Quantum processing unit

Abstract

A quantum processing unit comprising: first qubits and second qubits arranged alternatingly, wherein each two adjacent qubits are subject to a first two-qubit interaction, wherein each qubit is adjacent to at least two qubits; a first source configured to drive the first qubits with a first Rabi frequency; a second source configured to drive the second qubits with a second Rabi frequency; and a set of mediator qubits.

Description

[0001] PLANCKIAN S.R.L. 275 054 s27

[0002] Scuola Normale Superiore

[0003] Quantum Processing Unit

[0004] Technical field

[0005] The present disclosure relates to a quantum processing unit, in particular to a quantum processing unit that has an architecture that allows to drive the quantum processing unit globally .

[0006] Background

[0007] In the technical field of quantum computing, superconducting qubits, that is, qubits (short for: quantum bits) based on superconducting circuits have taken a very prominent role over the last decades. One of the reasons for this is that with superconducting circuits exceptional performance in executing precise control, that is, the ability to manipulate and govern the quantum states of the circuits with a high degree of accuracy and fidelity, has been demonstrated as well measurement operations has been demonstrated, see for example the paper by Arute et al., Nature 574, 500 (2019) . While this makes superconducting circuits / superconducting qubits a preferred choice for building quantum computing (QC) architectures from many perspectives, at least the following two challenges remain.

[0008] First, as widely recognized, the scalability of superconducting QC architectures (as well as most if not all other solid-state QC architectures) based on localized control of each logical qubit faces the hurdle termed in the scientific community "wiring problem": Since existing architectures require multiple control signals for each qubit, scaling up these existing architectures leads a wiring overload, which dampens the scalability as the solid-state device will simply have an overburdening amount of wires, rendering access to the qubits on the device difficult, see for example Kwon at al., J. Appl . Phys. 129 041102 (2021) .

[0009] Second, while high gate fidelities can be achieved on small superconducting devices, maintaining high gate fidelities while scaling up the number of qubits within a single processor presents a significant hurdle. Importantly, while reaching single-qubit operation fidelities as high as 99.99% is at present possible, reducing errors in two-qubit gates remains challenging and seem to persist at around 0.1%, see for example Singh et al., Phys. Rev. Research 6, 0132350 (2024) for a recent example. Here, one of the main limiting factors to enhance two-qubit gate fidelity in superconducting platforms is the "residual" longitudinal ZZ interaction between neighboring qubits, see for example Ni et al., Phys. Rev. Lett. 129, 040502 (2022) . Further, while methods have been developed to alleviate and even leverage ZZ coupling for implementing two-qubit gates, see Xu and Ansari, Phys. Rev. Applied 15, 064074 (2021) and Long et al., https: / / arxiv.org / abs / 2103.12305, such interaction generally remains undesirable within conventional superconducting computing frameworks.

[0010] In view of these existing approaches, there is thus a need for a quantum computing architecture, i.e., a quantum processing unit, that overcomes these challenges. That is, there is a need for a quantum processing unit that reduces the amount of wiring needed to realize quantum computation and allows to further alleviate the negative effects of unwanted two-qubit interactions negatively af fecting control over and the performance of the device .

[0011] One recent development in this regard is the work by Menta et al . , https : / / arxiv . org / abs / 2407 . 01182 , in which a globally driven superconducting quantum architecture is discussed . It is noted that in this speci fic architecture the number of physical qubits scales quadratically with the number of logical qubits .

[0012] Summary

[0013] The present disclosure has been made in view o f the above technical limitations of currently existing architectures , namely that they are facing the wiring problem threatening scalability of the architecture as well as persistent infidelities , in particular for two-qubit gates , that remain a severe roadblock towards f ault-tolerant quantum computing . According to an aspect of the present disclosure , a quantum processing unit is provided, the quantum processing unit comprising : first qubits and second qubits arranged alternatingly, wherein each two adj acent qubits are subj ect to a first two-qubit interaction, wherein each qubit is adj acent to at least two qubits ; a first source configured to drive the first qubits with a first Rabi frequency; a second source configured to drive the second qubits with a second Rabi frequency; and a set of mediator qubits .

[0014] In addition, preferred aspects of the present disclosure are defined in the dependent claims .

[0015] According to the invention, a quantum computing architecture addressing the wiring problem as well one source for the persistent high infidelities is provided . The resulting quantum processing unit drives the qubits globally, thereby drastically reducing the total number of wires necessary to run the computation, while using the two-qubit interaction between adj acent qubits of each row within the quantum processing unit to reali ze quantum computing . Moreover, the number of logical qubits provided by the quantum processing unit scales linearly in the number of physical qubits . In other words , the overhead necessary for addressing the wiring problem and the associated global driving of the qubits mani fests itsel f only in a pre- factor of scaling .

[0016] Brief description of the drawings

[0017] Embodiments of the present disclosure , which are presented for better understanding the inventive concepts , but which are not to be seen as limiting the disclosure , will be described with reference to the figures in which :

[0018] Fig . 1 shows an illustration a quantum processing unit according to the present disclosure ;

[0019] Fig . 2 shows an illustration of another quantum processing unit according to the present disclosure ;

[0020] Fig . 3 shows an illustration of a further quantum processing unit according to the present disclosure ;

[0021] Fig . 4 shows an illustration of moving quantum states of logical qubits through a quantum processing unit according to the present disclosure ; Fig . 5 shows an illustration indicative of the universality of quantum processing units according to the present disclosure ;

[0022] Fig . 6 shows an illustration of scalable quantum processing units according to the present disclosure ; and

[0023] Fig . 7 shows illustrations of further scalable quantum processing units according to the present disclosure .

[0024] Detailed description

[0025] As the present disclosure relates to the technical field of quantum computing, the following paragraphs will provide further details regarding the technology referred to and the terms used within this disclosure to facilitate the understanding of the present disclosure and the inventive concepts disclosed herein .

[0026] Quantum computing can generally be understood as technically implemented computing based on or exploiting quantum mechanical phenomena . Under certain conditions , in particular at small scales , classical theories of physical matter have to be replaced by quantum theories . One core element of these theories is that physical matter exhibits properties of both particles and waves . Quantum computing is built on the fact that leveraging this behavior can lead to what is a called a "quantum advantage" : For some calculations , there exist quantum algorithms that outperform classical algorithms , i . e . , algorithms performed on a classical computer, by a substantial margin, in some cases even exponentially faster . However, quantum computing as understood at present may not replace classical computing in general and for every type of calculation but only for speci fic technical applications for which a quantum algorithm outperforming known classical algorithms is known . Typical examples thereof include Shor' s algorithm for finding prime factors of an integer showing an exponential speedup compared to known classical algorithms and Grover' s algorithm for an unstructured search showing a quadratic speedup compared to known classical algorithms , both having a wide range of possible applications . Further fields where quantum computing is expected to outperform classical computing is the field of quantum simulation, i . e . , simulating a quantum system by using another quantum system governed by equivalent equations , originally proposed by Richard Feynman as well as speci fic optimi zation problems , in particular hybrid algorithms combining quantum computing aspects with classical optimi zation techniques . These two examples are followed by various industries as they could improve the performance and feasibility of many computationally very demanding tasks such as drug discovery and drug development , logistics as well as engineering .

[0027] At the same time , the fragile nature of quantum states leads to the possible computational advantage from quantum computing to be closely tied to a demanding engineering challenge as the quantum behavior of these states has to be preserved for a suf ficient amount of time . Due to the presence of noise disrupting the quantum behavior, the number of operations that can be performed on a quantum computer are limited and as a consequence , large-scale algorithms cannot be reali zed on the currently available Noisy Intermediate- Scale Quantum (NISQ) devices , i . e . , devices with non- negligible noise and for which scaling the number of qubits remains a challenge . The long-term goal of quantum computing is Fault-Tolerant Quantum Computing ( FTQC ) in which errors during operations are so limited that error correction becomes feasible , hence quantum computing becomes fault tolerant as these errors can reliably and ef ficiently be addressed .

[0028] As quantum computing originates in quantum physics , but relates to the field of computer technology, there is a need for a model or representation to bring quantum physics and computer technology together . The model currently most established is the QC model based on the classical circuit model . In the ( classical ) circuit model , a ( classical ) circuit is comprised of bits , having either the value 0 or the value 1 , to which gates are applied . In the QC model , each of these elements is replaced by its "quantum version" .

[0029] The quantum version of the bit is the qubit ( also referred to as quantum bit ) . It is , similar to a classical bit , a two- level ( or two-state ) system, however, a quantum-mechanical two-level system, possibly an ef fective two-level system . As a consequence of quantum physics , a qubit may be in any coherent superposition of both states 0 and 1 simultaneously .

[0030] A qubit or quantum bit may be considered as the basic unit of quantum information technology as well as the two-level quantum-mechanical system . It may refer to a physical qubit ( that is , physically implemented qubit ) and / or a logical qubit . As the present disclosure relates to a quantum processing unit ( QPU) , the main focus will be on the physical qubit , yet the present disclosure also discusses which part of the physical qubits of the quantum processing unit corresponds to the logical qubits used for the quantum computation and this is modi fied as part of the functioning of quantum processing unit . A quantum gate (or simply gate) may be considered as the basic quantum circuit operating on one or more qubits. Depending on whether one refers to the logical qubits or the physical qubits, the quantum gate may thus either refer to an operation on logical qubits, or to an operation in the context of the physical quantum-mechanical two-level system, i.e., a quantum gate operating on the physical qubits (as implemented by hardware, the physical quantum system) .

[0031] Quantum gates may operate on a various number of qubits. If it operates only on one qubit, the gate is also called a "single-qubit gate". Accordingly, "two-qubit gates" operate on two qubits. While also gates operating on three or more qubits are possible, for most applications, only single- and two-qubit gates are used.

[0032] Recent works the present disclosure takes inspiration from are a paper by Francesco Cesa and Hannes Pichler titled "Universal Quantum Computation in Globally Driven Rydberg Atom Arrays" and published in Phys. Rev. Lett. 131, 170601 (2023) , see also https: / / arxiv.org / abs / 2305.19220, and a paper by Roberto Menta et al. titled "A globally driven superconducting quantum computing architecture", see https: / / arxiv.org / abs / 2407.01182.

[0033] In the former, a universal quantum computer based on globally driven Rydberg atoms is presented. Core to Rydberg atoms is that two nearby atoms in an excited Rydberg state interact strongly, preventing the excitation of more than one atom within a certain distance, a phenomenon conventionally termed "Rydberg blockade". Building on this effect, Cesa and Pichler demonstrated that by arranging two different species of (9(7V2) Rydberg atoms in an almost regular rectangular lattice, one can implement a globally driven quantum computation on N qubits. In this model, the information is always localized in the N qubits of one of the vertical columns that form the lattice , also referred to as the " information carrier" in the following, while the remaining qubits are kept in a reference ( separable ) configuration . The computation proceeds through sequences of control pulses that operate collectively on either of the two species of qubits . These pulses serve to rigidly shi ft the position of the information carrier column in the lattice and activate single and two-qubit gates on its elements at speci fic locations in the device , both enabled by the Rydberg blockade interactions that connect the qubits .

[0034] While this concept introduces the possibility to perform quantum computing using a globally driven control , it increases the scaling of the required number of physical qubits from linear to quadratic . Moreover, this scheme relies as one of its crucial elements on the Rydberg blockade which may be seen as the characteristics of Rydberg atoms , making an extension to other QC platforms a far from trivial task .

[0035] The latter tackles this task and provides , among others , this extension from Rydberg atoms to superconducting qubits , in particular by engineering a blockade corresponding to the Rydberg blockade naturally emerging in Rydberg atoms .

[0036] Using the above , the following describes embodiments of the present disclosure in detail .

[0037] Fig . 1 shows an illustration of a quantum processing unit according to the present disclosure . Speci fically, Fig . 1 shows a quantum processing unit comprising first qubits ( shown as solid squares , also referred to as type A qubits ) and second qubits ( shown as dashed squares , also referred to as type B qubits ) arranged alternatingly, wherein each two adj acent qubits are subj ect to a first two-qubit interaction ( illustrated by the black springs ) , wherein each qubit is adj acent to at least two qubits ; a first source (V^(t) ) configured to drive the first qubits with a first Rabi frequency; a second source (I'BCO) configured to drive the second qubits with a second Rabi frequency; and a set of mediator qubits.

[0038] Here, "adjacent qubits" may mean "adjacent qubits of the first qubits and the second qubits". In a similar spirit, "each qubit" may mean "each qubit of the first qubits and the second qubits".

[0039] As regards "each qubit is adjacent to at least two qubits", it is noted that this may be understood in the sense that the qubits, i.e., the first and the second qubits, are arranged alternatingly in a (closed) loop, the loop being established by the interaction between the qubits. It is noted that this may in particular imply that the qubits are not only arranged such that the first two-qubit interaction can be realized but also such that no further two-qubit interaction between these first and second qubits are realized.

[0040] The first source and the second source as well as the third source, if applicable, may each be a classical source. The Rabi frequencies associated with these sources, i.e., first, second and third Rabi frequency, may be time-dependent. Further, each source may have a phase, i.e., first, second and third phase, which may be time-dependent as well.

[0041] In a superconducting quantum processing unit, i.e., a quantum processing unit in which the qubits are superconducting quantum processing qubits, that is, realized by means of superconducting circuits, the signal provided from the source to the qubits may be carried through a waveguide. Here, the "set of mediator qubits" may take part in enabling that, despite control being global control, multi-qubit gates can be executed.

[0042] Further, it is noted that, outside of interactions with the mediator qubits, each qubit, that is, each qubit of the first and second qubits, may be subject to (only) the (first) two- qubit interactions with (exactly) two qubits.

[0043] As shown in Fig. 1, the set of mediator qubits may consist of a special first qubit (the solid square with triangle and cross) being such that it is driven with at least twice the first Rabi frequency by the first source, the special first qubit may be subject to a second two-qubit interaction (illustrated in grey rather than black used for the first two-qubit interaction) with respect to three second qubits, named special second qubits (the dashed squares with triangles) , respectively. The first qubits may have a first qubit frequency and the special first qubit may have a qubit frequency different from the first qubit frequency, preferably equal to the sum of the first qubit frequency and an interaction strength of the second two-qubit interaction. The second qubits may have a second qubit frequency and the special second qubits may have a qubit frequency different from the second qubit frequency, preferably equal to the sum of the second qubit frequency and the interaction strength of the second two-qubit interaction. Further, one of the second qubits, including the special second qubits, (here indicated by the dashed square with a triangle and a cross) may be such that it is driven with at least twice the second Rabi frequency by the second source.

[0044] It is noted that "being such that it is driven with at least twice the / any Rabi frequency" includes the case of the Rabi frequency being exactly twice the (reference) Rabi frequency (with which the remaining qubits / the majority of qubits of that type) are driven, but also includes the case that this factor is larger than 2, for example 2.5 or 3 and the like, and is not particularly limited. Further, it is well conceivable that also a factor of less than 2 is conceivable, that is, the Rabi frequency is less than twice the (reference) Rabi frequency (with which the remaining qubits / the majority of qubits of that type) are driven.

[0045] In this context, it is noted that term " first / second / third qubits" may encompass all variations of the first / second / third qubit, that is, any "special first / second / third qubit" is encompassed as well. In particular, the term "first qubit" may encompass the "special first qubit", the "first special first qubit" and the "second special first qubit". Accordingly, the term "second qubit" may encompass the "special second qubits", the "first special second qubits" and the "second special second qubits". Further accordingly, the term "third qubit" may encompass the "special third qubit". Moreover, the term "special first / second qubit" may encompass the " f irst / second special first / second qubit". This is in particular the case if reference is made to all qubits of the quantum processing unit and not only to those forming the "loop".

[0046] The choice of the qubit frequency of the first / second special qubit (s) to be equal to the sum of the first / second qubit frequency and the interaction strength of the second two- qubit interaction may be advantageous to ensure proper functioning of the global control allowing to perform the single- and multi-qubit gates discussed herein.

[0047] The first qubit frequency and the second qubit frequency may be different. The fact that one of the second qubits , including the special second qubits , may be such that it is driven with at least twice the second Rabi frequency by the second source may relate to ensure that the single-qubit gates can be reali zed . Speci fically, as explained below, this qubit may be one of the qubits labeled QI to Q8, rather than one of the qubits that is part of the triplets labeled SI to S8.

[0048] Similar, the special second qubits subj ect to the second two- qubit interaction with the special first qubit may be qubits labeled QI to Q8, rather than one of the qubits that is part of the triplets labeled SI to SB .

[0049] The second two-qubit interaction may be the same as the first two-qubit interaction .

[0050] It is noted that the Rabi frequency of a qubit may, in particular i f the qubit is realized by a solid-state element such as a superconducting circuit , be a matter of fabrication of this element and will not require any modi fication of the source driving the qubits . In other words , these modi fications among the qubits do not af fect the possibility to drive the quantum processing unit in a global manner .

[0051] Further details of Fig . 1 , in particular with reference to the crosses , the triangles , the labels QI to Q8, and SI to S8, and other details will be discussed below . These details apply to the quantum processing units of Figs . 1 to 3 in a corresponding manner .

[0052] Fig . 2 shows an illustration of another quantum processing unit according to the present disclosure . Speci fically, as can be readily seen, the quantum processing unit shown in Fig . 2 corresponds to the quantum processing unit shown in Fig . 1 in all aspects except the set of mediator qubits . Hence , the discussion of common aspects will be omitted, and focus is given to the set of mediator qubits .

[0053] As shown in Fig . 2 , the set of mediator qubits may consist of a first special first qubit being such that it is driven with at least twice the first Rabi frequency by the first source and a second special first qubit being such that it is driven with at least four times the first Rabi frequency by the first source . ( For the term "at least four times" the same considerations as above in connection with "at least twice" apply . ) The first special first qubit may be subj ect to a second two-qubit interaction with respect to two second qubits , named first special second qubits , respectively . The second special first qubit may be subj ect to the second two- qubit interaction with respect to two second qubits , named second special second qubits , respectively, wherein the second special second qubits may be di f ferent from the first special second qubits , the second qubits may have a second qubit frequency and the first and the second special second qubits may have a qubit frequency di f ferent from the second qubit frequency, preferably equal to the sum of the second qubit frequency and an interaction strength of the second two-qubit interaction . One of the second qubits , including the first special second qubits and the second special second qubits , may be such that it is driven with at least twice the second Rabi frequency by the second source .

[0054] In other words , di f ferent from the set of mediator qubits shown in Fig . 1 being one special first qubit that is subj ect to the second two-qubit interaction with three special second qubits , in Fig . 2 instead two special first qubits are provided, each subj ect to the second two-qubit interaction with two ( di f ferent ) special second qubits . Further, as is discussed elsewhere in more detail, while the quantum processing unit as illustrated in Fig. 1 is able to execute three-qubit gates, in particular a three-qubit Toffoli gate, thus achieving the capability for universal quantum computing, the quantum processing unit as illustrated in Fig. 2 is able to execute, at two different positions within the quantum processing unit, two-qubit gates and in this manner achieves the capability for universal quantum computing .

[0055] Fig. 3 shows an illustration of a further quantum processing unit according to the present disclosure. Specifically, as can be readily seen, the quantum processing unit shown in Fig. 3 corresponds to the quantum processing unit shown in Figs. 1 and 2 in all aspects except the set of mediator qubits .

[0056] As shown in Fig. 3, the set of mediator qubits may consist of two third qubits (shown as dashed-dotted squares, also referred to as type C qubits) , one of these a special third qubit, wherein the (regular) third qubit may be subject to a second two-qubit interaction with respect to two second qubits, named first special second qubits, respectively, the special third qubit may be subject to the second two-qubit interaction with respect to two second qubits, named second special second qubits, respectively, wherein the second special second qubits are different from the first special second qubits. The second qubits may have a second qubit frequency and the first and the second special second qubits have a qubit frequency different from the second qubit frequency, preferably equal to the sum of the second qubit frequency and an interaction strength of the second two-qubit interaction. One of the second qubits, including the first special second qubits and the second special second qubits, may be such that it is driven with at least twice the second Rabi frequency by the second source. The quantum processing unit may further comprise a third source (c(t)) configured to drive the third qubits with a third Rabi frequency, and the special third qubit may be such that it is driven with at least twice the third Rabi frequency by the third source.

[0057] In other words, compared to Fig. 2 in which the set of mediator qubits is made of first qubits, here, a third type of qubits, the third qubits, driven by a third source, are used. These third qubits may have third qubit frequency, which may be different from the first and second qubit frequency .

[0058] In other words, while the first and second special first qubit of Fig. 2 are driven by the first source (as all the first qubits) , but each with a different Rabi frequency (at least twice the Rabi frequency or at least four times the Rabi frequency, respectively) , thus enabling separation between the two-qubit gates being executed, the third qubit and the special third qubit of Fig. 3 are driven by a separate source, the third source, to execute two-qubit gates, while the difference in Rabi frequency when driven between the third qubit and the special qubit enables separation between the two-qubit gates being executed.

[0059] It is noted that the specific position of the special qubits and their interaction with second qubits of the "loop" are not limiting. In fact, it is clear from the above that no such limitation is implied. Moreover, it is also clear from the above that the position of the second qubit being such that it is driven with at least twice the Rabi frequency compared to the other second qubits is variable and is not tied to any of the special second qubits. In the following, further details generally common to all three quantum processing units illustrated in Figs. 1 to 3 will be discussed.

[0060] In a quantum processing unit according to the present disclosure, logical qubits may be encoded in every other second qubit. These logical qubits may also be referred to as "information carrier" (IC) or "information carrier sites" (IC sites) . In Figs. 1 to 3, these logical qubits are highlighted by a thicker border and are labelled QI to Q8. Thus, it can be understood that Fig. 1 to 3 each show a quantum processing unit with eight logical qubits while in total consisting of 33 (in the case of Fig. 1) and 34 (in the case of Fig. 2 and 3) physical qubits: 16 first qubits and 16 second qubits forming the "loop" and one special first qubit (in the case of Fig. 1) , two special first qubits (in the case of Fig. 2) , or two third qubits (in the case of Fig. 3) .

[0061] Further, as can be seen in Figs. 1 to 3, one of the logical qubits may be the qubit being such that it is driven with at least twice the second Rabi frequency by the second source. As emphasized above, while in Figs. 1 to 3 this qubit is at the same time a special second qubit, this position is not limiting and any other second qubit may be the qubit that is such that it is driven with at least twice the second Rabi frequency by the second source. Any other logical qubit, in particular any logical qubit not subject to an additional two-qubit interaction with a special qubit, such as logical qubits Q5 to Q7 , can be this qubit.

[0062] In a quantum processing unit according to the present disclosure, the special qubits may be qubits in which the logical qubits are encoded. In more concrete terms, in the case of the Fig. 1, the special second qubits may be the qubits in which the logical qubits are encoded, as shown with qubits QI to Q3 . Further, in the case of Figs . 2 and 3 , the first special second qubits and second special second qubits , may be the qubits in which the logical qubits are encoded, as shown with qubits QI, Q2, Q4 and Q8.

[0063] In other words , the special qubits may be among the physical qubits used to encode the logical qubits , which may be used to perform the quantum processing .

[0064] In a quantum processing unit according to the present disclosure , triplets consisting of two first qubits and one second qubit may be provided between second qubits , which maybe alternatingly in a ferromagnetic state or in a paramagnetic state .

[0065] In other words , as can be seen in Figs . 1 to 3 , see also the labels SI to S8, between each two logical qubits , a triplet of a first qubit , a second qubit and a further first qubit is provided . The state of this triplet is either a ferromagnetic state ( indicated in the illustrations by " Ferro" ) or a paramagnetic state ( indicated in the illustrations by "Para" .

[0066] From this , it becomes clear that first qubits part of the special qubits ( special first qubits ) may be excluded from being part of the triplets . In other words , the triplets are formed from first and second qubits forming the " loop" of the quantum processing unit .

[0067] Here , a paramagnetic state of the triplet may mean that the states of the qubits of the triplet alternate between ground state and excited state , that is , the triplet is either in the state "ground state - excited state - ground state" or in the state "excited state - ground state - excited state" . Further, a ferromagnetic state of the triplet may mean that the states of the qubits of the triplet are the same states , that is , the triplet is either in the state "ground state - ground state - ground state" or in the state "excited state - excited state - excited state", i.e., all qubits are either in the ground state or the excited state.

[0068] For the paramagnetic state, the state in which only one qubit is in the excited state may be preferred. For the ferromagnetic state, the state in which all qubits are in the ground state may be preferred.

[0069] In a quantum processing unit according to the present disclosure the logical qubits may be encoded in the second qubits provided between the triplets. In other words, triplets and logical qubits alternate, as can clearly be seen in Figs. 1 to 3.

[0070] Further, as will become apparent from the discussion below, the triplets provided between the logical qubits serve the purpose of blocking quantum information from spreading from one logical qubit to an adjacent logical qubit.

[0071] To ensure the alternating arrangement of the states of the triplets, it may be preferable that the overall number of triplets, and thus the overall number of logical qubits, is even .

[0072] In a quantum processing unit according to the present disclosure, the first source may be configured to be off if the second source is on, and the second source may be configured to be off if the first source is on. Further, if a third source is present, the second source may be configured to be off if the third source is on, and the third source may be configured to be off if the second source is on.

[0073] That means, if the signal provided by the first source to the first qubits is non-zero, the signal provided by the second source is zero . Analogously, i f the signal provided by the second source to the second qubits is non- zero , the signal provided by the first source is zero . The third source , i f applicable , follows the first source : I f the signal provided by the third source to the third qubits is non- zero , the signal provided by the second source is zero .

[0074] In terms of the qubits , this is can be understood as follows : Adj acent qubits , that is , qubits subj ect to a ( shared) two- qubit interaction, are not driven at the same time / simultaneously .

[0075] This requirement may in particular assist in ensuring proper implementation of the various pulses constituting the various quantum operations .

[0076] In a quantum processing unit according to the present disclosure , the first source may be configured to drive the first qubits with a first oscillation frequency and a first phase , and the second source may be configured to drive the second qubits with a second oscillation frequency and a second phase , and, i f present , the third source may be configured to drive the third qubits with a third oscillation frequency and a third phase .

[0077] In a quantum processing unit according to the present disclosure , at least one of the first two-qubit interaction and the second two-qubit interaction is a ZZ-interaction .

[0078] Here " Z" refers to the Z-Pauli matrix and the ZZ-interaction may thus be a two-qubit interaction leading to crosstalk between the qubits and detuning the qubits .

[0079] It is noted that this type of longitudinal ZZ interaction is exactly the type of interaction that is unwanted in conventional approaches . As will be discussed in more detail below, the conventionally disadvantageous interaction is exploited to alleviate some of the degeneracies in the energy spectra of neighboring qubits , thus allowing to selectively impede speci fic energy transitions , thereby ef fectively emulating the blockade ef fect discussed by Cesa and Pichler in the framework of Rydberg atoms .

[0080] It is furthermore noted that neither the first not the second two-qubit interaction is not limited to a ZZ-interaction but may be any other interaction . Furthermore , the first and the second two-qubit interaction may be the same interaction, but may also be di f ferent interactions .

[0081] In a quantum processing unit according to the present disclosure , the first qubits may have a first qubit frequency, the second qubits may have a second qubit frequency, and, i f applicable , the third qubits may have a third qubit frequency .

[0082] A quantum processing unit according to the present disclosure may further comprise an initiali zation source configured to drive every fourth second qubit . Speci fically, the initiali zation source may be configured to drive every other second qubit that is part of the triplets . In other words , the initiali zation source may be configured to drive the second qubit of every other triplet .

[0083] In more details , it may be assumed that the qubits of the quantum processing unit are initially each in their ground state , as this state is the lowest energy state . As can be seen from Figs . 1 to 3 , quantum processing units according to the present disclosure may however assume a speci fic configuration for the triplets : Speci fically, as every second triplet may be in the paramagnetic state in which the middle qubit , a second qubit , is in its excited state , an additional source , the initiali zation source , may be provided to facilitate to state .

[0084] It is noted that this additional source does not impact the amount of wiring necessary substantially as it scales in the same manner as the first and second source . Moreover, since the initiali zation source drives a subset of the qubits driven by the second source , it may be conceivable that the initiali zation source may be implemented by providing switches to the second source ensuring that only this subset is driven . In other words , the initiali zation source may be implemented as a modi fication to the second source .

[0085] In a quantum processing unit according to the present disclosure , the second source may be configured to perform a single-qubit gate on the second qubit being such that it is driven with at least twice the second Rabi frequency by the second source .

[0086] In other words , the fact that one of the second qubits is ( fabricated) such that it is driven with at least twice the second Rabi frequency allows to perform quantum operations that af fect only this qubit without af fecting any of the remaining qubits of the quantum processing qubits .

[0087] In a quantum processing unit according to the present disclosure , the first source and the second source may be configured to perform a Tof foli gate on the special second qubits .

[0088] This may in particular the case when the set o f mediator qubits consist of one special first qubit , as illustrated for example in Fig . 1 . A Toffoli gate is a three-qubit gate which is also known as CCNOT gate ( "controlled-controlled-not" ) . From this, it can be understood that a Toffoli gate is a gate performing a NOT operation on a third (target) qubit if the first and the second (control) qubits are in state equal to "1". A Toffoli gate together with single-qubit gates can implement universal quantum computing.

[0089] Further, conventional quantum processing unit are based on the paradigm of single- and two-qubit gates (and not any multi-qubit gates involving more than two qubits) . This makes an implementation of a Toffoli gate a non-trivial task as it requires that the Toffoli gate is decomposed into several single- and two-qubit gates, specifically, at least five two- qubit gates. Different from that the Toffoli gate performed in a quantum processing unit according to the present disclosure may be a single-shot Toffoli gate. This means that decomposition in two-qubit gates can be omitted and instead single set of pulses acting on three qubits is implemented to perform the Toffoli gate. Such an implementation may be advantageous since it includes less quantum operations, hence less sources for error.

[0090] In a quantum processing unit according to the present disclosure, the first source and the second source may be configured to perform a two-qubit gate on the first special second qubits and / or on the second special second qubits.

[0091] This may in particular the case when the set of mediator qubits consist of two special first qubits, as illustrated for example in Fig. 2.

[0092] In other words, the first source and the second source may be configured to perform a two-qubit gate on the first special second qubits as well as to a two-qubit gate on the second special second qubits independently from each other.

[0093] In terms of Fig. 2 this can be readily understood: The two special first qubits interact with logical qubits QI and Q2 , and with logical qubits Q4 and Q8, respectively. This allows implementation of two-qubit gates between logical qubits QI and Q2, and between logical qubits Q4 and Q8, respectively. Further, the fact that these two special first qubits are (fabricated) such that they are driven with different Rabi frequency, both from each other and from the remaining first qubits, allows to perform operations, i.e., two-qubit gates, on logical qubits QI and Q2 separately from operations on logical qubits Q4 and Q8.

[0094] It is thus understood that in such a case, the capability of the quantum processing unit to perform universal quantum computation stems from being able to perform single- and two- qubit quantum operations.

[0095] In a quantum processing unit according to the present disclosure, the second source and the third source may be configured to perform a two-qubit gate on the first special second qubits and / or on the second special second qubits.

[0096] This may in particular the case when the set of mediator qubits consist of two third qubits, as illustrated for example in Fig. 3.

[0097] This case is conceptually similar to the case described in connection with Fig. 2. Different from the above case, the two third qubits are not first qubits driven by the first source, but a different type of qubit (third qubits) driven by a different source (third source) . Hence, there is no need for the third qubits to have a particular relationship in terms of Rabi frequency to the first qubits , since they are not driven by the same source . Similar to the above discussion, also in this case it is possible to perform operations involving the two third qubits independently since they are ( fabricated) such that their Rabi frequencies di f fer .

[0098] In addition, it is noted that also the provision of the third source does not af fect the overall wiring requirements substantially since it only requires wiring for two additional qubits .

[0099] Further details of the operations and that they allow for universal quantum computing will be provided below . It is moreover noted that the concepts underlying this find their correspondence in the above cited paper by Menta et al .

[0100] In a quantum processing unit according to the present disclosure , each first qubit and special first qubit may be connected to the first source by a first wiring, each second qubit and special second qubit may be connected to the second source by a second wiring, and, i f applicable the third qubit and the special third qubit may be connected to the third source by a third wiring . Here , wiring, that is , first , second and third wiring, may be reali zed by a waveguide . This may in particular the case i f the qubits are reali zed as superconducting qubits .

[0101] In a quantum processing unit according to the present disclosure , the first qubits , including any f irst special qubits , the second qubits , including any second special qubits , and, i f applicable , the third qubit and the third special qubit , are superconducting qubits . In other words , all qubits involved, that is , all qubits of the quantum processing unit, may be superconducting qubits, that is, may be implemented by means of superconducting circuits.

[0102] In a quantum processing unit according to the present disclosure, the first qubits, including any first special qubits, the second qubits, including any second special qubits, and, if applicable, the third qubit and the third special qubit, may be provided in a honeycomb structure. In other words, all qubits involved, that is, all qubits of the quantum processing unit, may be provided in a honeycomb structure. Such a honeycomb structure may be advantageous to ensure that the two-qubit interaction between adjacent qubits, which may typically depend on the distance, are (nominally) all identical.

[0103] In a quantum processing unit according to the present disclosure, the first source may be configured to drive the first qubits using a single signal provided (globally) to all of the first qubits, the second source may be configured to drive the second qubits using a single signal provided (globally) to all of the second qubits, and, if applicable, the third source may be configured to drive the third qubits using a single signal provided (globally) to all of the third qubits .

[0104] Here, the first qubits may include any special first qubits; correspondingly, the second qubits may include any special second qubits, that is, the first special second qubits and the second special second qubits, and the third qubits may include any special third qubits.

[0105] Fig. 4 shows an illustration of moving quantum states of logical qubits through a quantum processing unit according to the present disclosure. Specifically, while the above discussion has provided details how quantum processing units according to the present disclosure are able to perform single- and two / three-qubit gates at speci fic positions within the quantum processing unit , reali zing universal quantum computing may require that in such a case the states of the logical qubits can be moved to any position such that not only is it possible to perform single- and two / three- qubit gates but also to perform single- and two / three-qubit gates on every logical qubit .

[0106] To address this , in a quantum processing unit according to the present disclosure , the first source and the second source may be configured to simultaneously : rotate quantum states of odd ( even) logical qubits clockwise , rotate quantum states of even ( odd) logical qubits counter-clockwise , and trans form states of each triplet from the ferromagnetic state to the paramagnetic state and vice versa .

[0107] This operation is illustrated in Fig . 4 . The operation itsel f is denoted Ilexcand its inverse is accordingly denoted as n^. . As can be seen the starting point of this operation are two logical qubits , Q7 and Q8, in their corresponding states 14b) and l bh and three triplets alternating between the paramagnetic state |e>, |.g), |e> and the ferromagnetic state \g), Igh \g) -

[0108] As mentioned above , when performing the operation Ilexc, several things happened at the same time : First , the states of the triplets are inverted : a ) triplets previously in the ferromagnetic state are now in the paramagnetic state and triplets previously in the paramagnetic state are now in the ferromagnetic state ; b ) the quantum state previously in logical qubit Q7 is now in logical qubit Q8, that is , when considering Figs . 1 to 3 , rotated clockwise ; and c ) the quantum state l^s), previously in logical qubit Q8 is now in logical qubit QI, that is , when considering Figs . 1 to 3 , rotated counter-clockwise .

[0109] It can readily be understood that since all operations in quantum processing units according to the present disclosure are facilitated by sources acting on all physical qubits of the same type equally, the rotations of states 14b) and in clockwise and counter-clockwise apply to all states of all logical qubits . Similarly, the inversion of the states of the triplets applies to all triplets as well .

[0110] In summary, the odd logical qubits can be moved in one direction, the even logical qubits are then moved in the other direction, and the triplets invert their state . Similarly, as illustrated by the inverse operation n^., the even logical qubits can be moved in the one direction, the odd ones are then moved in the other direction, and the triplets invert their state .

[0111] From the above , it can be understood that the rotations possible are ef fectively two opposite rotations . This can also be understood as the quantum processing unit having two "conveyor belts" , one moving the odd logical qubits , the other moving the even logical qubits ( in the opposite direction) .

[0112] Accordingly, it can be understood that this type of rotation allows to move any individual qubit quantum state from any logical qubit to any other logical qubit of the quantum processing unit . Hence , having one physical qubit that allows to perform single-qubit gates at its position within the logical qubits is suf ficient to ef fectively perform singlequbit gates on any logical qubit . At the same time, such a type of rotation may limit the amount of relative ( re- ) positioning of quantum states of two

[0113] (or more) qubits.

[0114] Nevertheless, as sketched in connection with Fig. 5, the capability for universal quantum computing can be achieved if the quantum processing unit is able to perform a three-qubit gate, such as the Toffoli gate, or - at two locations - two two-qubit gates, such as CNOT gates which are also sufficient for universal quantum computing.

[0115] Specifically, Fig. 5 shows an illustration indicative of the universality of quantum processing units according to the present disclosure. This universality will be illustrated with respect to quantum processing units as the one shown in Fig. 1, i.e., those able to perform a Toffoli gate.

[0116] In this case, it is sufficient to show that any arbitrary swap operation between any two logical qubits can be implemented as this, together with the Toffoli gate for three fixed positions, leads to the capability to perform a Toffoli gate on any three qubit states.

[0117] Fig. 5 now illustrates, on the left side, the case of six logical qubits QI to Q6, wherein a Toffoli gate can be implemented on logical qubits QI, Q2 and Q3. First, one can use the Toffoli gate, in conjunction with single-qubit operations, to implement local swaps between any two logical qubits that are simultaneously connected to the mediator qubit, i.e., the special first qubit, thus generating swaps {QI, Q2 , {Q2,Q3} and {Q3, QI}. These swaps are shown, on the right side, in the N = 6 vertex graph formed by the six logical qubits as connected edges, i.e., the dashed lines. By now performing the operation Ilexcas illustrated in the middle of Fig. 5, additional swaps can be generated using the Tof foli gates . This process can be repeated recursively, leading to a fully connected graph which represents that all swaps can be implemented, thus demonstrating that universal quantum computing is possible on such a quantum processing unit .

[0118] For the cases illustrated in Figs . 2 and 3 , it i s noted that being able to implement two two-qubit gates , including the two swap gates , together with the above discussed rotation brings one into the same starting configuration as for the case illustrated in Fig . 1 . Hence , from this point the above discussion applies equally and hence also configurations as shown in and discussed in connection with Figs . 2 and 3 allow for universal quantum computing .

[0119] In order to ensure connectivity between the logical qubits in line with the above , in a quantum processing unit according to the present disclosure , the special second qubits may be among the qubits in which the logical qubits are encoded, and the special second qubits may be such that they comprise even logical qubits and odd logical qubits . This relates in particular to quantum processing units as the one illustrated in Fig . 1 .

[0120] Similarly, in a quantum processing unit according to the present disclosure , the first and second special second qubits may be among the second qubits in which the logical qubits are encoded, and the first and second special second qubits may be such that they comprise even logical qubits and odd logical qubits . This relates in particular to quantum processing units as the one illustrated in Figs . 2 and 3 .

[0121] In slightly more technical terms , this can be understood, with respect to the case illustrated in Fig . 1 , as follows : Given that the rotation of states of logical qubits treats even and odd logical qubits di f ferently, the Tof foli gate has to break this separation such that swaps from the even to the odd and vice versa become possible . As the Tof foli gate is able to induce swaps among all the logical qubits it acts on, having a combination of even logical qubits and odd logical qubits is suf ficient to achieve this .

[0122] In terms of Figs . 2 and 3 , this then follows similarly : Since the two two-qubit gates of these quantum processing units serve the same function, namely performing swaps , it is clear that also here having one two-qubit gate operating on an even logical qubit and an odd logical qubit breaks this separation, while the other two-qubit gate operating on a pair of even and even or odd and odd logical qubits allows to perform the swaps not requiring to break the separation .

[0123] As will be seen below, this can also be expressed in terms of the parity of the logical qubits .

[0124] Fig . 6 shows illustration of scalable quantum processing units according to the present disclosure . Speci fically, Fig . 6 illustrates that while Figs . 1 to 3 have all shown quantum processing units with eight logical qubits in speci fic configuration, this is in no way limiting and the quantum processing units can be scaled up to any si ze . In fact , as can be readily understood from the three drawings of Fig . 6 , this structure can be scaled to an arbitrary number of ( even) logical qubits by simply extending the " loop" . Further, when comparing Fig . 6 with Figs . 1 to 3 , it becomes clear that the relative position of the qubits is not speci fically limited : While Figs . 1 to 3 showed the set of mediator qubits as inner qubits , Fig . 6 shows possible configuration in which the set of mediator qubits ( in both cases being a special first qubit ) is an outer qubit and some of the first and second qubits are inner qubits . Moreover, it is clear that the extension to larger qubit numbers by simply elongating the quantum processing unit is not limiting, and other configurations are possible as well. In fact, it is the first two-qubit interaction forming the "loop" that may be seen as limiting the structure of quantum processing units according to the present disclosure.

[0125] Fig. 7 shows, in Figs. 7A to 7C, illustrations of further scalable quantum processing units according to the present disclosure .

[0126] As can be seen from Figs. 7A to 7C, each of these quantum processing units comprising first qubits (shown with solid lines) , second qubits (shown in dashed lines) and third qubits (shown in dotted lines) . Accordingly, a first, second and third source, each configured to drive the first, second or third qubits, respectively, is provided.

[0127] Further, in these quantum processing units every second qubit, i.e., the qubits of the second qubits, is a logical qubit, as indicated by the labels Qito Q8, and only a single qubit is provided by these logical qubits. In other words, the triplets labeled S to S8in the above description are replaced with single qubits alternating between first qubits and third qubits.

[0128] In other words, in quantum processing units as shown in Fig. 7, for each logical qubits only one additional qubit is provided, rather than three additional qubits as in the quantum processing units as shown in Figs. 1 to 3. Thus, the overall number of qubits required for a quantum processing units having N logical qubits can be reduced from 4N + 1 to 2N + 1, that is, can be effectively halved. Moreover, as can be seen from Figs . 7A to 7C, every other second qubit is a crossed qubit , that is , every other second qubit is such that it is driven with at least twice the second Rabi frequency by the second source . This allows to selectively address these two subsets of second qubits .

[0129] The mediator qubits as well as the general considerations for performing single- , two- and / or three-qubit gates discussed above remain applicable . In fact , as can be seen from Figs . 7A to 7C, in particular when comparing them to Figs . 1 to 3 , the structure of mediator qubits is identical and hence all of the above considerations can be applied correspondingly .

[0130] The main di f ference between the quantum processing units shown in Fig . 7 and in Figs . 1 to 3 lies in the way the movement of logical qubits is facilitated . As discussed in connection with Fig . 4 , movement of the logical qubits can be facilitated by simultaneously rotating quantum states of odd ( even) logical qubits clockwise , rotating quantum states of even ( odd) logical qubits counter-clockwise , and trans forming states of each triplet from the ferromagnetic state to the paramagnetic state and vice versa .

[0131] In the quantum processing units shown in Fig . 7 , this can be facilitated by recogni zing that this movement corresponds to a ( collective ) SWAP gate between pairs of logical qubits , wherein one of the second qubits is a crossed second qubit and the other second qubit is a "normal" second qubit . Such a SWAP gate can be implemented akin to the above discussed swap operations facilitated by the mediator qubits when considering the first or third qubit provided between the two second qubits as the mediator qubit . Importantly, by providing first qubits driven by the first source and third qubits driven by the third source and by arranging the first and third qubits alternatingly, the same movement as discussed above in connection with Fig . 4 can be achieved as this allows two di f ferent types of collective ( global ) SWAP gates resulting in rotating quantum states of odd ( even) logical qubits clockwise and in rotating quantum states of even ( odd) logical qubits counter-clockwise .

[0132] In line with the above , according to the present disclosure , a quantum processing unit may comprise : first qubits , second qubits and third qubits , wherein the second qubits and the qubits of the first and the third qubits are arranged alternatingly, wherein, within the qubits of the first and the third qubits , the first qubits and the third qubits are arranged alternatingly, wherein each two adj acent qubits are subj ect to a first two-qubit interaction, wherein each qubit is adj acent to at least two qubits ; a first source configured to drive the first qubits with a first Rabi frequency; a second source configured to drive the second qubits with a second Rabi frequency, wherein every other second qubit is such that it i s driven with at least twice the second Rabi frequency by the second source ; a third source configured to drive the third qubits with a third Rabi frequency; and a set of mediator qubits .

[0133] In other words , as regards the structural arrangement of the qubits , a quantum processing unit may comprise a structure formed by repeating units ABAC, wherein A represents a second qubit , B represents a first qubit , and C represents a third qubit . Further, the structure preferably consists of the repeating units .

[0134] In a quantum processing unit according to the present disclosure , the first source , the second source and the third source may be configured to simultaneously : rotate quantum states of odd ( even) logical qubits clockwise , and rotate quantum states of even ( odd) logical qubits counterclockwise .

[0135] In summary, the present invention provides quantum processing units that employ 0(N) qubits driven by a small (not more than four ) number of global controls to facilitate universal quantum computation on N qubits . In this manner, control of the quantum processing unit is not a matter of controlling each individual qubit on its own, but control can be addressed on the simpler global level .

[0136] Further, in particular when using solid-state device to reali ze the qubits , for example , superconducting circuits , this addresses one daunting issue of quantum computing : the wiring problem . Speci fically, the globally driving quantum processing unit requires far less wiring since it is not necessary to provide an individual wiring to each qubit to control this qubit .

[0137] Moreover, in particular in the case of superconducting circuits , in which unwanted ZZ interaction remains a severe problem in conventional architecture , this type of interaction be used to facilitate the blockade , akin to a Rydberg blockade suggested by Cesa and Pichler and then trans ferred to a quantum processing unit employing C?(7V2) qubits for universal quantum computation on N qubits by Menta et al . , to facilitate the operation of the present quantum processing unit . Thus , a predominately negatively seen aspect can be turned into a driving factor of a quantum computing architecture . In order words , in addition to the advantages provided by Menta et al . over conventional quantum computing architectures , the present disclosure substantially improves the scaling of the ratio physical qubits to logical qubits while maintaining the important characteristic of the quantum processing unit being globally driven.

[0138] Finally, it is noted that the present invention is not particularly limited to a quantum processing technology, specifically should not be seen as limited to the use of superconducting circuits.

[0139] Further technical details

[0140] In the following, further technical details relating to the above discussed quantum processing units are presented. These details serve in particular to provide further explanations on some aspects of the present invention. Specifically, these details serve to shed light on the underlying physics considerations on which the present invention is based. It is further noted that this discussion is of a specific model particularly suitable to explain the concepts of the present invention, but the present invention is not limited thereto.

[0141] In line with the above discussion, and focusing on quantum processing unit such as the one illustrated in Fig. 1, but without any loss of generality, in particular regarding the quantum processing units such as the ones illustrated in Figs. 2 and 3, a quantum processing unit according to the present disclosure comprises a (closed loop) formed by 4N (superconducting) qubits, i.e., physical qubits, coupled via a first two-qubit interaction, e.g., a ZZ-interaction, of (possibly uniform) coupling strength illsutra (black springs in the figure) . The loop contains N sites, identified by the symbols Q1,Q2, ... ,QN , which play a special role in the model and will also be referred to as Information Carrying (IC) sites. For all j e {1 N} the IC site Qj is separated from its adjacent counterpart Qj + 1 by a sector Sj which contains three sites. These sites may also be referred to as non-active sites. Additionally, there is an extra (superconducting) qubit placed inside the loop, connected to the first three IC sites QI, Q2, and Q3, via a further, potential the same interaction, (represented by gray elements to emphasize the loop geometry) . The qubits in the model belong to two distinct types (families) : first qubits (also: A-type qubits) , represented as solid squares, and second qubits (B- type qubits) , represented as dashed squares, which alternate in the loop in an ABABA pattern.

[0142] As shown in Fig. 1, all the IC sites host a B-type qubit, while the intermediate sectors Sj are formed by one B-type and two A-type qubits. All qubits of the same type may share the same level spacing h )A / B, except for the elements marked with triangles, which require local corrections due to their higher number of nearest neighbors (e.g., three) compared to the other qubits in the setup. This correction may lead to a level spacing of h a>A / B+ ) . To eliminate the possibility of a residual "swap" term between nearest-neighbor qubits, it may be necessary to have )A )B. Each family of qubits is collectively driven by the same time-dependent external source, termed (t)- Thus, the control is global. The device also includes two inhomogeneities: the additional A- type element inside the loop and one of the three B-type elements connected to it (specifically, the one located at the IC site Q2) . These qubits, referred to as crossed qubits, are used to perform multi- and single-qubit operations. They maintain the nominal level spacing of their respective families but have an augmented coupling with the external source, i.e., the Rabi frequency. It is, however, emphasized that the position of this crossed B-type qubit is not limited to one of the three B-type qubits connected to the additional A-type qubit, as also explained above. As shown by Menta et al., this difference allows independent control of the normal qubits and the crossed qubits of a given family, despite the global control pulse acting simultaneously on them. Both the A-and B-type crossed qubits are marked with a triangle.

[0143] Further, a third control line, KnitCO, acting on a subset of the B-type qubits, may be provided for the initialization of the quantum processing unit. This control line is active only at the initial stage of the processing. Regarding the readout procedure, a possible implementation is discussed below.

[0144] Adopting the same convention of Menta et al., one can write the Hamiltonian of the setup W(t) Ho+ / / drive (0, where

[0145] Eq. (1) describes the local energy contribution of the qubits and their ZZ interactions which are fixed by the geometry of the model, while

[0146] Eq. (2) is the time-dependent driving contribution induced by the control lines, which may be classical control lines. In these

[0147] (x v Z) equations G> represent the Pauli matrices acting on the Hilbert space of the i-th qubit, expressed in the local energy basis {I^X k>} . The summation in the interacting part of Hoencompasses all nearest-neighbor interactions, i.e., all the black segments / springs of Fig. 1. The parameter (^>d,x denotes the oscillation frequency of the driving pulse while and 0%(t) define the time-dependent Rabi frequency and phase of such control. In the following, it is assumed that these quantities take constant values on disjoint timewindows so that, at each time, only one species (either A or B) is externally driven. For simplicity in writing WdriveCO we omit the driving term associated with the control line Vinit(t), which operates solely at the very beginning of the processing on a specific subset of the physical qubits. Additionally, it is not explicitly stated that when in the expression for Hothe site index i identifies a dashed element marked with a black triangle, the corresponding qubit's level spacing becomes h(o)B+ £) . instead of ho)B. . The same substitution is performed when the site index i identifies a solid element marked with a black triangle, i.e., for the A-type crossed qubit. Similarly, whenever in the expression WdriveCO the index i identifies a crossed qubit, is replaced by . Apart from these adjustments, it is important to note that areindependent of the site index i, indicating that they are associated with a control pulse acting globally on all qubits of / -type in the model. It is emphasized that this Hamiltonian is exactly the same as the one studied by Menta et al. and hence also the same as studied by Cesa and Pichler. Therefore, all the relevant dynamical features of the model apply also in this case. In the next paragraph such features are described. Details can be found in the reference papers to which explicit reference is made, as well as in the below discussion.

[0148] Dynamical features . The first ingredient is to emulate the Rydberg blockade effect of Rydberg using the ZZ interactions of the model. Specifically one can show that under the condition T]BR■= » 1, by properly detuning the driving frequenciesMd,x from the nominal level spacing of the qubits, one is able to selectively induce transitions among states of the system only when two nearest-neighbor sites do not simultaneously occupy their excited levels. The second fundamental ingredient is the presence of two different values of the Rabi frequency for qubits of the same family, i.e. regular qubits and crossed qubits. Crucially, one can prove that this difference is sufficient to drive independently these two types of qubits even though the control is global, see also Menta et al. for details. Specifically, one is able to perform generic unitaries

[0149] (unitary operations) defined as:

[0150] Eq. (3) where / xand / rare the subsets of / that includes all its crossed and the regular (non-crossed) elements respectively. For £ e { / x, / r}, the operator acts uniformly on all the qubits in via a control-unitary transformation. Such transformation, depending on whether the neighbouring sites of i E % are all in the ground state, applies to such an element a single-qubit rotation Ri(0,ri) ■=e-ie / 2n-a<-^ parametrized by the 3D unit vector n and by the angle 0 6 [0, 2TT] .

[0151] In particular, given i E the operator which enters in the definition of W^0,n), is the projector on the subspace of the nearest-neighbouring / -type qubits of such site which are in the ground state \g), and Q^) the orthogonal complement of P(i)- If the index i identifies a regular / -type qubit we have P{iy.= \gg){gg\ and Q{i}:= |ee)(ee| + \eg){eg\ + \ge){ge\, where \gg), \e9)r \9e)r and |ee), represent the energy levels of the two qubits of / -type that exhibit a ZZ coupling with such site. On the contrary if i identifies a dashed (solid) element with a black triangle then there are three interacting B (A) -type sites so that ■= \eee){eee\ + \eeg)(eeg\ + ...+ \gge){gge\.

[0152] Note that the unitaries of Eq. (3) where one and only one of the parameters O', d" differs from zero, correspond to scenarios where selective operations on either / xor / rare performed .

[0153] Information encoding and exchange operations . In the setup of Fig. 1 the logical information is encoded in the IC sites Q1,Q2 QN, which are thus also referred to as logical qubits, with the intermediate sectors S1,S2, ... ,SN , also referred to as triplets or triples, acting as separators. It is noted that this may be seen as a peculiarity of the model which has not analogue in the works by Menta et al., and Cesa and Pichler, where instead the location of the N logical qubits can rigidly drift along the entire 2D array of physical qubits of the device.

[0154] Specifically, a generic N-qubit logical state \lP) = ^2> ■■■> kfi) is expressed in one of the two possible well-formed configurations, I'FjFF) or I'FjFF). Both these vectors have the central crossed A-type qubit in the ground state, and the intermediated sectors Sj in an alternating sequence of "ferromagnetic" (e.g., IO:= \999» or "paramagnetic" (e.g., IO:= \geg)) phases. This may require N to be an even number. Further, it is noted that the specific geometry is not relevant for the functioning of the quantum processing unit. Any deformation of the closed loop is possible, as long as the A-type crossed qubit is connected to two even- (or odd-) indexed qubits and to an odd- (or even-) indexed one . In particular IV7; FP) ( resp . I'FjFF)) sets Sj in |F)sy if j is odd (even) , and in |F)sy if j is even (odd) , so that

[0155] Eq. (4) and \V;PF) = \V;FP)^P}. Note that, when the system is initially configured with all qubits in the ground state, the control KnitCO can be used to initialize the system in a vector of the form I^FP), see Fig. 1. This process brings all the associated qubits into the | e) state. As mentioned in our architecture, single and multi-qubit gates can be activated on specific sites which host, or which are directly coupled to, crossed qubits (e.g., QI, Q2 and Q3) . This implies that a fundamental prerequisite to perform QC may be the ability to coherently exchange positions of the logical qubits. This can be achieved through a sequence of eight alternating global pulses, i.e. Ilexc with n^r := WAr(n,X) acting as a conditional-bit-flip only on the regular A-type qubits, and nB:= WB(TT, X; n, x) acting as a conditional bit-flip on all B-type qubits, including the crossed one. As shown below, when acting on a FP (resp. PF) well-formed state I'PjFP) (I'FjFF)) the transformation Ilexcruns in parallel N two-qubit swap gates ^QjQj+ionthe couples {Q1,Q2},{Q3,Q4} . {QN-1,QN}, (resp. {Q2, Q3}, {Q4, Q5} . {QN ,Q1}) while exchanging the ferromagnetic and paramagnetic phases of the intermediate sectors producing a PF (resp. FP) output configuration, see also Fig. 4. The resulting motion of the logical qubits acquires the following structure. One has that with the vector |'F(£2 'Oe ) ( resp . |^2,oe)) obtained by applying to \lP) a Z-step clockwise rotation of the internal states of the sites Qj with odd ( even) index j, and a contemporary I - step counterclockwise rotation of the internal states of the sites Qj with even ( odd) index j . Note that since the direction of the rotations depends on the initial location of the ferromagnetic and paramagnetic phases as explained below, multiple application of Ilexcpulses do not cancel out . For instance using I times the trans formation on the input state I'Fj FP) leads to for I even, and

[0156] Ilexcl1^; FP) = l^yo -Oe’ PP^ for odd . Of course , for I = N the system goes back to the initial configuration . Exploiting this feature , one can move any IC qubit in any other IC qubit location . Further, suppose one aims to apply the single qubit trans formation R(0, ri) to the logical qubit located in position Qj of the loop . Given the previous explanations , this task is straightforward : one simply needs to use the trans formation IleXCwith I such that the state of Qj rotates into the position Q2 which is hosting the B-type crossed- qubit . Once there , one applies the pulse UBx ■= WBx(0, n) and ( in case needed) , reverse the exchange operation via the sequence ( or using Z -times the inverse of Ilexcas discussed in connection with Fig . 4 and defined as IleXC: = nBnexcnB. This demonstrates that one is capable of performing any single-qubit unitary on any elements of the logical state . In the next paragraph, it is described how a three- qubit Tof foli gate can be implemented, thereby achieving a universal gate set . One-shot Toffoli gate. In conventional QC architectures, the Toffoli gate is performed using two-qubit gates. Here, a three-qubit one-shot Toffoli gate involving a total number of four qubits is implemented: one A-type crossed qubit, which acts as a mediator, and the three B-type logical qubits located in the IC sites QI, Q2 and Q3. One convenient way to implement a Toffoli is by decomposing it into a controlled- controlled-Z (CCZ) gate combined with two single qubit Hadamard gates.

[0157] Since it has already been discussed how to perform singlequbit gates, all that remain is a discussion how a CCZ gate can be realized. The fundamental observation here is that the unitary ZAx ■= WAx(2n,ri) induces a (— 1) phase factor on the state of the system if and only if the three B-type qubits that are connected to it are all in the ground state \g), which apart from a global NOT, is exactly a CCZ applied on the three B-type qubits.

[0158] The transformation ZAx is obtained from Eq. (3) for / = A, setting O' = 0 and d" = 2n , independently from the choices of n' and n" . It corresponds to where the projector P{Ax)r due to the fact that the A-type crossed-qubit has three connections, can be written as P{AX):=IS'S'S')(ggg|, while the complementary projector is Q{AX) -= +

[0159] |eeg)(eeg| + ... + |gge)(gge|.

[0160] Accordingly setting n = (l,0,l) / V2, the transformation

[0161] 7j3^2:=WB*(7i,ri)WB(n, X)ZJ4XJ / / B(TT, x)WBx(n,n),

[0162] Eq. (5) corresponds to a Toffoli gate where QI and Q3 act as controllers and Q2 as the controlled qubit. In the above expression PVgC^x) stands for WB(n, x; n, x) . It is noted that this discussion can be generalized to an arbitrary number of qubits connected to the central A-type crossed qubit: indeed in principle a N-qubit Toffoli gate can be realized in our setup as well.

[0163] Universality. In the previous sections we have seen that, using the global controls V^Ct) and V_B (t), the architecture of Fig. 1 allows us to implement any single-qubit operation on each site that encodes the logical information, the exchange gate Ilexcas well as a Toffoli gate T13^2• To prove that this is sufficient to guarantee universal QC in the model, we now show that using the above resources we can induce arbitrary swap gates U^QP+1between any two IC sites, that is, logical qubits. These operations will then be used to convert T13^2into Toffoli gates that couple all possible triples of the model, thus leading to universal QC .

[0164] It should first be noted that the Toffoli gate, in conjunction with single-qubit operations, can be employed to implement local swaps between any two qubits that are simultaneously connected to the central A-type crossed qubit. Starting from T13^2we can generate swaps gates between each of the couples {QI, Q2), {Q2, Q3], and {Q3, QI] which we represent as edges of a N vertex graph formed by the active qubits of the quantum processing unit, see also Fig. 5 and corresponding discussion above. Next, Ilexcis applied to rotate the active qubits, e.g., inducing the mapping (QI, Q2, Q3) (Q2, QI, Q4), and repeat the whole procedure obtaining two new swaps among the couple {Q1.Q4} and {Q2,Q4} which allows us to draw two new edges in the graph. Note further that by combining the swaps generated in this manner, one also induce an extra new swap between {Q3, Q4}, that is, UQ3Q4= ' leading to the subgraph formed by the sites QI, Q2, Q3 and Q4 fully connected . Proceeding along this way, by recursion one can show that all the other vertices can also be included in a fully connected graph, meaning that we can generate all possible swaps hence showing that this set of operations is suf ficient for universal quantum computing .

[0165] Di scussi on . The model we proposed represents an improvement in terms of scalability in comparison to the model discussed by Menta et al , and Cesa and Pichler, which need of C?(7V2) physical qubits to implement N logical qubits . Indeed, in the present disclosure , a (9(N) scaling is obtained .

[0166] It is furthermore important to stress that all considerations regarding the physical requirements for the implementation of this setup remain consistent with those presented by Menta et al . , as the physical obj ects involved are identical . In other words , current technology is able to provide the building blocks required for the quantum processing units discussed in the present disclosure .

[0167] With respect to the work by Menta et al . , it is noteworthy that the usage of a one-shot Tof foli gate could improve the execution time as well as the fidelity of many quantum algorithms .

[0168] Further, it is noted that while the present discussion focuses on implementations based on superconducting circuits , i . e . , superconducting circuits , the concepts of the present disclosure can be readily implemented in di f ferent physical platforms , such as Rydberg atoms or spin qubits . Further information containing additional technical details will be provided in the following sections . As a remark, these sections may us a slightly di f ferent notation in that operators are indicated by corresponding symbols , "hats" named in the technical field, however, the same quantities and concepts are referred to and discussed as above .

[0169] Further technical details on the quantum processing unit

[0170] In the following, additional technical details in order to clarify the results presented in the main text. In Section I, we give a characterization of the exchange operator IIexcdiscussing how it acts on the well-formed states of the model. In Section II, we prove the universality of our globally driven architecture. In Section III, we provide a concise overview of the initialisation and readout procedures. Finally, in Section IV, we propose possible alternative variants to the architecture described in the main text.

[0171] I. CHARACTERIZING THE EXCHANGE OPERATOR

[0172] In this Section we discuss in details the action of the exchange operator IIexcdefined in the main text, i.e.

[0173] Recall that induces to the site i of regular (non-crossed) .1-lype a controlled rotation (times a phase — i), depending on whether all its first neighbouring sites (which are always of B-type) are in the ground state \g) . Specifically, if even one of the neighboring sites of i is in the excited state |e), then l i p will act as the identity on such an element. Similarly does the same on all the crossed and non-crossed B-type qubits (in this case the transformation depends on the first neighboring sites which are of A-type).

[0174] In particular we are interested in determining how flexcacts on the well-formed states |rh; FP) and |rh; PF) that encode the logical information in the model. As indicated in Eq. (4) of the main text, given 4m the probability amplitudes of a A-qubit logical state expressed in the computational basis {\g), e)}, such states have the form with where for j g {1, • • • , A}, |F)s3and |P)s3, define the ferromagnetic and paramagnetic vectors of the sector Sj. It is worth remembering that in our setup, all the qubits associated with the IC sites Q / s are B-type qubits. Recall also that each sector Sj is formed by three neighboring sites: the first and last hosting a regular (non-crossed) .4-l vpe qubit, which we name / I1 1and A®, and the second hosting instead a B-type qubit which we name B - , i.e.

[0175] M := (A«, BP, AP)) . (7)

[0176] Accordingly we can conveniently express the vector |F)B. and |P)s3as

[0177] A. Evolution of the well-formed state under llexc

[0178] To study the action of Hexcon the well-formed states we shall proceed step-by-step analysing the role of the individual pulses that compose it. To simplify the analysis we shall focus on the individual components (5) and (6), then invoke linearity to reconstruct the evolutions of the vectors ’P; FP) and |tP; PF). For this purpose it is useful to observe the following facts: tot i) The Information Carrying (IC) site Qj admits as neighbouring sites the third element Aj_1of the sector Sj-i, and the first element A)^ of the sector Sj+±, i.e.

[0179] / |(3) _ Q _ A A1) j + 1 ’ (9)

[0180] Accordingly the operator I I / ; will act on Qj as if and only if both A)^ and A)^ are in the ground configuration. No direct transformation on Qj is induced by l i p . Note that the first three IC sites Qi, Qz, Q3, also have the central crossed ,1-lype element as neighboring site. However, since this element is always initialized in the ground state \g) and IIexcincludes no transformations that acting on such term, the central crossed .1-lype element has no role in controlling the evolution of Qi, Q^. Q3 under the transformation (1). ii) The central site of the sector Sj admits as neighbouring sites the first Aj11and the last A® of the same sector

[0181] AjX)- B<2)- A® . (10)

[0182] Accordingly the operator I I / ; will act on B^ as if and only if both .4^1and A® are in the

[0183] (2) ground state. No direct transformation on Bj7is induced by l i p . hi) The first site A^of the sector Sj admits as neighbouring sites the second element B^ of the same sector and the IC site Qy-i,

[0184] Accordingly the operator l i p will act on .4^1as ifan(l only if both Qj-i and B^ are in the ground state. No direct transformation on A^ jsinduced by I I / ;. iv) The third site A)7of the sector Sj admits as neighbouring sites the second element Bj7of the same sector and the IC site Qj+i,

[0185] B<2)- A® - Qj+1. (12)

[0186] ~ ( 3) (2}

[0187] Accordingly the operator l i p will act on A7if and only if both Qj+i and Bj7are in the ground state. No direct transformation on Aj7is induced by I I / ;.

[0188] Equipped with the above observations we can now proceed with the step-by-step analysis of the evolution induced by the unitary operator (1):

[0189] 1. First pulse (Hpir):This transformation acts directly only on the regular A-type sites. In our model they are only present inside the sectors Sj which in the input state are either in a paramagnetic (IP)S' ) or a ferromagnetic (|F)s3) configuration. Note that the presence of the |e)s(2> element in the formula of |P)s3prevents l i p from modifying such part of the vectors (5) and (6), i.e.

[0190] To evaluate the effect of l i p on the components |F)sr. , we need to take into account the state of the neighboring IC sites that are directly connected to them, i.e. the vectors \kj)' Qjand \kj+i)Qj+1. In this case we have

[0191] Equations (13) and (14) together determine the evolution of fc; FP; g) and fc; PF; <;), and hence of rk; FP) and rk; FP), under the action of the first component l i p of flexc. Second pulse (I I / ;): Let us next apply I I / ; on the transformed vectors which emerge from the application of the first flAroperator. In this case, only the IC site Qj and the central elements of sectors Sj are directly affected by the evolution.

[0192] To begin with note that where we use the fact that the B-type qubit inside paramagnetic sectors gets flipped because is sor- ruonded by \g} qubits. To evaluate the action of I I / ; on the vectors (14) we need to include their first tot neighbouring sites: the third A-type qubit of the sector Sj-i (i.e. AJA J and the first .1-lype qubit of of the sector Sj+± (i.e. A^J. By construction, these qubits were originally assigned to paramagnetic phases, and hence (due to Eq. (13)), have remained in the state \g) they started from. Accordingly, they do not prevent the action of I I / ; on the B-type qubits of the vectors (14); the only control being exerted by and A®. Therefore we can write

[0193] Note that according to the above expression after the action of 11 / ; I I p all the IC qubits Qj are in the ground state so they never blockade the action of the next l i p pulse. As usual combining (15) and (16) we obtain the evolution of fc; FP; g) and fc; PF; <;), and hence of ’F; FP) and rf; FP), under the action of the first two components I I / ; l i p of flexc. Third pulse (HAP) : As for the first pulse, this operator acts directly only on the regular .1-lype sites. The first thing to note here is that the neighboring sites of the .1-l ype qubits of the vector (15) are IC sites that are in the ground state. Then we can write which is a paramagnetic vector where the \g) and |e) components have been inverted. The evolution of (16) under l i p is instead obtained as follows

[0194] Combining (17) and (18) we obtain the evolution of k; FP; g) and Ay PF; <7), and hence of rf; FP) and rh; FP), under the action of the first three components l l p l l pd l p of flexc. Fourth pulse (HB):When acting on the term (17), the action of I I / ; is blocked by the excited state of the .1-lype qubits. Therefore in this case the evolution is trivial:

[0195] To evolve the term (18) observe that both left and right neighboring sites of the compound are .1-lype elements of (19) which are in the excited state. Accordingly the sites Qj and Qj+i are not effected by the new I I / ; pulse and remain in the |<7) state. The internal B-type element of the compound is instead

[0196] As usual combining (19) and (20) we obtain the evolution of fc; FP; g) and fc; PF; <7), and hence of / _ _ X 2 rh; FP) and rf; FP), under the action of the first four components ( l l / ; l l p ) of IIexc. Fifth pulse (flAr):Recall that from the previous Section we have learned that all Qj states are still the ground state. This implies that the A-type qubits of (19) have all first neighbouring sites in \g). The action of l i p on them is hence simple:

[0197] To evolve (20) observe that the internal .1-l vpe elements of the compound are controlled by the B)7element (the other B-type sites being in the ground). Therefore

[0198] Combining (20) and (21) we obtain the evolution of k; FP; g) and fc; PF; g), and hence of rh; FP) and / _ _ \ 2 _ ' rh; FP), under the action of the first five components l i p ( 11 / „> I I p ) of IIexc.

[0199] ( 2) Sixth pulse (HB): The action of HB on (20) is simple as both the neighboring sites of B)7are in the ground state:

[0200] To evolve (21) observe that neighbouring sites of the compound are .1-lype qubits that are in a \g) state. Hence the action of I I / j on the internal B-type elements of the vector are only controlled by the .1-lype element of the compound itself, i.e. A'1 1 Hence we can write

[0201] Combining (22) and (23) we obtain the evolution of k; FP; g) and fc; PF; <y), and hence of rh; FP) and . \ 3

[0202] |^; FP), under the action of the first six components I nBllAr) of IIexc. Seventh pulse (flAr):Th® action of l i p on (22) is simple as the internal B-type qubits prevents the operator from modifying the state, i.e.

[0203] To evolve (23) note that the neighboring B-type sites are only internal elements of the compound itself. Accordingly we can write 5)

[0204] Combining (24) and (25) we obtain the evolution of k; FP; g) and fc; PF; g), and hence of rf; FP) and / _ _ \ 3 rh; FP), under the action of the first seven components l i p I l l / ; l l v I of IIexc.

[0205] 8. Eigth pulse (nB): The action of nBon (24) is simple as the internal .4-l vpe qubits are both in the ground state:

[0206] (nsnAJ |P)SJ= nexc|p)Sj= (-06nAr|p)Sj(26)

[0207] = (-?:)6nAdff)A<i) le)Bm \g)Af< > = H)7Iff)Ao) lff)

[0208] To evolve (25) note that the external first- neighboring .4-l vpe sites of the compound are in the <7) state so do not prevent the action of the pulse. Therefore

[0209] Combining (26) and (27) we obtain the evolution of |fc; FP; g) and |fc; PF; g) under the eight components

[0210] The last equation shows that whenever two consecutive IC sites of the system are separated by a ferromagnetic region |F)B. , their internal state are swapped by l l, .;r. while |F)B. gets replaced by the paramagnetic term P)s3. Equation (26) instead says that under nexc, the components |P)s3of (5) and (6) gets transformed into |F)B . Accordingly we can write where dropped an irrelevant global phase which does not depends upon the input state of the system. A final simplification arises by observing that the sequence of the swapping gates corresponds to a single-step, clock-wise rotation on the internal state of the IC qubits Qj with j odd, and a single-step, anti-clock-wise rotation on the internal state of the IC qubits Qj with j even. Similarly

[0211] ' ^QN' QIcorre8Ponds to a single-step, clock- wise rotation on the internal state of the IC qubits Qj with j even, and a single-step, anti-clock-wise rotation on the internal state of the IC qubits Qj with j even. As an illustrative example consider for instance the case with N = 6 elements.

[0212] At the level of the vectors (4) this leads to the identities reported in the main text.

[0213] B. Concatenation rule

[0214] An important aspect of the evolution (32), is that iterative applications of flexcon a well-formed state do not cancel out. In particular we have ( 2) where now the logical state of FP) (resp. |1Ir(2>(,oj PF)), is obtained by applying a two-step, clockwise rotation on the internal state of the active IC sites Qj with j odd (even), and a two-step, anti-clock-wise rotation on the internal state of the IC qubits Qj with j even (odd). The reason for this is that, due to Eq. (32), the odd and even IC sites of f[exc|1I'; FP) have been exchanged. Therefore when we act with flexcon such configuration, despite the fact that the system is now in a PF well-formed vector, the net effect is still to induce an extra single-step clock-wise rotation on the IC qubits with j odd, and an extra single-step anti-clock-wise rotation on the IC qubits with j even. Building upon this for t integer, it then follows that we can write

[0215] } I^Oo Oe ’ FP^ for I? even, } ’I' QJ PF) for I? even.

[0216] / „ X t

[0217] Observe that of course for t = N, the unitary ( IIexc) acts as the identity transformation, i.e. due to the fact that ’P / o = ’P. Accordingly the action of I IIexc) is ’’inverted” by I IIexc)

[0218] An alternative way to realize such effect is to use the transformation that effectively acts as the inverse of flexcwhen operating on well-formed states (the proof of this assertion follows from the same derivation presented here). A direct consequence of (34) is that, given any target values j,j' € {1, • • • , IV}, we can use our control pulses to induce a transformation that brings the input state of the j'-th IC site of any well-formed state into the / -th IC site.

[0219] II. UNIVERSAL QUANTUM COMPUTING

[0220] Here we prove that, using the encoding provided by well-formed states |1H; FP) and |^; PF) the setup allows for universal QC. The starting point of the analysis are the following facts:

[0221] 1) Using iterative application of the transformation flexcwe can induce cyclic rotations among the IC sites.

[0222] 2) We can realize all possible single qubit transformation on any IC sites of the model.

[0223] 3) We can realize the Toffoli gate TI,3^2 which has Qi and Q3as the controller qubits and Q2as the controlled one.

[0224] Thanks to these properties, universality can be proved by simply showing that one can induce individual, two-qubit swaps among all possible couples {Qj. Q'A of IC qubits. Indeed if we attain such a task, then we can convert Ti,3-»-2 in an arbitrary Toffoli transformation Tj1that couples each possible triple {Qji, Qj2, Qja} of the system. Then we can invoke the fact that universal QC is granted as soon as you can induce arbitrary single-gate transformations (point 2) of the above list), and arbitrary Toffoli gates.

[0225] A. Inducing all possible swap transformations

[0226] Here, we show that using the properties 1), 2), and 3), we can generate all the individual two-body swap gates among the IC sites of the model. This problem can be mapped into a graph problem. The idea is to represent each IC site of the setup as individual vertex of a graph and to draw an edge between two of them if and only if there is a sequence of operations that, using the properties 1), 2), and 3), allows us to implement the swap gate between the corresponding IC elements. In this context, proving the thesis means being able to show that in the end the graph is fully connected.

[0227] Let us start from some preliminary observations. Given a, b, c qubits, the CNOT gate U^c withabeing the controller and c the controlled element, can be realized by concatenating two Toffoli transformations Ta,b^cplus two local operations on the b qubit, i.e.

[0228] This can hence be transformed into a CNOT gate U^a where c is the controller and a the controlled qubit, by using extra local operations on a and c, i.e. (38) with Haand / / , bei we can then realize a swap gate among a and c, i.e.

[0229] Since Ta,b^cis symmetric with respect to the exchange between a and 6. the previous analysis can also be used to show that using Ta,b^cand local operations, also the swap gate Ub^apis attainable. From that we can finally construct L',)('a|' by simple concatenation of the previous two, i.e.

[0230] Ujap= U™apU6scwapU™ap. (40)

[0231] Accordingly we can say that fa,b^c+ local ops {^:ap, cwap> ^rp} • (41)

[0232] Thanks to this result, from the properties 2) and 3) we can conclude that in our graph problem we can draw at least three edges among the sites Qi, Q% and Q3. Let us now use flexc to induce a rotation of these sites. Recalling that the even and odd sites of the model counter-propagate, we can ensure that after this transformation, the sites (Qi, Q%, Q3) are mapped to (say) (Qzj Qi , Qi)- In conjunction with this enables us to realize the Toffoli gate T2,I->4- Invoking (41), this implies that we can also acquire all the swap gates between Qi, Q% and Q \ . Additionally, we can have an extra swap between Q3 and Q -, as a consequence of the composition rule (40). This implies that in our problem, the first four sites are fully connected. Applying further rotations will increase the number of edges. To show that in the end, we can fully connect the entire graph observe that, if we started from a FP well-formed state, iterative applications of ftexc will force the odd sites to rotate clock-wise. In particular, after an even number of applications of nexc, in the position originally occupied by Qi and Q3, we will have a generic couple of consecutive odd elements C^'-i and <?2j+i- Accordingly, using we can now generate 72j-i,2j+i^2j' ■ with 2j' being some even index that is not important to determine at this level. Hence, using (41) and (40) we can conclude that we will be able to connect the two odd sites Q2J-1 and (?2j+i with an edge. Since j is arbitrary, this implies that in our model, the subgraph associated with the odd sites is fully connected. A similar argument can be used to conclude that also the subgraph associated with the even sites is also fully connected. Note also that the odd and even subgraphs are connected by at least one edge (e.g. the one associated with the swap gate between Qi and Qz). Invoking the percolation property (40), this single connection can then be used to easily verify that any other edges connecting the two subgraphs is also achievable, concluding the thesis.

[0233] III. INITIALIZATION AND READ-OUT

[0234] A. Initialization

[0235] As mentioned in the main text we can initialize the system into a well-formed state starting from a configuration where all the sites (including the IC ones) are in the ground state. For this purpose it is indeed sufficient to use the control I4nit(t) to induce a 7T-pulse that brings the associated qubits from \g) to e). Since such elements are internal B-type qubits of alternating sectors of the device, this will force those sectors to assume a paramagnetic phase P). In the case of the scheme of Fig. 1 of the main text this, will produce a FP well-formed state with |^o) being the logical state where all the IC qubits are in the ground state.

[0236] B. Read-out

[0237] Since the read-out of multiple qubits can be done only locally on each qubit, we cannot perform the read-out directly on the quantum processing unit (i.e. the architecture design of Fig. 1 of the main text). One idea can be to use a register, which is simply an additional conveyor belt-like wire with no .4-lype crossed-qubit inside, but still with a single B-type crossed qubit to perform single-qubit gate on it. Such additional area may be used to host all the quantum information moving from the processing unit, at the moment of the read-out. Since this additional read-out area is not involved in the computation, we can use local control lines on the Q)ead~out, . . . , Q'):"1elements of the additional setup. The two conveyor belt systems (the processing unit and the read-out register) are coupled via an additional G-type qubit (controlled by an additional source Vc'(t)), which is in turn ZZ coupled to the two B-type crossed qubits of the processing unit and read-out area, respectively. Such G-type qubit allows to implement a two-qubit operation between the qubits of the two islands, the logical state of processing unit and the “empty” qubit of the read-out register.

[0238] Once the computation is over, let us call the final well-formed state rE; FP; <7), the protocol for the read-out procedure can be schematised as follows a) Initialization of the read-out register.

[0239] The whole state of the read-out area is initialized as explained above for the processing unit area, see Eq. (5). b) Transfer of the quantum information into the read-out area.

[0240] Combining single-qubit local operations on the two B-type crossed qubits of the two islands, with two-qubit gates performed on the G-type inter qubit which couples the two islands, a SWAP gate can be performed [1], This allows us to transfer (swap) a logical qubit form the processing unit the read-out area (and vice-versa). c) Total resetting of the processing unit area. and. transfer completion.

[0241] As explained in the main text and in Sec. II, through sequences of global pulses (1), we are able to bring each logical state in the position corresponding to the B-type crossed element of the processing unit area. Once done, we repeat the step b) and subsequently c) until the final well-formed state of the processing unit area will be completely moved into the read-out area and vice-versa |^'; FP;fl) o |^0; FP;fl)read-out.

[0242] Upon completion of the aforementioned step-by-step procedure, each logical state within the read-out area can be measured independently. The objective of this section is to emphasise the potential for a separate globally driven read-out area, equipped with N local control lines (probes) used to make measurements.

[0243] IV. ALTERNATIVE DESIGNS

[0244] In this Section we discuss two alternative ways of implementing multi-qubit gates, which allow to perform a universal quantum computation. These alternative implementations are based on two-qubit gates, which can be performed by simply cutting one of the three connections to the central crossed .1-lype element [2], In this case, (regular, crossed or double-crossed) qubits inside the conveyor-belt QC will mediate a two-qubit CZ gate, instead of a three-qubit CCZ gate.

[0245] A. Proof of universality

[0246] Suppose we are able to perform two-qubit gates between the logical qubits of two pairs of 10 site, say (Qi, Q3) and (Qi, Qz)- Two possible ways of independently decide which of the two pairs we are controlling will be presented later on. The combination of the CZ gate with single qubit operations allows to implement any possible two-qubit gate, including the swap gates and Thanks to Eq. (40), we are also able to implement UQ^3, making the graph formed by the three IO sites fully connected, which is exactly what we proved for the case of the Toffoli gate. The same proof for the universality presented in Section II can thus be applied to the present case.

[0247] B. Independent two-qubit gates

[0248] Referring to the discussion above, how do we select which pair of qubits, (Qi, ^) or (Qi, ^),weare acting on? It does not suffice to connect such qubits via two .1-lype crossed elements. Indeed, since the control is global, the pulse Z< := (2TT, n) (see main text) would perform a CZ gate simultaneously on the two pairs, and the control would not be independent. A possible way of breaking this symmetry is to employ an additional control line. In this alternative design, the two qubits placed inside the loop belong to a third species named C, and are controlled by a third control line Vcft). To independently control them, we require one of the two, say the one connecting Qi and Qx- to be crossed (double Rabi frequency) [2], Then, the pulse TTc>r(27r, n) implements a CZ gate between the qubits at sites Qi and Q3, while the pulse Wpx (2TT, n) implements a CZ gate between the qubits at sites Qi and Q3.

[0249] It turns out that it is possible to maintain the two species scheme. Given three qubits of the same species %, with Rabi frequencies equal to Qx, 2QXand 4QX, an independent control of the three can be performed, i.e. referring to the qubits with frequency 4QXas double-crossed (X) elements, we can perform where TVxr(0';n'), Wxx (O", n") and TVxx(0" / , n'") apply only to the regular, crossed and double-crossed qubits, respectively. In the case of Fig. 2, there is only one crossed element of the A type. We stress that all angles and vectors in the above equation are completely independent. Following the discussion of the previous implementation [2], it is straightforward to prove that, by making one of the two qubits inside the loop crossed and the other double-crossed, we achieve a universal computation.

[0250] The prove of this fact can be obtained as a generalization of the argument presented in Ref. [2]. Let us start observing that applying a control pulse VA(t) on a time- window T with constant values of QA)^) and we can induce the evolution

[0251] WA (0, n±; 20, n±; 40, n±) := WA? (0, n±) WAX (20, n±) TU4x(4$, n±) , (44) with 0 = A \ T (T being the duration of the time interval) and n being a vector orthogonal to z. Observe that, due to the choice we made for the Rabi pulses of the various subsets, crossed and double-crossed A- type qubits experience rotations with angles which are twice and four times that of regular A-type qubits. Observe next that given n and m orthogonal unit vectors, the usual single-qubit rotation identities [2] can be generalized as follows: which at the level of the transformation (44) give us where now n and rn are orthogonal vectors in the xy plane, i.e. n ■ mA = 0 = n • z = mA ■ z. Equation (46) shows that there exists a proper concatenation of the control pulses VA)^) that enable one to apply, selectively, correlated control operations WAx(40, n_|_) on just the two subsets or crossed and doublecrossed elements. Since in the previous expression 0 is an arbitrary phase and n. is an arbitrary vector in the xy plane, we can directly invoke the results of Ref. [2] to claim that a proper concatenation of the transformations (46) will generate arbitrary rotations of the form WA* (0" , n") WAx(0'" , n'" ) where now 0" , O'" , and n" n!" are no longer restricted. Using them we can compensate for the crossed and double-crossed terms of (44) ultimately leading to the thesis. We can hence assume that, with the control line VA(£)wecan independently operate on the double-crossed A-type qubits without affecting the regular and crossed ones, and vice-versa.

[0252] Another equivalent alternative design is depicted in Fig. 3, where two qubits (one regular and the other one crossed) of a third species (G-type qubits) are introduced instead of the .4-l vpes qubits of Fig. 2. Such two G-type qubits which mediate the two-qubit gates are driven by a third control source Vc(t), and in order to control them independently the Rabi frequency must be different (one of them is regular with Rabi frequency Qp, the other one is crossed with Rabi frequency 2QG>). In other words, as we have seen above [2], the evolution

[0253] Wc(0', n'; 0", n") := WCr(0', n')WCx (0", n") (47) can be performed.

[0254] Finally, we stress that for the universality proof above, we considered connections between the pairs (<?1, (?2) and (Qij Qa)* this was done merely for a sake of simplicity to reduce the proof the the Toffoli’s proof of Sec. II. In the Figs. 2 and 3 this cannot be done by construction. Therefore the pairs employed are (Qi, Qs) and (Q4, Qs)- However the proof can be generalized to the case of any pair of qubits provided we connect two qubits of the same parity and two qubits of different parity (like the arrangement we used in the figures, i.e. the (Qij Qa) and (Q4, Qs) pairs).

[0255] [1] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information (Cambridge University Press. Cambridge. 2010).

[0256] [2] R. Menta. F. Cioni. R. Aiudi. M. Polini, and V. Giovannetti. Globally driven superconducting quantum computing architecture. Phys. Rev. Research 7. L012065 (2025) [arXiv:2407.01182] .

Claims

Claims1 . A quantum processing unit comprising : first qubits and second qubits arranged alternatingly, wherein each two adj acent qubits are subj ect to a first two-qubit interaction, wherein each qubit is adj acent to at least two qubits ; a first source configured to drive the first qubits with a first Rabi frequency; a second source configured to drive the second qubits with a second Rabi frequency; and a set of mediator qubits .2 . The quantum processing unit according to claim 1 , wherein the set of mediator qubits consists of a special first qubit being such that it is driven with at least twice the first Rabi frequency by the first source , the special first qubit is subj ect to a second two-qubit interaction with respect to three second qubits , named special second qubits , respectively, the first qubits have a first qubit frequency and the special first qubit has a qubit frequency di f ferent from the first qubit frequency, preferably equal to the sum of the first qubit frequency and an interaction strength of the second two-qubit interaction,the second qubits have a second qubit frequency and the special second qubits have a qubit frequency di f ferent from the second qubit frequency, preferably equal to the sum of the second qubit frequency and the interaction strength of the second two-qubit interaction, and one of the second qubits , including the special second qubits , is such that it is driven with at least twice the second Rabi frequency by the second source .3 . The quantum processing unit according to claim 1 , wherein the set of mediator qubits consists of a first special first qubit being such that it is driven with at least twice the first Rabi frequency by the first source and a second special first qubit being such that it is driven with at least four times the first Rabi frequency by the first source , the first special first qubit is subj ect to a second two-qubit interaction with respect to two second qubits , named first special second qubits , respectively, the second special first qubit is subj ect to the second two-qubit interaction with respect to two second qubits , named second special second qubits , respectively, wherein the second special second qubits are di f ferent from the first special second qubits , the second qubits have a second qubit frequency and the first and the second special second qubits have a qubit frequency di f ferent from the second qubit frequency, preferably equal to the sum of the second qubit frequency and an interaction strength of the second two- qubit interaction, andone of the second qubits , including the first special second qubits and the second special second qubits , is such that it is driven with at least twice the second Rabi frequency by the second source .4 . The quantum processing unit according to claim 1 , wherein the set of mediator qubits consists of a third qubit and a special third qubit , the third qubit is subj ect to a second two-qubit interaction with respect to two second qubits , named first special second qubits , respectively, the special third qubit is subj ect to the second two- qubit interaction with respect to two second qubits , named second special second qubits , respectively, wherein the second special second qubits are di f ferent from the first special second qubits , the second qubits have a second qubit frequency and the first and the second special second qubits have a qubit frequency di f ferent from the second qubit frequency, preferably equal to the sum of the second qubit frequency and an interaction strength of the second two- qubit interaction, one of the second qubits , including the first special second qubits and the second special second qubits , is such that it is driven with at least twice the second Rabi frequency by the second source , the quantum processing unit further comprises a third source configured to drive the third qubits with a third Rabi frequency, andthe special third qubit being such that it is driven with at least twice the third Rabi frequency by the third source .5 . The quantum processing unit according to any one of claims 1 to 4 , wherein logical qubits are encoded in every other second qubit .6 . The quantum processing unit according to claim 5 , when dependent on any of claims 2 to 4 , wherein one of the logical qubits is the qubit being such that it is driven with at least twice the second Rabi frequency by the second source .7 . The quantum processing unit according to claim 5 or 6 , when dependent on any of claims 2 to 4 , wherein the special second qubits , when dependent on claim 2 , or the first special second qubits and second special second qubits , when dependent on claim 3 or 4 , are qubits in which the logical qubits are encoded .8 . The quantum processing unit according to any one of claims 5 to 7 , wherein triplets consisting of two first qubits and one second qubit provided between second qubits are alternatingly in a ferromagnetic state or in a paramagnetic state .9 . The quantum processing unit according to claim 8 , wherein the logical qubits are encoded in the second qubits provided between the triplets .

10. The quantum processing unit according to claim 8 or 9, wherein the first source and the second source are configured to simultaneously: rotate quantum states of odd (even) logical qubits clockwise, rotate quantum states of even (odd) logical qubits counter-clockwise, and transform states of each triplet from the ferromagnetic state to the paramagnetic state and vice versa.

11. The quantum processing unit according to any one of claims 1 to 10, wherein the first source is configured to be off if the second source is on, and the second source is configured to be off if the first source is on, and when dependent on claim 4, the second source is configured to be off if the third source is on, and the third source is configured to be off if the second source is on; and / or the first source is configured to drive the first qubits with a first oscillation frequency and a first phase, and the second source is configured to drive the second qubits with a second oscillation frequency and a second phase, and when dependent on claim 4, the third source is configured to drive the third qubits with a third oscillation frequency and a third phase; and / orat least one of the first two-qubit interaction and the second two-qubit interaction is a ZZ-interaction; and / or the first qubits have a first qubit frequency, and the second qubits have a second qubit frequency, and when dependent on claim 4 , the third qubits have a third qubit frequency; and / or the quantum processing unit further comprises an initiali zation source configured to drive every fourth second qubit ; and / or the second source is configured to perform a singlequbit gate on the second qubit being such that it is driven with at least twice the second Rabi frequency by the second source .12 . The quantum processing unit according to any one of claims 1 to 11 , when dependent on claim 2 , wherein the first source and the second source are configured to perform a Tof foli gate on the special second qubits .13 . The quantum processing unit according to any one of claims 1 to 11 , when dependent on claim 3 , wherein the first source and the second source are configured to perform a two-qubit gate on the first special second qubits and / or on the second special second qubits .14 . The quantum processing unit according to any one of claims 1 to 11 , when dependent on claim 4 , wherein the second source and the third source are configured to perform a two-qubit gate on the first special second qubits and / or on the second special second qubits .15 . The quantum processing unit according to any one of claims 1 to 14 , wherein each first qubit and special first qubit are connected to the first source by a first wiring, and each second qubit and special second qubit are connected to the second source by a second wiring, and when dependent on claim 4 , the third qubit and the special third qubit are connected to the third source by a third wiring; and / or the first qubits , including any first special qubits , and the second qubits , including any second special qubits , and, when dependent on claim 4 , the third qubit and the third special qubit , are superconducting qubits ; and / or the first qubits , including any first special qubits , and the second qubits , including any second special qubits , and, when dependent on claim 4 , the third qubit and the third special qubit , are provided in a honeycomb structure ; and / orthe first source is configured to drive the first qubits using a single signal provided globally to all of the first qubits , and the second source is configured to drive the second qubits using a single signal provided globally to all of the second qubits , and when dependent on claim 4 , the third source is configured to drive the third qubits using a single signal provided globally to all of the third qubits ; and / or when dependent on claims 2 and 5 , the special second qubits are among the qubits in which the logical qubits are encoded, and the special second qubits are such that they comprise even logical qubits and odd logical qubits , and when dependent on claim 3 or 4 and on claim 5 , the first and second special second qubits are among the second qubits in which the logical qubits are encoded, and the first and second special second qubits are such that they comprise even logical qubits and odd logical qubits .16 . A quantum processing unit comprising : first qubits , second qubits and third qubits , wherein the second qubits and the qubits of the first and the third qubits are arranged alternatingly, wherein, within the qubits of the first and the third qubits , the first qubits and the third qubits are arranged alternatingly,wherein each two adj acent qubits are subj ect to a first two-qubit interaction, wherein each qubit is adj acent to at least two qubits ; a first source configured to drive the first qubits with a first Rabi frequency; a second source configured to drive the second qubits with a second Rabi frequency, wherein every other second qubit is such that it is driven with at least twice the second Rabi frequency by the second source ; a third source configured to drive the third qubits with a third Rabi frequency; and a set of mediator qubits .17 . The quantum processing unit according to claim 16 , wherein the first source , the second source and the third source are configured to simultaneously : rotate quantum states of odd ( even) logical qubits clockwise , and rotate quantum states of even ( odd) logical qubits counterclockwise .

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