312]

[0313] Other combinations of Rydberg lasers will create the corresponding internal-state-dependent van-der-Waals interaction. That is, via tuning the frequency of the two respective Rydberg lasers to the twodifferent polarizable particles, the desired internal- state-dependent Van-der-Waals interaction can be chosen.

[0314] Furthermore, the van-der-Waals interaction may be used to implement a three-particle entangling gate (or CZZ gate) by placing three polarizable particles 44 within the Rydberg blockade radius and using phase modulation of the Rydberg lasers (see e.g., Evered et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268-272 (2023)). With this an interaction of the from e~ien ninkmay be created, wherein 0 may be chosen as n in the following.

[0315] As detailed above for the two-particle entangling gate, with state-selective excitation to the Rydberg manifold 41 an intemal-state-dependent three-particle entangling gate denoted by

[0316] r-.a, B,v —inn^n^nL

[0317] E

[0318]

[0319] i,j k =e 1 1 k

[0320] may be realized.

[0321] D.2 Physical realization of internal-state-dependent transfer with polarizable particles trapped in optical potential wells

[0322] The operations introduced in D.l may be used for realizing the operations used in the embodiments of Figs. 1, 3, 4a, 4b, 4c, 5a, 5b, 14.

[0323] D.2,1 Single-particle operations

[0324] The operation T“j(n) =

[0325]

[0326] ( / ? X)+(7T)ML^R / ?£(7T)MR^L / ?£(TT) exchanges the a -populations between the sites i and j, and is a physical realization of the operation denoted by C(TLC.

[0327] D.2, 2 Two- or more- particle operations

[0328] A combination of internal-state-dependent shuttling and internal-state-dependent van-der-Waals interactions allows for constructing intemal-state-dependent transfer operations.

[0329] Single-site control

[0330] Tunneling may be conditioned on the internal state of the population of polarizable particles 44 at a single site. The operation C^T^keffects tunneling of ^-population (|0) for fl =4 and |1) for fl =T)between sites j and k conditioned on the a -population on site i. It can be realized by the sequence of physical operations (internal-state dependent shuttling and van-der-Waals interaction):

[0331]

[0332] — t— n“c^+Hc')

[0333] which realizes the effective unitary operation e

[0334]

[0335] 2A7'1 k' '.

[0336] Two-site control

[0337] Tunneling may be conditioned on the population on two sites. The operation C^T^j effects tunneling of state y between sites i and j when the state of the site i is a and the state of site j is?. It may be implemented by the sequence of physical operations

[0338]

[0339] = K)^i2KfK)^i2Kf

[0340] Tunneling with two-site control may be generalized to when the tunneling sites k and 1 are different from the sites with the control population i and j using the three-particle internal-state-dependent entangling gate:

[0341] c

[0342]

[0343] °fTh =

[0344] Negated two-site control

[0345] Tunneling may be conditioned on not having a certain two-site state. The gate

[0346]

[0347] • Ttj effects tunneling of state y between sites i and j when not the state of the site i is a and the state of site j is / 3, i.e., when the site i is not a or the state of site j is not / 3. It may be implemented by the sequence of physical operations

[0348]

[0349] Note the minus in one of the arguments differing from the previous formula.

[0350] D.3 X-gate with polarizable particles trapped in optical potential wells

[0351] The implementation in Figs. 1 and 4a uses single-site controlled tunneling of the component particles 4 at the component sites 7 indexed by cl and c2 to the output site 4 indexed by o, controlled by the internal state of the input particle 2 at the input site 5 denoted by index i. The operations to be performed are T^o and TC2;0. According to section D.2.2, this amounts to applying two intemal-state-dependent van-der-Waals interaction gates to the population at input site 5 and a respectivecomponent site 7, i.e., these sites must be within the Rydberg blockade radius at least for the operation. Further, it amounts to applying two internal-state-dependent shuttling operations between the respective component sites 7 and the output site 4. Similar physical operations may be used for the embodiments of Figs. 4b and 4c.

[0352] The implementation of the X gate according to Fig. 5a and the implementation of the X-gate-like sequence on the left four-site register for the CNOT gate in Fig. 14, the internal-state-dependent transfer may be physically realized by the operation T (TT) (a single-particle operation described in D.2.1). Similar physical operations may be used for the embodiment of Fig. 5b.

[0353] D.4 CNOT-gate with polarizable particles trapped in optical potential wells

[0354] The CNOT gate implementation of Fig. 3 uses two-site controlled tunneling of the component particles 7 at the component sites 7 indexed by cl and c2 to the output site 4 indexed by o, controlled by the combined internal state of the input particles 2 at the input sites 5 denoted by il and i2. The operations to be performed are C

[0355]

[0356] Qi2T^o, Cji2Tc\0, Cji2TcT2 oand C^i2T^0. They may be physically realized by using intemal-state-controlled van-der-Waals interaction and internal-state-dependent shuttling, especially using the three-particle internal-state-dependent entangling gate, according to section D.2.2.

[0357] The CNOT gate implementation of Fig. 14 uses (on the right four-site register) two-site controlled tunneling and negated two-site controlled tunneling between two sites controlled by the population at these two sites.

[0358] The operations to be performed for internal-state-dependent transfer are

[0359]

[0360] and C^T^s. They may be physically realized by using intemal-state-controlled van-der-Waals interaction and intemal-state-dependent shuttling, especially comprising two-particle internal-state-dependent entangling gates, according to section D.2.2.

[0361] The operations to be performed for population-dependent tunneling in the embodiment of Fig. 14 are ^?6iT^5,6

[0362]

[0363] and They may be physically realized by using intemal-state-controlled van-der- Waals interaction and internal-state-dependent tunneling, especially comprising two-particle intemal-state-dependent entangling gates, according to section D.2.2.Note that in the embodiment of Fig. 14, intemal-state-dependent (physical) operations are used for the population-dependent transfer. In an alternative embodiment, the population-dependent transfer may be achieved by trapping the polarizable particles 44 in a tilted optical lattice. Two Raman laser frequencies are adjusted to bring transitions inducing hopping between neighboring sites into resonance. Thereby, hopping may be induced from the vibrational ground state of a first site to the vibrational ground state of a second site, or from a vibrational ground state of a first site to an excited vibrational state of a second site. In the latter case, a de-excitation of the excited vibrational state may be engineered, for instance due to a dissipative process (preferably induced by band-changing collisions with a background gas). A first Raman laser beam may excite the polarizable particle 44 to an excited internal state of the polarizable particle 44 (for instance an electronically excited state), while a second Raman laser beam de-excites the internal state of the polarizable particle 44. The resonance frequencies may be chosen such that a hopping of the polarizable particle 44 is favorable. The direction of the hopping can be downhill or uphill the tilted optical lattice, which is controllable by tuning the frequencies of the Raman lasers. When on-site collisional interactions are so large such that double occupations are energetically unfavorable (hardcore boson limit), the atoms tunnel only until there is another atom "blocking the way". An exemplary physical system which may be adapted for the current purpose is described in Sharma et al., Driven-dissipative control of cold atoms in tilted optical lattices, Phys. Rev. A 103, 04332 (2021).

[0364] E, Implementation in non-spatial sites

[0365] The sites may also be realized by external states which are not spatial degrees of freedom. For instance, momentum lattices may be used to realize non-spatial sites. The rearrangement operation comprising internal-state-dependent transfer and population-dependent transfer then comprises no spatial transfer but a transfer between different momentum states. Other non-spatial sites are conceivable.

[0366] F, Apparatus for implementing a quantum operation

[0367] An apparatus 50 for implementing a quantum operation modifying a logical input state to obtain a logical output state may be configured to perform the quantum computational method described herein.

[0368] Fig. 19 shows a schematic representation of such an apparatus 50. The apparatus 50 comprises a quantum processing unit 20 comprising a plurality of sites 4, 5, 7, 10, 18, a control unit 48 forproviding and rearranging the population in the plurality of sites 4, 5, 7, 10, 18 and a logical unit 49 connected to the control unit 48.

[0369] The logical unit 49 is configured to instruct the control unit 48 to implement the quantum operation modifying the logical input state to obtain the logical output state. Implementing the quantum operation may comprise:

[0370] • providing, using the control unit 48, the at least one input particle 2 in at least one input site 5 of the quantum processing unit 20, wherein the logical input state is encoded into an internal state of the at least one input particle 2, wherein the internal state of the at least one input particle 2 is a superposition of at least two basis states (e.g. \B1), |B2));

[0371] • providing, using the control unit 48, at least two component particles 3 in at least one component site 7 of the quantum processing unit 20, wherein distinct component states (e.g. | Ci), |C2)) chosen from a set of at least two component states (e.g. {|, | C2)}) are encoded into the respective internal states of each of the at least two component particles 3, wherein the set of component states (e.g. { | Cx), | C2)}) is linearly independent, wherein the at least two component particles 3 are indistinguishable particles;

[0372] • performing, using the control unit 48, a rearrangement operation controlled by the internal state of the at least one input particle 2, wherein, conditioned on each of the at least two basis states (e.g. \B1), \B2)) of the internal state of the at least one input particle 2, the rearrangement operation populates an output site 4 with a respective component particle 3 of the at least two component particles 3;

[0373] • outputting at least the internal state of the population 6 at the output site 4 as logical output state.

[0374] A register 1 of the quantum processing unit 20 may be a collection of sites comprising all sites 4, 5, 7, 10, 18 which are used for implementing a specific quantum operation. The quantum processing unit 20 may comprise several registers 1. Different registers 1 may share sites. Sites as well as registers may be created, moved or discarded at runtime, also during the implementation of a specific quantum operation.

[0375] The quantum operation may be given as a mapping 51 between basis states and component states. In other words, a mapping 51 which maps each basis state to one of the component particles 3, wherein the mapping 51 is based on the quantum operation to be implemented, may be provided. The mapping 51 may be stored in a (classical) logical unit 48 and trigger a sequence of control signals of a (classical) control unit 49 which control the quantum processing unit 20 to perform the quantum operation. Then,conditioned on a respective basis state, the rearrangement operation populates the output site 4 with the respective component particle 3 to which the respective basis state is mapped by the mapping 51.

[0376] The sites 4, 5, 7, 10, 18 of the quantum processing unit 20 may be electrostatically defined quantum dots 21 and the at least one input particle 2 and the at least two component particles 3 may be spin particles 22, preferably electrons or holes, confined in the electrostatically defined quantum dots 21.

[0377] The sites 4, 5, 7, 10, 18 of the quantum processing unit 20 may be optical potential wells 38 and wherein the at least one input particle 2 and the at least two component particles 3 may be polarizable particles 44, preferably atoms, trapped in the optical potential wells 38.

[0378] The sites may as well be non-spatial sites such as sites of a momentum space lattice.

[0379] LIST OF REFERENCE SIGNS

[0380] 1 register

[0381] 2 input particle

[0382] 3 component particle

[0383] 4 output site

[0384] 5 input site

[0385] 6 population at the output site after performing the rearrangement operation

[0386] 7 component site

[0387] 8 initial arrangement

[0388] 9 modified arrangement

[0389] 10 target / reference site

[0390] 11 reference particle

[0391] 12 internal-state-dependent transfer

[0392] 13 first population-dependent transfer

[0393] 14 second population-dependent transfer

[0394] 15 discarding operation

[0395] 16 intermediate arrangement

[0396] 17 isolated final state

[0397] 18 intermediate output site

[0398] 19 ancilla particle

[0399] 20 quantum processing unitquantum dots

[0400] spin particles

[0401] plunger gate

[0402] barrier gate

[0403] source gate

[0404] drain gate

[0405] micromagnet

[0406] barrier

[0407] SiGe layer

[0408] Si layer

[0409] reservoir

[0410] first internal-state-dependent transfer

[0411] second internal-state-dependent transfer

[0412] population-dependent transfer

[0413] internal-state-dependent transfer and cascaded transfer undo internal-state-dependent transfer

[0414] swap operation

[0415] optical potential well

[0416] ground manifold

[0417] excited manifold

[0418] Rydberg manifold

[0419] first set of internal states

[0420] second set of internal states

[0421] polarizable particle

[0422] left static tweezer

[0423] right static tweezer

[0424] transport tweezer

[0425] control unit

[0426] logical unit

[0427] apparatus for implementing a quantum operation

[0428] mapping

Claims

CLAIMS1. A quantum computational method for implementing a quantum operation modifying a logical input state to obtain a logical output state, comprising the following steps:• providing at least one input particle (2), wherein the logical input state is encoded into an internal state of the at least one input particle (2), wherein the internal state of the at least one input particle (2) is a superposition of at least two basis states;• providing at least two component particles (3), wherein distinct component states chosen from a set of at least two component states are encoded into the respective internal states of each of the at least two component particles (3), wherein the set of component states is linearly independent, wherein the at least two component particles (3) are indistinguishable particles;• performing a rearrangement operation controlled by the internal state of the at least one input particle (2), wherein, conditioned on each of the at least two basis states of the internal state of the at least one input particle (2), the rearrangement operation populates an output site (4) with a respective component particle (3) of the at least two component particles (3);• outputting at least the internal state of the population (6) at the output site (4) as logical output state.

2. The quantum computational method according to the preceding claim, comprising:• providing a mapping (51) which maps each basis state to one of the component particles (3), wherein the mapping (51) is based on the quantum operation to be implemented,wherein, conditioned on a respective basis state, the rearrangement operation populates the output site (4) with the respective component particle (3) to which the respective basis state is mapped by the mapping (51).

3. The quantum computational method according to one of the preceding claims,• wherein at least one, preferably each, of the basis states of the internal state of the at least one input particle (2) is physically alike to a respective component state of one of the at least two component particles (3), and / or• wherein there are several input particles (2), wherein the internal states of the individual input particles (2) are a superpositions of at least two single-particle basis states, and wherein at least one, preferably each, of the single-particle basis states ofthe internal states of the individual input particles (2) is physically alike to a respective component state of one of the at least two component particles (3).

4. The quantum computational method according to one of the preceding claims, wherein there is one input particle (2), wherein the at least two basis states of the internal state of the input particle (2) comprise a first basis state and a second basis state, wherein the at least two component particles (3) comprise a first component particle (3) and a second component particle (3), wherein the first component particle (3) is prepared in the first basis state and wherein the second component particle is prepared in the second basis state, wherein performing the rearrangement operation comprises:• conditioned on the first basis state of the internal state of the input particle (2), populating the output site (4) with the second component particle (3), and / or • conditioned on the second basis state of the internal state of the input particle (2), populating the output site (4) with the first component particle (3).

5. The quantum computational method according to one of the preceding claims, wherein there are two input particles (2), wherein the internal state of the two input particles (2) comprises a superposition of a first, a second, a third and a fourth basis state, wherein • for the first basis state, both the first input particle (2) and the second input particle (2) are in a first single-particle basis state, and• for the second basis state, the first input particle (2) is in the first single-particle basis state and the second input particle (2) is in a second single-particle basis state, and • for the third basis state, the first input particle (2) is in the second single-particle basis state and the second input particle (2) is in the first single-particle basis state, and • for the fourth basis state, both the first input particle (2) and the second input particle (2) are in the second single-particle basis state,wherein a first component particle (3) of the at least two component particles (3) is prepared in the first single-particle state and a second component particle (3) of the at least two component particles (3) is prepared in the second single-particle state, wherein the rearrangement operation comprises:• conditioned on the first basis state and / or the fourth basis state of the internal state of the two input particles (2), populating the output site (4) with the first component particle (3), and / orconditioned on the second basis state and / or the third basis state of the internal state of the two input particles (2), populating the output site (4) with the second component particle (3).

6. The quantum computational method according to one of the preceding claims, wherein populating the output site (4) with the respective component particle (3), comprises • transferring the respective component particle (3) to the output site (4), preferably from a component site (7), and / or• keeping the respective component particle (3) at the output site (4), and / or• transferring at least one further component particle (3) of the at least two component particles (3) out of the output site (4), preferably to the component site (7) or a further component site (7).

7. The quantum computational method according to one of the preceding claims, wherein the at least one input particle (2) and the at least two component particles (3) are indistinguishable particles.

8. The quantum computational method according to one of the preceding claims, wherein performing the rearrangement operation involves• performing an intemal-state-dependent transfer (12, 32, 33) of the at least one input particle (2) and / or of at least one of the two component particles (3), and / or • performing a population-dependent transfer (13, 14, 34) of the at least one input particle (2) and / or of at least one of the two component particles (3).

9. The quantum computational method according to the preceding claim, wherein performing the rearrangement operation involves• performing an intemal-state-dependent transfer (12, 32, 33) of the at least one input particle (2), thereby creating a modified arrangement of the at least one input particle (2), and• performing a population-dependent transfer (13, 14, 34) of at least one of the component particles (3) depending on the modified arrangement of the at least one input particle (2).

10. The quantum computational method according to claim 8 or 9, wherein• the internal-state-dependent transfer (12, 32, 33) of the at least one input particle (2) and / or the at least one of the at least two component particles (3) involves using a Pauli spin blockade, and / or• the population-dependent transfer (13, 14, 34) of the at least one input particle (2) and / or the at least one of the at least two component particles (3) involves using a Coulomb repulsion.

11. The quantum computational method according to one of claims 8 to 10, wherein• the internal-state-dependent transfer of the at least one input particle (2) and / or the at least one of the least two component particles (3) involves using an intemal-state- dependent van-der-Waals interaction and / or using intemal-state-dependent shuttling, and / or• the population-dependent transfer of the at least one input particle (2) and / or the at least one of the at least two component particles (3) involves using a collisional repulsion and / or a van-der-Waals interaction and / or internal-state-dependent shuttling.

12. The quantum computational method according to one of the preceding claims, wherein the at least one input particle (2) and / or the at least two component particles (3) are fermionic particles and / or spin particles (22), preferably electrons or holes, confined in electrostatically defined quantum dots (21).

13. The quantum computational method according to the preceding claim, wherein the at least two component states and / or the at least two basis states are represented by different spin states or pseudo-spin states of the spin particles (22).

14. The quantum computational method according to one of the preceding claims, wherein the at least one input particle (2) and / or the at least two component particles (3) are fermionic polarizable particles, preferably fermionic atoms, trapped in optical potential wells (38).

15. The quantum computational method according to claim 10 and one of claims 12 to 14, wherein the Pauli spin blockade is performed between a reference particle (11) and a chosen input particle (2), preferably by:• providing the chosen input particle (2) at an input site (5);• providing the reference particle (11) at a reference site (10), wherein the internal state of the reference particle (11) is prepared in one of the at least two basis states of the internal state of the chosen input particle (2);• inducing to transfer the chosen input particle (2) from the input site (5) to the reference site (10), especially preferably wherein by the Pauli spin blockade mechanism the chosen input particle (2) transfers conditioned on its internal state being mapped to the two-particle singlet state of the internal states of the reference particle (11) and the chosen input particle (2) by the dominant transition matrix element.

16. The quantum computational method according to one of the preceding claims, wherein the at least one input particle (2) and / or the at least two component particles (3) are polarizable particles (44), preferably atoms, trapped in optical potential wells (38).

17. The quantum computational method according to the preceding claim, wherein the at least two component states and / or the at least two basis states are represented by a first set (42) and a second set (43) of internal states of the polarizable particle (44), preferably wherein transitions within each set (42, 43) of internal states are performed and no transition between different sets (42, 43) of internal states are performed.

18. The quantum computational method according to one of the preceding claims, comprising:performing a discarding operation (15) after performing the rearrangement operation, preferably• wherein the discarding operation disentangles the population (6) at the output site (4) and optional further sites from the population outside of the output site (4) and the optional further sites, and / or• wherein the discarding operation involves a measurement of the internal state of the population (6) at the output site (4) and on optional further sites.

19. An apparatus (50) for implementing a quantum operation modifying a logical input state to obtain a logical output state, wherein the apparatus is configured to perform the quantum computational method according to one of the preceding claims.

20. An apparatus (50) for implementing a quantum operation modifying a logical input state to obtain a logical output state, preferably according to the preceding claim, comprising:• a quantum processing unit (20) comprising a plurality of sites (4, 5, 7, 10, 18),• a control unit (48) for providing and rearranging the population in the plurality of sites (4, 5, 7, 10, 18),• a logical unit (49) connected to the control unit (48),wherein the logical unit (49) is configured to instruct the control unit (48) to implement the quantum operation modifying the logical input state to obtain the logical output state, wherein implementing the quantum operation comprises:• providing, using the control unit (48), the at least one input particle (2) in at least one input site (5) of the quantum processing unit (20), wherein the logical input state is encoded into an internal state of the at least one input particle (2), wherein the internal state of the at least one input particle (2) is a superposition of at least two basis states;• providing, using the control unit (48), at least two component particles (3) in at least one component site (7) of the quantum processing unit (20), wherein distinct component states chosen from a set of at least two component states are encoded into the respective internal states of each of the at least two component particles (3), wherein the set of component states is linearly independent, wherein the at least two component particles (3) are indistinguishable particles;• performing, using the control unit (48), a rearrangement operation controlled by the internal state of the at least one input particle (2), wherein, conditioned on each of the at least two basis states of the internal state of the at least one input particle (2), the rearrangement operation populates an output site (4) with a respective component particle (3) of the at least two component particles (3);• outputting at least the internal state of the population (6) at the output site (4) as logical output state.

21. The apparatus (50) according to the preceding claim, wherein the sites (4, 5, 7, 10, 18) of the quantum processing unit (20) are electrostatically defined quantum dots (21) and wherein the at least one input particle (2) and the at least two component particles (3) are spin particles (22), preferably electrons or holes, confined in the electrostatically defined quantum dots (21).

22. The apparatus (50) according to the claim 20, wherein the sites (4, 5, 7, 10, 18) of the quantum processing unit (20) are optical potential wells (38) and wherein the at least one input particle (2) and the at least two component particles (3) are polarizable particles (44), preferably atoms, trapped in the optical potential wells (38).

Images