Quantum processing unit

Abstract

A quantum processing unit comprising : rows comprising first qubits and second qubits, the first qubits and the second qubits being arranged alternatingly within each row, the first qubits and the second qubits forming columns, respectively; 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 two adjacent qubits within each row are subject to a first two-qubit interaction; one in every two columns of first qubits comprises a first special qubit provided between two adjacent second qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits, and one in every two columns of second qubits contains one second qubit, named second special qubit, such that it is driven with at least twice the second Rabi frequency by the second source.

Description

[0001] Quantum Processing Unit

[0002] Technical field

[0003] 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.

[0004] Background

[0005] In the technical field of quantum computing, superconducting qubits, that is, qubits (short for: quantum bits) based on super conducting 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.

[0006] 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. 129041102 (2021).

[0007] 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., arXiv:2103 .12305, such interaction generally remains undesirable within conventional superconducting computing frameworks.

[0008] 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 affecting control over and the performance of the device.

[0009] Summary

[0010] The present disclosure has been made in view of 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 fault-tolerant quantum computing.

[0011] According to an aspect of the present disclosure, a quantum processing unit is provided, the quantum processing unit comprising: rows comprising first qubits and second qubits, the first qubits and the second qubits being arranged alternatingly within each row, the first qubits and the second qubits forming columns, respectively; 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 two adjacent qubits within each row are subject to a first two-qubit interaction; one in every two columns of first qubits comprises a first special qubit provided between two adjacent second qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits, and one in every two columns of second qubits contains one second qubit, named second special qubit, such that it is driven with at least twice the second Rabi frequency by the second source.

[0012] According to another aspect of the present invention, a quantum processing unit is provided, the quantum processing unit comprising: rows comprising first qubits, second qubits and third qubits, the first qubits being arranged alternatingly with a set of the second qubits and the third qubits within each row, the set of the second qubits and the third qubits alternating between the second qubits and the third qubits, the first qubits, the second qubits and the third qubits forming columns, respectively, 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; a third source configured to drive the third qubits with a third Rabi frequency, wherein two adjacent qubits within each row are subject to a first two-qubit interaction; each column of first qubits comprises a first special qubit provided between two adjacent second or third qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits or to each of the two adjacent third qubits, each column of second qubits contains one second qubit, named second special qubit, such that it is driven with at least twice the second Rabi frequency by the second source, and each column of third qubits contains one third qubit, named third special qubit, such that it is driven with at least twice the third Rabi frequency by the third source.

[0013] In addition, preferred aspects of the present disclosure are defined in the dependent claims.

[0014] 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 adjacent qubits of each row within the quantum processing unit to realize quantum computing.

[0015] Brief description of the drawings

[0016] 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:

[0017] Fig. 1 shows an illustration a quantum processing unit;

[0018] Fig. 2 shows an illustration of another quantum processing unit;

[0019] Fig. 3 shows an illustration of a further quantum processing unit;

[0020] Fig. 4 shows an illustration of a still further quantum processing unit;

[0021] Fig. 5 shows an illustration of energy gaps of three qubits coupled via two-qubit interactions; and

[0022] Fig. 6 shows an illustration of steps involved in inducing a collective motion within a quantum processing unit. Detailed description

[0023] 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.

[0024] 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.

[0025] However, quantum computing as understood at present may not replace classical computing in general and for every type of calculation but only for specific 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 specific optimization problems, in particular hybrid algorithms combining quantum computing aspects with classical optimization 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 .

[0026] 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 sufficient 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 realized 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.

[0027] 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 efficiently 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 effective two-level system. As a consequence of quantum physics, a qubit may be in any coherent superposition of both states 0 and 1 simultaneously. A qubit or quantum bit may be considered as the basic unit of quantum information technology as well as the two-level quantummechanical 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 modified as part of the functioning of quantum processing unit.

[0030] 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 "singlequbit 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] One recent work the present disclosure takes inspiration from is 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. Therein, 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 (7QV2) 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 shift the position of the information carrier column in the lattice and activate single and two-qubit gates on its elements at specific locations in the device, both enabled by the Rydberg blockade interactions that connect the qubits.

[0033] 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.

[0034] Using the above, the following describes embodiments of the present disclosure in detail.

[0035] Fig. 1 shows an illustration a quantum processing unit. Specifically, Fig. 1 shows a quantum processing unit comprising: rows comprising first qubits (shown as solid squares) and second qubits (shown as dashed squares), the first qubits and the second qubits being arranged alternatingly within each row, the first qubits and the second qubits forming columns, respectively; a first source (^(t)) configured to drive the first qubits with a first Rabi frequency; a second source (B(t)) configured to drive the second qubits with a second Rabi frequency, wherein two adjacent qubits within each row are subject to a first two-qubit interaction (shown as the horizontal springs); one in every two columns (that is, every second column) of first qubits comprises a first special qubit (indicated by the cross) provided between two adjacent second qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits (shown as the vertical springs), and one in every two columns (that is, every second column) of second qubits contains one second qubit, named second special qubit (also indicated by the cross), such that it is driven with at least twice the second Rabi frequency by the second source.

[0036] There may be N rows, wherein N is a natural number and corresponds to the number of logic qubits of the quantum processing unit.

[0037] The first and second two-qubit interactions may be always-on two-qubit interactions, that is, they are not subject to any time-dependent control but act at any given time.

[0038] The rows may be arranged in parallel. The columns may be arranged in parallel. Thus, the rows and columns may form a two- dimensional ladder, as shown in Fig. 1. It is noted that this is not to be understood as limiting, in fact, the illustration shown in Fig. 1 is guided to ensure ease of understanding. Hence, a quantum processing unit according to the present disclosure may also take different forms in which the rows and columns are not arranged in parallel, thus not resulting in a two-dimensional ladder.

[0039] The first source may be a classical source; the first Rabi frequency and the first phase may be time-dependent. Similar considerations apply to the second source: The second source may be a classical source; the second Rabi frequency and the second phase may be time-dependent.

[0040] The first column of first qubits having a first special qubit may be a second column of first qubits. In other words, the very first column of first qubits may not have a first special qubit. The first special qubits may enable two-qubit gates within their respective column, as will be discussed in more details below.

[0041] The second special qubits may enable single-qubit gates within their respective column, as will also be discussed in more details below.

[0042] Further, it is noted that the "first special qubits" may be understood to be encompassed in the "first qubits". In general, a "first / second / third special qubit" may be understood to be encompassed in the "first / second / third qubits".

[0043] Further, as can be seen from Fig. 1, the first special qubits are provided as additional first qubits in columns of second qubits.

[0044] The Rabi frequency of the first special qubits may be exactly twice the first Rabi frequency. Similarly, the Rabi frequency of the second special qubits may be exactly twice the second Rabi frequency.

[0045] It is noted that the Rabi frequency of a qubit may, in particular if 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 modification of the source driving the qubits. In other words, these modifications among the qubits do not affect the possibility to drive the quantum processing unit globally.

[0046] The second (vertical) interaction may be the same interaction as the first (horizontal) interaction.

[0047] The relevancy of the optional black triangle will be explained further below.

[0048] Fig. 1 also shows the separation of the states of the qubits during operation of the quantum processing unit in which the quantum processing is separated column-wise into a paramagnetic phase in which the states of the columns alternate between "all qubits of a column in the ground state \g )" and "all qubits of a columns in the excited state |e)"(here exemplary shown on the left), a ferromagnetic phase in which all qubits of all columns are in the same state, here the ground state \g ) (here exemplary shown on the right), and the state of the logical qubits I1}7) of the N qubits, also referred to as the "information carrier", at the interface between these two phases (emphasized by the dotted- dashed surrounding).

[0049] In summary, the present disclosure is directed at a quantum processing unit in which (7QV2) qubits, provided in rows and columns and subject to specifical two-qubit interactions such that globally driven control of an extensive subset of the total qubits can facilitate universal quantum computing.

[0050] In this regard, this quantum processing unit is not dissimilar to the concepts discussed by Cesa and Pichler, however, do not rely on the Rydberg blockade and optics to drive the atoms, but are agnostic to the platform used, in particular are based on solid-state elements, e.g., superconducting circuits, as physical qubits, and wiring, e.g., waveguides, to facilitate the global driving.

[0051] Importantly, as discussed in more detail below, the two-qubit interaction referred to above and employed to not only achieve similar effects as those of the Rydberg blockade but also realize two-qubit gates, can be engineered to be the two-qubit interaction that negatively affects the performance of conventional superconducting QC architectures, thus turning a conventional problem into a useful resource.

[0052] Fig. 2 shows an illustration of another quantum processing unit. The underlying idea of this quantum processing unit is the observation that the quantum processing unit as shown in Fig. 1 contains several columns of qubits that do not contain any "special qubit" and as such mainly serve to ensure that during operation of the quantum processing unit, information is confined to the information carrier and does not "leak" into the paramagnetic phase or the ferromagnetic phase. As detailed below, by introducing a third species of qubits, the "third qubits", the number of columns not containing a special qubit and thus the overall number of physical qubits can be reduced significantly - asymptotically, it can be reduced to the half.

[0053] Specifically, Fig. 2, in which same elements are depicted in the same manner as in Fig. 1 and thus corresponding explanations are not repeated, shows a quantum processing unit comprising: rows comprising first qubits, second qubits and third qubits, the first qubits being arranged alternatingly with a set of the second qubits and the third qubits (shown as dotted squares) within each row, the set of the second qubits and the third qubits alternating between the second qubits and the third qubits, the first qubits, the second qubits and the third qubits forming columns, respectively; 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; a third source configured to drive the third qubits with a third Rabi frequency, wherein two adjacent qubits within each row are subject to a first two-qubit interaction, each column of first qubits comprises a first special qubit provided between two adjacent second or third qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits or to each of the two adjacent third qubits, each column of second qubits contains one second qubit, named second special qubit, such that it is driven with at least twice the second Rabi frequency by the second source, and each column of third qubits contains one third qubit, named third special qubit, such that it is driven with at least twice the third Rabi frequency by the third source.

[0054] As with the first and the second source, also the third source may be a classical source; the third Rabi frequency and the third phase may be time-dependent. Similarly, the Rabi frequency of the third special qubits may be exactly twice the third Rabi frequency.

[0055] In other words, in particular when comparing Figs. 1 and 2, one can understand the quantum processing unit discussed in connection with Fig. 2 as a variant of the quantum processing unit of Fig. 1 with the following modification: Every second column of second qubits is replaced with a column of third qubits that is controlled by a third source, and now each column of qubits contains a first / second / third special qubit, respectively. Thus, the "empty" columns can be removed, thereby reducing the overall number of qubits significantly - at the price of introducing an additional source.

[0056] More plastically, the column arrangement of (F=first qubits, S=second qubits, T=third qubits) "...FSFSFSFS..." has been replaced with "...FSFTFSFT...".

[0057] In the following, further details of the two quantum processing units will be discussed. Since the quantum processing unit of Fig. 1 may be seen as the simpler version, the focus will be on this quantum processing unit, however, all details equally apply to the quantum processing unit of Fig. 2, as will also explained below.

[0058] In an embodiment of 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. Similarly, in an embodiment of the present disclosure the second source may be configured to drive the second qubits using a single signal provided globally to all of the second qubits. Further, in an embodiment of the present disclosure - if present - the third source may be configured to drive the third qubits using a single signal provided globally to all of the third qubits.

[0059] That is, it is emphasized that the source provides a common signal to all qubits connected to this source and thus control of the qubits is not individual on the level of each qubit being provided with an individual signal, but rather the individual control of logical qubits is achieved by moving (by means of the global pulses) the qubits at the right position within the quantum processing unit and then carrying out the quantum operations, in particular one- and two-qubit gates, at this position.

[0060] Here, it is noted that "source" is used as a term indicated that this element is a source of a control signal, for example, an electrical pulse signal. Thus, the term "source" can be equally replaced with "control source" or "controller".

[0061] In an embodiment of the present disclosure, the first source may be configured to be off if the second source is on. Similarly, in an embodiment of the present disclosure the second source may be configured to be off if the first source is on. Further, in an embodiment of the present disclosure - if appropriate - the first 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 first source is on.

[0062] In other words, if the first source is on, the second source and the third source are off; if the second or third source is on, the first source is off; if the second (third) source is on, the third (second) source can be on. From a different perspective, this can be understood as follows: sources controlling adjacent columns cannot be on at the same time.

[0063] That means, if the signal provided by the first source to the first qubits is non-zero, the signal provided by the second source to the second qubits (and the signal provided by the third source to the third qubits, if applicable) is zero.

[0064] Similarly, if the signal provided by the second source to the second qubits is non-zero, the signal provided by the first source to the first qubits is zero, while - if applicable - the signal provided by the third source may be non-zero.

[0065] Similarly, if the signal provided by the third source to the third qubits is non-zero, the signal provided by the first source to the first qubits is zero, while the signal provided by the second source may be non-zero.

[0066] This may, in other words, also be formulated as: at most the first source, or at least one of the second source and the third source (if applicable) is driving their respective qubits at a time.

[0067] In an embodiment of the present disclosure, the first source may be configured to drive the first qubits with a first oscillation frequency and a first phase, and / or the second source may be configured to drive the second qubits with a second oscillation frequency and a second phase, and / or - if present - the third source may be configured to drive the third qubits with a third oscillation frequency and a third phase.

[0068] In an embodiment according to the present disclosure, at least one of the first two-qubit interaction and the second two-qubit interaction may be a ZZ-interaction.

[0069] Here "Z" refers to the Z-Pauli matrix and the ZZ-interaction is thus a two-qubit interaction leading to crosstalk between the qubits and detuning the qubits.

[0070] It is noted that this type of longitudinal ZZ interaction is exactly the type of interaction that is unwanted in conventional approaches. As will 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 specific energy transitions, thereby effectively emulating the blockade effect discussed by Cesa and Pichler in the framework of Rydberg atoms.

[0071] This building block thus not only allows to transfer the concepts developed by Cesa and Pichler to other platforms, in particular the platform of superconducting circuits, but also allows to turn a problematic aspect of this platform into an advantage. In an embodiment according to the present disclosure, one of the two second qubits subject to the two-qubit interaction with a first special qubit may be the second special qubit of the respective column. Similarly, in an embodiment according to the present disclosure, - if applicable - one of the two third qubits subject to the two-qubit interaction with a first special qubit is the third special qubit of the respective column.

[0072] This aspect can also be seen in Figs. 1 and 2 in which the second / third special qubit is provided below the first special qubit, if such a first special qubit is provided in an adjacent column.

[0073] In an embodiment according to the present disclosure, the first qubits may have a first qubit frequency (co^), and / or the second qubits have a second qubit frequency (uB), and / or - if applicable - the third qubits may have a third qubit frequency (uc).

[0074] That is, the first qubits may (all) have the same first qubit frequency, and / or the second qubits may (all) have the same second qubit frequency, and / or the third qubits may (all) have the same third qubit frequency. The first qubit frequency, the second qubit frequency, and / or the third qubit may be different from each other.

[0075] Here, qubit frequency may refer to the frequency corresponding to the energy difference between the two (quantum) states defining the qubit.

[0076] In an embodiment according to the present disclosure, the qubit frequency of each second qubit adjacent to the first special qubit may be different from the second qubit frequency, preferably equal to the sum of the second qubit frequency and an interaction strength (g^ ) of the second two-qubit interaction. Similarly, in an embodiment of the present invention, - if applicable - the qubit frequency of each third qubit adjacent to the first special qubit may be different from the second qubit frequency, preferably equal to the sum of the third qubit frequency and the interaction strength of the second two-qubit interaction. This interaction strength may be positive.

[0077] In an embodiment according to the present disclosure, each first qubit and special first qubit may be connected to the first source by a first wiring, and / or each second qubit and special second qubit may be connected to the second source by a second wiring, and / or - if applicable - each third qubit and special third qubit may be connected to the third source by a third wiring.

[0078] This wiring, i.e., the first / second / third wiring, may be waveguide. This may be in particular appropriate if the quantum processing unit is based on solid-state devices such as superconducting circuits.

[0079] In an embodiment according to the present disclosure, a set of logical qubits may be encoded in a column of first qubits, or in a column of second qubits, or - if applicable - in a column of third qubits. This may correspond to the "information carrier" discussed elsewhere in this document and can also be seen in Figs. 1 and 2 discussed above in detail.

[0080] In an embodiment according to the present disclosure, states of the first qubits, the second qubits and the second special qubits, and - if applicable - the third qubits and the third special qubits on one side of the column encoding the set of logical qubits (the "information carrier") may be in a paramagnetic phase and states of the first qubits, the second qubits and second special qubits, and - if applicable - the third qubits and the third special qubits on the other side of the column encoding the set of logical qubits are in a ferromagnetic phase.

[0081] The first special qubits may not be subject to this separation into "paramagnetic phase" and "ferromagnetic phase".

[0082] As also discussed elsewhere, "paramagnetic phase" may mean that qubits of the same column are in the same state, that each qubit is either in the ground state or the excited state, and that the state of the columns alternate, while "ferromagnetic phase" may mean that all qubits of all columns are in the same state, e.g., the ground state.

[0083] In an embodiment according to the present disclosure, the first source, and / or the second source, and / or - if applicable - the third source may be configured to move a quantum state from one column.

[0084] In an embodiment according to the present disclosure, the first source, and / or the second source, and / or - if applicable - the third source may be configured to move the quantum state of the column encoding the set of logical qubits, while maintaining the paramagnetic phase and the ferromagnetic phase on the two sides of the column encoding the set of logical qubits.

[0085] More details how moving the set of logical qubits can be realized and how to this can be realized while maintaining the paramagnetic phase and the ferromagnetic phase will be discussed below, in particular in connection with Fig. 6.

[0086] In an embodiment according to the present disclosure, the first source and the second source may be configured to perform a single-qubit gate on a second special qubit of a column of second qubits, and / or - if applicable - the first source and the third source may be configured to perform a single-qubit gate on a third special qubit of a column of third qubits.

[0087] Specifically, it is the different Rabi frequency of the second / third special qubit with respect to the second / third qubit that allows to "single out" the rows at which the second / third special qubit is provided even though the control signal provided by the second / third source is provided globally such that the qubit at this row is subject to a different operation that the remaining qubits of this column, leading to the single-qubit gate. It is noted that the presence of the paramagnetic phase / ferromagnetic phase on the sides of the "information carrier" can ensure that the states of these parts of the quantum processing unit remains unaffected while applying the single-qubit gate at the "information carrier" column.

[0088] In an embodiment according to the present disclosure, the first source and the second source may be configured to perform a two- qubit gate on the two second qubits adjacent to the first special qubit of the column, and / or the first source and the third source may be configured to perform a two-qubit gate on two third qubits adjacent to the first special qubit of the column.

[0089] Specifically, it is the interplay between the added first special qubits between two second / third qubits and the second two-qubit interaction between both pairs of adjacent qubits that allows to "single out" the two involved rows of second / third qubits even though the control signal provided by the first / second / third source is provided globally such that the qubits at these rows are subject to a different operation that the remaining qubits of this column, leading to the two-qubit gate. It is noted that also here the presence of the paramagnetic phase / ferromagnetic phase on the sides of the "information carrier" can ensure that the states of these parts of the quantum processing unit remains unaffected while applying the two-qubit gate at the "information carrier" column.

[0090] In an embodiment according to the present disclosure, an interaction strength of the first two-qubit interaction is greater than the first Rabi frequency, and different from the second Rabi frequency, and / or - if applicable - different from the third Rabi frequency.

[0091] In an embodiment according to the present disclosure, the quantum processing unit may further comprise a plurality of quantum actuators, wherein one quantum actuator is connected to each first qubit, each first special qubit, each second qubit, and each second special qubit, and / or - if applicable - each third qubit and each third special qubit, respectively, via the two- qubit interaction. A quantum actuator may be seen a qubit which does not directly participate in the logical computation and can be initialised either in the ground state or excited state. Please note that first special qubits can be seen as an example of quantum actuators .

[0092] In an embodiment according to the present disclosure, the first qubits, including the first special qubits, and / or the second qubits, including the second special qubits, and / or - if applicable - the third qubits and the third special qubits, may be solid-state devices, in particular may be superconducting qubits.

[0093] Figs. 3 and 4 show illustrations of further quantum processing units. As the extension depicted therein (with respect to Figs. 1 and 2, respectively) is the same, the focus of the following will lie on Fig. 3, while these considerations apply correspondingly to Fig. 4. Specific differences between these two cases will be highlighted as appropriate.

[0094] As can be seen from Fig. 3, the qubits of quantum processing unit of Fig. 1, which correspond to the "Processing Area" of Fig. 3, have been extended by qubits of an "Initialization Area" (on the left) and an "Read-Out Area" (on the right). These areas, as the names suggest, facilitate initialization of the quantum processing before the quantum computation (in the sense of the operations that manipulated the quantum states such that a subsequent read-out / measurement can obtain a computation result) takes place in the Processing Area as well read-out after the quantum computation has been completed.

[0095] The qubits in these two added areas follow generally the same rules of alternating between first and second qubits (or alternation between first and second / third qubits, wherein second and third qubits alternate as well, as in Figs. 2 and 4) and of "paramagnetic phase" and "ferromagnetic phase".

[0096] As further detailed below, the very first column (the first column of the Initialization Area) and the very last column differ in their qubit frequency (as shown by the solid black dot) to facilitate initialization and read-out, respectively.

[0097] Further, in Fig. 3 initialization is facilitated by means of an initialization source W nitC ) while in Fig. 4 the third source can take over this role.

[0098] That is, in an embodiment according to the present disclosure, the quantum processing unit may further comprises: two initialization columns at a first side of the rows; and an initialization source configured to drive first initialization column, wherein - in the case that third qubits are present in the quantum processing unit - a first initialization column may be a column of third qubits, and the initialization source is the third source, and - in the case that third qubits are not present in the quantum processing unit - the first initialization column may be a column of second qubits.

[0099] Here, the first initialization column may refer to the very first column of the quantum processing unit.

[0100] In an embodiment according to the present disclosure, - in the case that third qubits are present in the quantum processing unit - the qubit frequency of the third qubits of the first initialization column may be different from the third qubit frequency, preferably equal to the difference between the third qubit frequency and an interaction strength of the second two- qubit interaction, and - in the case that third qubits are not present in the quantum processing unit - the qubit frequency of the second qubits of the first initialization column may be different from the second qubit frequency, preferably equal to the difference between the second qubit frequency and an interaction strength of the two-qubit interaction.

[0101] In an embodiment according to the present disclosure, the quantum processing unit may further comprises: two readout columns at a second side of the rows different (opposite) to the first side, wherein a second readout column may be a column of second qubits, and the qubit frequency of the second qubits of the second readout column maybe different from the second qubit frequency, preferably equal to the difference between the second qubit frequency and an interaction strength of the two-qubit interaction .

[0102] Here, the second readout column may refer to the very last column of the quantum processing unit.

[0103] As mentioned above, further details how the Initialization Area and the Read-Out Area function, will be provided below.

[0104] Fig. 5 shows an illustration of energy gaps of three qubits coupled via two-qubit interactions. Specifically, Fig. 5 serves to illustrate how the first (horizontal) two-qubit interaction alleviates some of the degeneracies in the energy spectra of the qubits such that a pulse provided by a source can only drive specific state transitions.

[0105] In more detail, Fig. 5 shows the case of three qubits, (in order) a first qubit, a second qubit and another first qubit and a second source driving the second qubit. This can be extended without any loss of generality to any of the situations occurring in the present disclosure.

[0106] As can be seen from Fig. 5, the energy gap between the ground state \g) of the second qubit to the excited state |e) of the second qubit depends - as a consequence of the first (horizontal) two-qubit interaction - on the states of the adjacent first qubits. Specifically, if the system is driven with a frequency of (B— 2gAB, as shown in the bottom the transition between ground state and excited state only takes place if both adjacent qubits are in their respective ground state as well as all other transitions are severely detuned from this transition frequency. This results in behavior effectively identical to the Rydberg blockade utilized by Cesa and Pichler and thus allows to maintain the structure of "paramagnetic phase - information carrier - ferromagnetic phase" while moving the information carrier through the quantum processing unit and performing the operations constituting the quantum computation.

[0107] Fig. 6 shows an illustration of steps involved in inducing a collective motion within a quantum processing unit. While Fig. 6 shows the case of a single row, this process can be extended without loss of generality to an arbitrary number of rows.

[0108] In Fig. 6, initially (that is, in the top image of the four images) the first two qubits correspond to the paramagnetic phase, the third qubit is the information carrier, and the fourth to sixth qubits correspond to the ferromagnetic phase. The qubit of the information carrier is assumed to be in a state |vp)= \g)+ P|e), that is, a generic superposition between the two states of the qubit.

[0109] In a first step, the unitary operation n^r, the information is distributed from the third qubit over to the third and the fourth qubit and the resulting combined of these two qubits is a\ge)+ P\eg). Note that in this step the last qubit changes its state as well, that is, temporarily the ferromagnetic phase is broken and - apart from the information carrier - the whole row is in a paramagnetic phase.

[0110] In a second step, the unitary operation nB, the information transfer from the third and the fourth qubit to the fourth qubit is completed: The state of the third and fourth qubit are separable, the third qubit is its ground state, and the fourth qubit is in the state . While this moved the information carrier by one column to the left, it is evident that the states of the various qubits are different from the initial states.

[0111] Thus, to finalize the collective motion, in a third step, the unitary operation n^r is once more carried out, thereby inverting the state of the fourth qubit and reestablishing the paramagnetic and ferromagnetic order on the sides of the new position of the information carrier. As indicated by the arrows on the right side of Fig. 6, the operations can be inverted to move the information carrier to the left. Further, note that details on the operations n^r and nBare provided below.

[0112] In summary, the present invention provides quantum processing units that employ (7QV2) qubits driven by a small (not more than three) 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.

[0113] Further, in particular when using solid-state device to realize the qubits, for example, superconducting circuits, this addresses one daunting issue of quantum computing: the wiring problem. Specifically, 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.

[0114] 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, 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 .

[0115] 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.

[0116] Further technical details 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.

[0117] In line with the above discussion, in particular of Figs. 1 and 3, a quantum processing unit according to the present disclosure may have a two dimensional "ladder" structure with N columns, which may preferably be parallel. These columns house two groups of qubits, which may preferably be superconducting qubits, arranged in an alternating fashion. These two groups of qubits are also referred to as "A" and "B", which corresponds to the first qubits and the second qubits of the above discussion. The central part, also termed the "Processing Area" comprises 4N — 3 columns of qubits. Additionally, the quantum processing unit may comprise two columns on the left for initialization and two columns for read-out, which are accordingly also termed "Initialization Area" and "Read-Out Area", resulting in a total of 4N + 1 columns in some embodiments.

[0118] As explained also above, in the Processing Area, one in every two columns formed by B qubits also incorporates an additional A qubit. This A qubit acts as a "coupler" between adjacent rows of the ladder, facilitating two-qubit logic gates. These additional qubits are in N — 1, bringing the total number of qubits up to (4N + 1)N + N — 1= 4N2+ 2N — 1.

[0119] All qubits within a group may share the same level spacing: h)Afor A-type qubits and ha>Bfor B-type qubits. There may be exceptions to this rule, marked with a black circle or triangle in some figures, which may have level spacings slightly detuned from the nominal values to compensate for the anomalous (different) number of nearest-neighbouring interacting elements. In this model, qubits within the same row are interconnected via a nearest-neighbor coupling, which may be a ZZ coupling, with uniform strength hgAB, which may correspond to the black springs in the figures. A similar interaction also links additional intra-row coupler qubits A with adjacent B elements in the same column.

[0120] Importantly, all A-type qubits are collectively driven by a single time-dependent external electrical signal, VA^ , through a dedicated line. Similarly, all B-type qubits, including those with a black circle or triangle, are collectively driven by signal Vg(t) through a dedicated line.

[0121] Additionally, the model may include a third control line acting as a switch, carrying a control pulse Vinit(t). This pulse may be active for a finite time at the beginning of the computation, selectively operating only on qubits in the first column when other pulses are inactive. Finally, the cross symbols in the scheme identify a subset of superconducting qubits A and B that, while maintaining their nominal level spacings, couple with external electrical signals ^(t) and Vg(t) at augmented Rabi frequencies. Crossed elements of the same type may never appear in the same row or column. Additionally, any two columns that contain crossed elements of the same type are always separated by at least one column of the same type without any crossed qubits. It is noted that feature, which is instrumental for implementing single-qubit and two-qubit gates, can be engineered during circuit fabrication and needs no extra control.

[0122] In view of the definitions introduce so far, the total Hamiltonian of the two-dimensional ladder can be written as / / (t):= #o+ ^drive(t), where

[0123] Eq. (1) describes the local energy contribution of the qubits and their interactions which are fixed by the model, e.g., the geometry of the model, while

[0124] Eq. (2) is the time-dependent driving contribution induced by the control lines, which may be classical control lines. In these equations represents the Pauli matrices acting on the Hilbert space of the i-th qubit, expressed in the local energy basis \gt), le^. The summation in the interacting part of Hoencompasses all nearest-neighbor interactions detailed in the introductory paragraphs of this section.

[0125] The parameter <*>d,x denotes the oscillation frequency of the driving pulse while and <frx(t) define the time-dependent Rabi frequency and phase of such control. For simplicity in writing / / driveCO, we omit the driving term associated with the control line Vinit(t), which operates solely at the very beginning of the computational process on the leftmost B column of the model.

[0126] Additionally, we do not explicitly state that when in Eq. (1) the site index i identifies an element with a black solid circle, the corresponding qubit's level spacing becomes h{a>B— gAB) instead of ha)B, and when it identifies an element with a black triangle, it becomes h(a)B+ gAB^) instead of ha>B. Similarly, whenever in Eq. (2) the index i identifies a crossed element, . In addition to these adjustments, it is important to note that and <*>d,xareindependent of the site index i, indicating that they are associated with a control pulse acting globally on all qubits of type x in the model. Throughout our analysis, we assume that only one of the control pulses l^(t) or Vg(t) is active at any time, ensuring that two nearest-neighbor columns of the ladder are never simultaneously driven (i.e. Furthermore, in the time intervals where the control s active, it is assumed that the corresponding nz(t) and <frx(t) values are constant. Under these conditions there exists a particular choice of the frequency <*>d,x of the driving pulse which allows us to emulate a blockade regime such as the one developed by Cesa and Pichler.

[0127] A physical insight into the origin of this effect can be obtain considering the energy spectrum of the Hamiltonian Hofor the simplified case where we have only a single row of three alternating qubits, either ABA or BAB - the ABA case is shown in Fig. 5.

[0128] For this system, one notices that, under rotating wave approximation (RWA), if we apply a robust pulse at frequency a>dx:= a>x— 2gABto the central element, only the transition between states \ggg) and \geg) can be activated, while other transitions are inhibited due to off-resonance. In other words, whenever at least one of the external qubits of the triple resides in the excited state |e), the system remains unaffected by the specified control pulse. Depending on the circuit design, indeed, the coupling gABcan be either positive or negative. However, its sign does not influence the blockade effect. Indeed, the interaction detunes the \eg) -> |ee) and the \ge) -> |ee) transitions with respect to the non-interacting case, effectively "blocking" these transitions and making the |ee) state inaccessible with the external pulse that we use. It is further noted that gAB~ kHz - MHz while the natural frequency of the qubits is a)x~ GHz. Thus, for a two-qubit system, the |ee) state is the highest in energy even in presence of interactions.

[0129] Hence, by selecting a specific drive frequency, we achieve an e- e blockade analogous to the one suggested by Cesa and Pichler for the Rydberg atoms set-up. We stress that, since within the present disclosure we are not interested in driving two nearest- neighbour qubits at the same time, the study of the simplified three-qubit setups analysis above is sufficient for generalizing the result to a single wire of alternating A- and B-type of qubits. A formal proof of this fact can be obtained by describing the system evolution in the time-dependent reference frame defined by the controls. In particular, using the transformation Urf(t) , under RWA the Hamiltonian model becomes

[0130] Eq. (3) which if hgABis the largest energy scale, exactly mimics the effective Rydberg blockade Hamiltonian employed by Cesa and Pichler. A derivation of Eq. (3) is presented below, whereas here it is stressed that it applies to all the qubits of the scheme, including the off-detuned elements indicated by black triangles or circles in Fig. 1.

[0131] Since in our model assume constant values on disjoint time windows, the dynamical evolution induced by the Hamiltonian Hrf(t') can be expressed in terms of ordered products of the form

[0132] Eq. (4) where for ZJ® isthe unitary operator associated with the Z-th window where the pulse 7 / Z(t) is active. Specifically, given }, the temporal duration of the time window, and £1® , the corresponding constant values of the control with H® — stricting to the blockade regime, i.e. imposing the condition r / BR■= the interaction term appearing in A. can be considered as a constraint that prevents any dynamical evolution on the / -type qubits unless both of their nearest neighbor qubits of the opposite type / are in the ground state \g). Formally this implies that each one of the unitaries Uxappearing in (4) reduces to a product of identical control-unitary operations, where each qubit of the / -type is controlled by its first neighbouring sites of / -type. In particular, given i 6 / , define P^) the projector on the subspace of the nearest-neighbouring x_type qubits of such site which contains no excitations, and the orthogonal complement we can write

[0133] Eq. (5) where 1, is the identity operator of the i-th site, and the single-qubit unitary evolution induced by the non-interacting part of the Hamiltonian Hxon the same site. Specifically, = ular (non-crossed) / -type qubits, the crossed ones in view of their augmented Rabi frequencies. Relaying on the group properties of the transformations of Eq. (5), one can prove that by using sequences of Eq. (4) that only involve a finite set of Uxactivated by the same control line Vx(t), one can induce arbitrary evolutions of the form

[0134] Eq. (6) where / xand / rare the subsets of / that includes all its crossed and the regular (non-crossed) elements respectively. For 6 { / x, / r}, is a control-unitary transformation that applies to all qubits in a uniform, single-qubit rotation Ri(0,n)'■= e~l9^2na(l)parametrized by the 3D unit vector n and by the angle 6 E [0,2TT]. Of particular relevance are the unitaries of Eq. (6) where one and only of the parameters d', 6" differs from zero, which correspond to scenarios where we selectively operate on either or xr. Among these, obtained for 6'= 0, 6" = 2TT is highlighted.

[0135] Following the paper by Cesa and Pichler, one can encode the quantum information on the two-dimensional ladder by placing one logical qubit per each row of the array. As indicated in Fig. 1, at each computational step, these qubits reside on a single column of the process unit of the device.

[0136] Such an information carrier column (ICC) (or simply "information carrier" as also used herein) can be formed either by A-type qubits or by B-type qubits. However, if the ICC is formed by B- type qubits, the crossed elements of Fig. 1, which act as intrarow couplers, are not involved in the encoding. These special qubits will always be set into the in the ground state |g) of their local Hamiltonian. Most importantly each qubits of the ICC is positioned at the interface between a string of qubits in a "ferromagnetic" phase on the right hand side (\ggg...g)), and a "paramagnetic" phase on the left (\...gegeg ...)). For instance, assuming that at a given computational step a (possibly entangled) state \lP) of the N-logical qubit is located in the k- th column of the ladder, the global state of the model is described by the many-body quantum vector: where for k = 1,...,2N + 3, the ket |•••)krefers to the state of the fc-th column of the quantum processing unit (for simplicity, the states of the crossed elements are omitted). Notice that in Eq. (7) all the qubits except those in the ICC are factorized. Starting from a full ferromagnetic phase where all the qubits of the device are in the ground state \g), one can bring the system into a state of the form of Eq. (7) by using the control pulse VfnitG) to selectively promote the first column in Fig. 1 in . Formally, this produces a vector I1 / 7;k) where the ICC is located in the first B-type column of the quantum processing unit, and a logical state I1 / 7):= Thanks to the of Eq. encoding (7) one can shows that under the e-e blockade conditions, there exist control unitary evolutions as in Eq. (4) that enable universal QC. The key ingredients that make this feasible are: a) the ability to rigidly move the ICC at any position in the ladder (including a last column of the readout area where measurements can be performed at the end of the computation); b)the implementation of arbitrary single-qubit gates on the crossed element of the B-type column where the ICC is located, while leaving the rest of the qubits unaffected; c) the ability to activate a non-trivial two-qubit entangling gate (e.g., a control-Z operation) between the j-th and G + 1)-th logical qubits when the ICC is located in the B-type column that contains the crossed A-type qubit, which acts as a coupler between the j- th and G + 1)-th rows of the ladder.

[0137] To achieve the task a), we identify a sequence t / shift of the transformations of Eq. (6) that shifts the horizontal position of the ICC without affecting its internal state, i.e., t / shift \W;k) = |^;k + 1). This can be done through two different types of n-pulses: one that acts only on the non-crossed elements of A (i.e., IlAr:= WAr(n;x)), and the other on the full set of B qubits (i.e. nB■= WB(ji,TV,X,X)). Specifically, if the ICC of \V;k) is of B-type we take t / shift=kIArnBnAr, while if it is of A-type we use ( / shift = nBnArnB. Note also that since in both scenarios t7shift coincides with its own inverse, the same operation can also be used to induce a collective motion of the interface in the opposite direction, i.e.,

[0138] The operations needed for implementing the single-qubit gates of point b) can be realized composing sequences of the form Z^?tV / Bx(0 / 2;—n1)Z^?tV / Bx(0 / 2;n1) with n±orthogonal to z and Z%rfas previously defined. Indeed, thanks to the fact that the crossed B-type qubit are located on columns which are at least three columns apart from other, when acting on I1 / 7;k) the above transformation will effectively correspond to apply the singequbit rotation / ?(0,n±) on the crossed element of the ICC. Implementing an entangling two-qubit gate between two adjacent rows is much simpler. Once the ICC is in the proper position, we only need to use a single V / J4X(2TI,Z)= Ilie?ix[(? evolution that leaves unchanged every normal A-type qubits while inducing a conditional-phase shift on the crossed ones.

[0139] In summary, these considerations combine two main ideas: the employment of a two-qubit interaction between two nearest- neighbours, usually unwanted in superconducting architecture of QC, and the concepts of Cesa and Pichler to perform universal quantum computation through external global pulses. It is remarked that the number of physical qubits scales like N2, where N is the number of logical qubits. Moreover, it is stressed that using superconducting qubits (or any solid-state devices) allow to obtain an improved scalability with respect to Rydberg atoms setups, which is a clear advantage in view of future experimental implementations. Finally, it is noted that via numerical simulation it was confirmed that the blockade condition r / BR» 1 is fundamental in order for the protocol to be effective. In some studies, it was managed to detune the qubits with respect to their bare frequency up to 1 GHz, with a step in frequency of order of ~10 MHz. Since typical values of QABI^- range between 10 KHz and 100 MHz the condition for the repulsive interaction can thus be achieved with current technology. Regarding the blockade approximation, via numerical simulations it was found that a value of QAB greater than is needed. On the other hand, the Rabi frequency can cover a wide range of values, starting from 5 MHz, thus making this feasible as well.

[0140] In the following, additional input is provided explaining how the embodiment using two types of qubits can be extended to three types of qubits. To this extent, first, it is briefly discussed why "2-species architecture" needs to have a large number of columns in the device, and then, second, the "3-species architecture" is introduced, explaining why this allows to reduce the number of columns (and thus qubits) of the 2-species architecture .

[0141] The "2-original species" version of the quantum processing uit assumes the presence of two different types (or species) of qubits (A and B). As detailed elsewhere, these two species are controlled by two independent pulses (1^( and VgCt), respectively), which operate globally on the associated subgroup of qubits. In a device capable of performing universal quantum computation on N logical qubits, these species are organized into alternating columns of an ("almost regular") two- dimensional lattice formed by N rows and 4N + 1 columns. The model also contains N — 1 extra A-type qubits that act as couplers between adjacent rows, for a total number of physical qubits given by:

[0142] Eq. (I)

[0143] In order for the quantum processing unit to work, some of the qubits of the device need to be promoted to "crossed qubits": these are qubits which exhibit an enhanced value of the corresponding Rabi frequency that mediates the coupling with the associated control pulse. These special qubits play an essential role for the realization of logical gates. Specifically, the crossed B-type qubits are employed to realize single-qubit gates, while crossed A-type qubits are employed to implement two-qubit gates.

[0144] To ensure the correct functionality of the device, one needs to impose the following constraints:

[0145] 1. (At least) one crossed B-type qubit is needed for each logical qubit. Hence, the minimum number of crossed B-type elements is N

[0146] (Working with less than N crossed B-type elements may be possible but for sure will require a rather complex routing procedure of the information in the ladder.) 2. The crossed B-type elements must be placed in different columns. This requirement arises because, in the scheme, all B-type qubits (crossed or not crossed) are driven with the same control pulse Vg(t). This symmetry is a direct consequence of our architecture's distinct feature: the pulses are global and not local. To break this symmetry, we use two strategies:

[0147] • Augmented Rabi Frequency: Crossed B-type elements have an augmented Rabi frequency compared to non-crossed ones. This ensures that the net effect of Vg(t) on crossed B qubits is different from its effect on regular (non-crossed) B qubits.

[0148] • Special Encoding Strategy: Our architecture assumes that, at each computational step, the N logical qubits are all located in the same column of the device. This special column, which we may call the "Information Carrier Column" (ICC) of the system, acts as an interface between two different global states (the "paramagnetic" phase on the left and the "ferromagnetic" phase on the right).

[0149] Thanks to these specific choices, despite the fact that all the N crossed B-type elements experience the same effect when Vg(t) is active, only those located in the ICC will undergo the designed evolution. If two (or more) crossed B-type qubits would be in the same column, instead of performing the desired singlequbit gate on a single element of the register, one would end up applying it simultaneously to two (or more) logical qubits.

[0150] 3. To fulfill the previous two requirements, it would be sufficient to have N B-type columns. However, the quanutm processing device depicted in Fig. 1 contains roughly twice that number of B-columns. The reason for this "redundancy" (i.e. increase in the number of columns) is the last and least-obvious requirement that we must impose. To ensure that the logical information remains confined in the ICC without diffusing into the two-dimensional lattice, it has to be ensured that crossed B-type elements are sufficiently well separated. If this does not happen, then when we switch on the control Vg(t), the specific information encoding we use will induce undesired interactions among neighbouring crossed B-type elements, spreading the information over the entire two-dimensional lattice. The minimal distance requirement is that between two B-type columns each containing a crossed element, there must be at least one B- type column with no crossed elements.

[0151] The requirement of having a sufficiently large distance between columns with crossed elements also applies, in principle, to columns containing A-type qubits: in this case, however, the geometry of the model automatically ensures that this happens (indeed, crossed A-type qubits can only be found as intra-row elements, which are always well separated).

[0152] The 2-species design presented in Fig. 1 fulfils all the three requirements described above. As a matter of fact this is arguably the most efficient arrangement of this type.

[0153] The 3-species variant of the scheme, presented in Fig. 2, has been devised to reduce the number of columns in the ladder by a factor of 2, as compared to the 2-species design illustrated in Fig. 1.

[0154] As the name suggests, the design in Fig. 2 features three types of superconducting qubits A, B, and C. As in the 2-species case, each family of qubits is controlled by dedicated classical control lines, i.e., VA(t), VB(t), and Vc(t), respectively.

[0155] These qubits are organized, as in the 2-species design, in an ("almost regular") 2D ladder and are coupled by two-qubit couplings. The ladder contains 2N + 3 columns, each housing a given type of qubit. As shown in the figure, these columns follow a ABACABACA... ordering, so that B and C columns always alternate and have two A columns as first neighbours, one on the left and one on the right. The only exception to this rule may occur for the C and B columns at the outer edges of the device, which have only one A-type neighbouring column. Like in the case of the 2-species device, the 3-species device also features extra N — 1 intra-row elements formed by A-type qubits. Furthermore, also in this 3-species device we have crossed qubits for each of the three species, and qubits characterized by different values of the local energy gaps, identified in Fig. 2 by qubits with black circles and black triangles. These latter "detuned" elements belong to B and C species and occur at sites where a qubit is either coupled to only one other qubit (black circle) or at sites where a qubit is coupled to three other qubits (black triangles).

[0156] The main advantage of the 3-species model with respect to the 2- species one, is that in the former design we can reduce the separation between the various crossed elements. Indeed, there is no problem having a crossed B qubit near a crossed C qubit since they are activated by independent pulses. This allows us to drop the previous requirement of the "large separation", i.e. item 3 in the above list of requirements Notice that in the design with 3 species, the total number of physical qubits is given by

[0157] Eq. (II) which is approximately half of the number reported in Eq. (I) for the 2-species design. In the 3-species design, have thus 2N(N - 1) less qubits.

[0158] It is finally pointed out that the initialization area for the 3-species model does not need a dedicated control£n£t(t) as in the case of the 2-species model. Indeed, due to the effective augmented distance between the columns that are activated by the same control, one can use the Vc(t) line to play the role that was played by V£n£t(t)in the 2-species model.

[0159] While the above discussion has been limited to 2- and 3-species model, it can be readily understood that techniques according to the present disclosure are not limited to only two or three species, but more species such as four, five, etc. are easily conceivable. For example, it is conceivable that in the 2-species model alternating columns of the first species are of a first and third species, while alternating columns of the second species are of a second and a fourth species. That is, an order ABCDABCD..., in which A, B, C and D represent the four species, is conceivable. Clearly, each species would then require their only wiring, i.e., controls l^(t), Ks(t), Vc(t) and VD(t) may be required. Returning to the discussion of the 2- and 3-species model, it can be understood that in the 2-species model a separation between two columns in which single qubit operations are performed needs to four columns, i.e., only every fourth column contains such a crossed qubit.

[0160] In the 3-species model this separation can halved by introducing the third species, which allows for more flexibility in the control, thus allowing to limit the range of unwanted side effects .

[0161] An alternative to this scheme is to introduce, in addition to the crossed qubits have twice the Rabi frequency of the "normal" qubits, double-crossed qubits having four times the Rabi frequency of the "normal" qubits. In a similar fashion as the crossed qubits can be controlled independently (within the present disclosure) from the "normal" qubits, also the doublecrossed qubits can be controlled independently from the other types of qubits.

[0162] Based on this, in the 2-species model every second crossed qubit can be replaced with a double-crossed qubit and the "empty" columns, i.e., the column with neither crossed nor doublecrossed qubit and one adjacent column without any connection to any "special" qubit can removed without affecting the functionality .

[0163] In other words, according to an aspect of the present disclosure, there may be provided a quantum processing unit comprising: rows comprising first qubits and second qubits, the first qubits and the second qubits being arranged alternatingly within each row, the first qubits and the second qubits forming columns, respectively; 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 two adjacent qubits within each row are subject to a first two-qubit interaction, each columns of first qubits comprises a first special qubit provided between two adjacent second qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits, one in every two columns of second qubits contains one second qubit, named second special qubit, such that it is driven with at least twice the second Rabi frequency by the second source, and one in every other two columns of second qubits contains one second qubit, named further second special qubit, such that it is driven with at least four times the second Rabi frequency by the second source.

[0164] Consequently, in this manner the improved scaling of the 3- species model can also be achieved within the 2-species model.

[0165] Further information containing additional technical details will be provided in the following sections. As a remark, these sections may us a slightly different 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.

[0166] Furthertechnicaldetailson the quantum processing unit

[0167] Inthefollowing,additionaltechnicaldetailsinordertoclarifyandilluminatetheresultspresentedinthe above.First,somefundamentalsforunderstandinghow tomanipulatethestateofasuperconductingqubit usingexternalclassicalcontrolsisdiscussed.Thenitisdiscussedhow tocoupletwosuperconductingqubits vialongitudinalZZ couplingand how such interaction,in addition totheoff-resonancepulses,can induce an e-eblockaderegime. SomerealizationsofZZ interaction known in theliteraturearementioned.The nextsection addressesthedynamicalevolutionoftheladderintherotatingframe,underthee-eblockade. Then,an analysisofhow toperform quantum computation follows.Itisnoted thatwhilethesesections focuson thecaseofsuperconductingqubitsand aZZ interaction,thisisnottobeconstrued aslimiting thepresentinvention,rather,superconducting qubitsare,asoneofthefrontrunnersin thedevelopment ofquantum computingplatforms,awell-established platform todemonstratetheinventiveconceptsofthe presentdisclosure.

[0168] I. FUNDAMENTALS OF SUPERCONDUCTING QUBITS

[0169] A superconducting qubit [SI,S2]isa superconducting circuitwhich can beeffectively described asa two-levelsystem,bytheHamiltonian where% = A,B denotesthequbittype, isthePauli-zoperatorand a>x= (Ee— Eg) / h istheenergy splitting between theexcited state |e)and theground state \g}ofthe qubit. The latterdependsonly on the geometricparametersofthequantum chip. Indeed,fora simplequbitdesign,thefrequency is given by wx= (y / 8EjEc — Ec) / h whereEc = e2 / (2Cx)isthetotalcharging energy,Cy = Ca+ Cjis thetotalcapacitance,includingtheshuntcapacitanceCaandtheself-capacitanceofthejunction Cj,while Ej= Ic<b0 / 27ristheJosephsonenergy,withIcbeingthecriticalcurrentofthejunctionand$Q= ft / (2e)being thesuperconductingmagneticfluxquantum.Itturnsoutthattherearesomekindofsuperconductingqubits knownassplittransmons(SQUID),whichenablefrequencytunabilityusingexternallyappliedmagneticfield [si].

[0170] Byconnectingthe%-typesuperconductingqubittoanexternalelectricpotentialsourceUx(t),weareable toperform unitaryrotationson thequbit.Hereweoutlinehow thisisachieved.Followingtheconnection withtheexternalclassicalsource,weintroduceanadditionalinteractionterm intheHamiltonian,i.e. whereV := (Cd / CE)^ / (2^ 7c;)isaconstantdependingontheconstructionparametersofthecircuit. Wecangenerallyassumethatthetime-dependentpartofthevoltagewouldtakeagenericoscillatingform, where<xqjXisthefundamentalfrequencyofthedrivingpulse,<Sx(t)isanenvelopefunction,andfinally<t>x(t) isthephaseofthedrivingthatin thefollowingweshallassumetobetime-dependent.Itisinstructiveto moveinto aframerotatingwith thedrivefrequency <xqjX.Thisisdoneby definingtheunitary operator Frf(i)= such thatthenew statein therotating frameis |Qrf(t))= Urf(t)|Q(t)). The time-evolutioninthisnew frameisgivenbytheSchrodingerequation,

[0171] Simplemanipulationsshow thatin thisframewehave IE = = wx— <X\JJXisthe detuning.Thedrivingterm insteadyields i)(cos(wdiXt)d-fo)+sin(wdiXt)o-(:c)) with Qx(t):= VSx(t) / hthetime-dependentRabifrequency.NoticethatQx(t)dependson V,which itself relieson the capacitance between the qubitand the driving source. Thisallowsusto change Cdand consequently V in ordertomodify thenon-time-dependentpartoftheRabifrequency.Thisprocedureis theoneadopted torealizecrossedqubits(seethemaintext).Performingthemultiplication and dropping fastrotatingtermsthatwillapproximatelyaveragetozero (i.e.termswith 2wdjX),knownasrotatingwave approximation (RWA),theHamiltonianHdrive xcanbewrittenas whichallowsRabitransitionsbetweenthegroundandtheexcitedstatesandvice-versa.

[0172] A. TheZZ coupling impliestheblockaderegime

[0173] In thissection,we show thatin the rotating frame induced by the unitary transformation I7rf(t) := eZCT4 “a.s* / 2theHamiltonian H(t)ofthe 2D ladder,underRWA reducesto Hrf(t)ofthemain text. Followingtheapproachdetailedintheprevioussectionmovingintherotatingframewehave withHQand / / ,INV,(7)themany-bodyHamiltoniansdescribedinthemaintextwhichweexpresshereas withoutexplicitlystatingwhichtypeofqubitisassociated withthei-th site,butmaintain theconvention thatiftheindexhappenstoidentifyaregular%-typequbitthenWt= wx,ifinsteadidentifiesa%-typequbit with ablackcirclethen w,= w*:= wx— andfinally ifitidentifiesa%-typequbitwith ablacktriangle thenw;= wx:= wx+ £).

[0174] Notice hence that despite the presence ofthe ZZ couplings one has =Ho and that AccordinglythefirstcontributionofEq.(6)canbeexpressedas

[0175] ObservenextthatbyimposingwdjX= wx— 2£,intherotatingframethefrequenciesofthequbitsbecomes proportionaltothenumberofZZconnectionstheysharewiththeirneighbouringsites,i.e. AccordinglywecanrewriteEq.(9)asasum oftriplesoftheform const. (11) whereinthelastpassageweusedtheidentity

[0176] Thedrivingterm ofEq.(6)insteadyields

[0177] Replacing(11)and (14)andinEq.(6)gives / / ,(ofthemaintext.

[0178] B. RealizationsofZZ coupling

[0179] Oneimplementationofthearchitecturediscussedhereinmayusesuperconductingqubitsthatinteractwith theirneighborsviaaZZ coupling.Inmodernquantum architectures,thiskindofinteractionisexploited toimplementC-Phasegatesandithasbeenalreadyinvestigatedinnumerousexperiments,usingdifferent strategies.Inthissection,wepresentthreeofthepossiblewaysonecanrealizethisinteraction.

[0180] 1. Cavitymediatedcoupling

[0181] OnewaytoreachaneffectiveZZinteractionbetweentwosuperconductingqubitsistoconnectedthem dispersivelythroughacavity,i.e.awaveguidelinewhoseresonancefrequencywcisdetunedwithrespectto theoneofthequbits.ThiswasexperimentallydemonstratedinRef.[S3],wheretheyreachedaZZinteraction ofmagnitudeupto160MHz.TheresultisaneffectiveZZinteractionofstrength whereAx beingthelevelspacingbetweenthestateswithjandk excitationforthe -typequbit;gAandgBcorrespondtothecouplingsbetweenthecorrespondingqubitand thewaveguide.ThemainissueofthisschemeisthattheresonatorgivesrisealsotoaSWAP interaction(of thetype betweenthequbits.However,thiscanbeneglectedifoneconsiderthequbits fardetunedfrom eachother. 2. Qubitmediatedcoupling:Josephsonjunctionsdirectlyconnected

[0182] AnotherpossibilitytorealizetheZZcouplingistoplaceathirdsuperconductingqubits(calledcoupler) thatmediatestheinteractionbetweenthetwoqubits.ThecouplercanbeplacedsuchthattheJosephson junctionsofthethreequbitsaredirectlyconnectedtoeachother.Themainadvantageofthisarchitectureis thatitprovidestheabilitytocompletelybringtheSWAP interactiontozeroforacertaincouplerfrequency. ThisschemewasimplementedexperimentallyinRef.[S4]andstudiednumericallyinRef.[S5].Theinteraction engineeredinthiswaydependsonthecircuitelementsas with Ejand Ec beingrespectivelytheJosephson andchargeenergyofthequbitsand EjtheJosephson energyofthecoupler.Onlysmallcouplingshavesofarbeenmeasured(< 10MHz),butnumericalcalculations hintatthepossibilityofengineeringstrongerinteractionsupto300MHz,whileatthesametimekeeping theSWAP interactionclosetozero.However,thisreliesonacouplerqubithavingaverylow Josephson energy(20MHz),whichmightnotbefeasibleexperimentally.Moreover,itcouldbechallengingtoscalethis approachtoalargenumberofqubitsasthesystem becomesincreasinglysensitivetofluxnoise,asitwas pointedoutinRef.[S6],

[0183] 3. Qubitmediatedcoupling:Josephsonjunctionscapacitivelyconnected

[0184] Thelaststrategywebrieflymentionhereissimilartothecavity-mediatedapproach,butwiththedistinction thataqubitreplacestheroleofthewaveguideasthecoupler.TheeffectiveZZcouplingwascalculatedup tofourthorderinRefs.[S7,S8].Thefirstnon-zerocontributionforthecouplingstrengthisapproximately withA| / j= A\~ is 1'1Ganharmonicityofthe%-typequbitandgisthecapacitivecouplingbetween thequbitsandthecoupler.ThisschemewasinvestigatedexperimentallyinRef.[S9],wheretheyreacheda couplingstrengthof~ 50MHz,andnumericallyinRef.[S6].Itisworthmentionthatthistypeofbuilding blockistheoneemployedinmodernquantum architectures,e.g.intheSycamorequantum processor[S10].

[0185] II. DYNAMICAL EVOLUTION OF THE 2D LADDER

[0186] Inthissectionwediscussindetailsthedynamicalevolutionofthe2D ladder.Thematerialisorganizedas follows:inSec.IIA wesolvetheequationofmotionforarbitrarysequencesofcontrolpulsesunderthee-e blockaderegimeprovingthattheyallowsustorealizeacertainproductsoftheoperatorsWx(0',n';d".n") defined in Eq.(6)ofthemain text;in Sec.IIB wepresentsomegenericpropertiesofcontrolunitaries Wx(0',n';6",n");in Sec.IIC wefinallyshow thatanypossibletransformationsWx(0',n';0",n")can be implementedbyproperlyselectingthesequencesofourglobalcontrols.

[0187] A. Effectivedynamicsundere—eblockadeconditions

[0188] AsdetailedinthemaintextinourmodeltheclassicalcontrolpulsesVA(£)andVB(t)assumeconstantvalues ondisjointtimeintervals.Formallythisimpliesthatwecandividethetemporalevolutionintoacollection ofdisjointtimewindows71,71,•••,71whereonlyoneofthecontrolsisnonzero.Accordinglyineachof thosetimewindows,theHamiltonian77rf(t)ofEq.(3)ofthemaintextcanbetreatedastime-independent, i.e. with xe€ {A,B}representingthecontrolthatisactiveon the€-th timeinterval,and with QXf*and ^xl theconstantvaluesassumedbythefunctionsQXf(t)and respectively.Thereforethetotalevolution inducedbyHrf(t)canbeexpressedasthesequenceinEq.(4)ofthemaintext,i.e. where4xp[...]standsforthetime-orderedexponential,7tot:= 7iU E U '''U isthetotaltimeintervalof thedynamics,and istheunitaryevolution associatedwith thef-th timeinterval(-77beingitstemporalduration).Inthee-e blockaderegimewherethecouplingconstant£isthemainenergyscaleofthemodel thetransitionsinducedbythedrivingpartofH*?aresuppressedwhenevertheyinvolveinputorfinalstates whereatleastoneofthequbitofy^-typethatarefirstneighboursofthey / -lypeisin theexcited state, leadingtoEq.(5)ofthemaintext.Morepreciselyweshallseethatinthelimit / / m;2>1eachelement ofthesequence(19)canbeexpressedintermsofcontrolunitarygatesoftheform uptoaglobalunitarygatethatcanbepostposedtotheveryendoftheprocessandwhichplaysnorolein thesystem.InEq.(21),P^ istheprojectoronthesubspaceofthenearest-neighbouringyy-typequbitsof thei-thsitewhichcontainsnoexcitations, istheorthogonalcomplementofP^,whilefinally isthe single-qubitunitaryevolutioninducedbytheRabicontributionof ,i.e.

[0189] Werecallthatiftheindex iidentifiesaregular(possibly crossed)y-typequbitwehaveP^ := \gg / (gg\ and := |ee)(ee|+ \eg)(eg\+ |ge)(ge|,where\gg),\&g),\ge),|ee),representtheenergylevelsofthetwo qubitsofy-typethatexhibitaZZ couplingwithsuchsite.OnthecontraryifiidentifiesaB qubitwitha blackcirclethen,thesiteiscoupledwithonlyasinglesitesothatP^ := |g)(g\and := |e)(e|.Finally ifinstead iidentifiesaB qubitwith ablacktrianglethen therearethreeinteractingA-typesitessothat P(i):= \ggg)(ggg\andQ{i}:= |eee)(eee|+ \eeg)(eeg\H- I-\gge)(gge\.

[0190] ToproveEq.(21)itisconvenienttoexpresstheunitary (20)in theinteraction picture,identifyingthe interactionpart oftheHamiltonian (18)withtheterm thatisproportionaltotheRabifrequency,and thefree-contributionHfreewiththeterm thatinsteadisproportionaltotheZZcoupling,i.e.

[0191] Accordinglywecanthenwrite isthedurationofthetimewindow 7)and theinteractionterm expressedininteractionpicture.TodetermineH^' (i)observethatthevariouscontributionsofHfreeallcommutesothatwecanwrite where istheidentityoperatorontheHilbertspaceofthei-thandj-thqubitthatform aninteractingZZ couple Observealsothatwhen actingon agenericproductstate|J)oftheladderqubitsformed by elementsofthecomputationalbasis{|g),|e)}(e.g.statesoftheform |ei<7ie2C3••• withM beingthetotal numberofqubitin thesystem),thetransformation ex11willevolveitbyaddingaphasecontribution foreachoftheZZinteractingcoupleswhichareinthe|ee)configuration.Moreexplicitlywecanwrite withM(J)theintegerthatcountsthenumberofZZinteractingcoupleswhichinthesequence|J)areinthe |ee)state.Toseehow e~iHfreettransforms itisusefultorecallthatthe\gi)(ej|whichentersinEq.(23) isanoperatoronthefullladderthat,forallsitesbutthei-th,actsastheidentity.Inparticular,indicating with agenericelementofthecomputationalbasis{|g),|e)}oftheentirecollectionofthequbitsofthe system excludedthei-thwecanwrite

[0192] From Eq.(27)ithencefollowsthat countingrespectivelythenumberofZZinteractingcoupleswhichinthesequences

[0193] \giJ^)and|ejJ^)areinthe|ee)state.Observethatbyconstructionwehavethat isalwaysgreater than orequalto and thatthetwocoincideifand only ifthesiteswith which thei-th qubitis

[0194] ZZ-coupledareallinthegroundstate,aconditionwhichcanexpressedas theprojectorsappearinginEq.(21).Accordinglywecanwrite anoperatormadeofasum oftermswhichoscillatewithfrequencieswhichareintegermultiplesof£.Atthe leveloftheoperatorH^t(i)thistranslatesintotheidentity Underthee-eblockaderegime, / / mi= |^ / Q^I 1,theoscillatorypartofH^' (t)evolvesovertime-scales thataremuchsmallerthanthetypicaltime-scalesdeterminedbythefirstcontribution.Enforcingatemporal coarse-grainingovertimeintervalsTc.g.suchthat|QX2|2>Tc,g, 1 / |£|thisleadsto sothat with asin Eq.(22)-noticethatinwritingthelastidentitywedonotneed toabouthow toorder thevarioustermssincetheyallcommute.ReplacingthisintoEq.(24)wehenceobtainthatforeachtime window T / onehas

[0195] (38)

[0196] Noticingthat commuteswith (36)and hencewith alltheoperatorsoftheform (37)(seee.g.

[0197] Eq.(32)),thisfinallyallowsustoexpresstheglobalevolutionoperator(19)as provingthethesis.

[0198] B. Generalproperties

[0199] A convenientwaytorewriteEq.(21)istointroducetheparameters

[0200] Recallingthatthecrossedelementsoftheyy-typequbitshavetwicetheRabifrequencyoftheregularones, wecanhencewritethetransformation(37)as wheregivenx€{Al, andy1andy theregular(noncrossed)andcrossedsubsetsofx,wedefined with

[0201] Ri(0,n) := exp[—i(0 / 2)n•a;]= cos(0 / 2)ii—isin(0 / 2)n•dy, d;:= (444444 > (43) thesingle-qubitunitaryrotationassociatedwiththeunitvectornandtheangle0.

[0202] Thetransformations(42)obeysomeveryusefulproperties: 1.Setting6'= 0(0"= 0),theunitaryevolution Wx(0',n';0",n")reducestothemappingWJxJJn") (lJxr(0',n'))thatonlyactsonthecrossed(resp.regular)elementsofthe%-typequbits,i.e.

[0203] (thisisatrivialconsequenceofthefactthat +Q{i)= iyqistheidentityoperatorontheassociated space).

[0204] 2.ForfixedyythecrossedandregularcontributionsofWx(0',n';0"n")commute,i.e.

[0205] 3.Forfixed£6 {%r,%x}thetransformationsW$(0,n)form agroupthatobeysthesamecomposition rulesoftheSU(2)definedbyunitarymatricesRJJn).Inparticularwehavethatgiventheangles 01,0-2andtheunitvectorsni,112onehas with theangle0$and theunitvectorn3satisfying theidentity lj(02,r^RjJ,nJ = Rj03,n3). FurthermoretheinverseofI4J(Jn)correspondstoI4J(—0,n)= I4J(J—n),i.e.

[0206] 4.Asin thecaseofthesinglequbitrotationsRJJn),forfixed n,thetransformationsI4J(Jn)are periodicin0withperiod4TF,i.e.

[0207] Inparticularfor0= 4TFthetransformationcoincideswiththeidentity.Noticehoweverthatfor0= 2TF theevolutioncorrespondstoanontrivialphase-transformationontheJtypequbits,

[0208] 5.Theproperty(46)fory1andy translatesthefollowingcompositionrulefortheglobaloperations,i.e. with03,n3,andJ', determinedthetheidentitiesRJ0JnjRjJ,nJ = Rjj,nj and RJJ',njRjJ',n'J = RJ03,n3),respectively.

[0209] InSec.IIA wehaveseenthatine-eblockaderegime,eachindividualpulsedunitaryUxthatcomposefJot implementsaspecifictypeofcontrolunitarygates(42),i.e.thesubsetoftransformations whichactsimultaneouslyony1andy inducingsingle-qubitrotationsaroundagenericunitvectorn of theajy-plane(i.e.z•ni = 0)andwithcorrelated angles0and 20.Hereweshow thattheseconstraints can beovercomeby properly combiningsequencesofthesespecialtypeofpulses:specifically weprove thatthesequences(19)generatedbytheclassicalcontrolsoftheladder,canproduceanytransformation Wx(6',n'-,6"n"')with 6',6",and n',n"arbitrarilyselected.Thefirstingredienttoattainthisresultisa propertythatholdsforPaulirotationsalongorthogonalunitvectors,i.e. n •m = 0 => Ri(7F,—n)R;(0,m)R;(7F,n)= Ri(—0,m)= Rix(0,m) W . (52)

[0210] ThisidentityisadirectconsequenceofthefactthatRj(7T,n)= —in•<7i,andthatthePaulioperators(n•<? / ) and (m •cq)anti-commute,i.e.(n•cq)(m •cq)(n•cq)= —m •cq.RecallingthatR;(27r,n)= —iq wecan henceconcludethat

[0211] Considerhencea4-pulsesequence(19), wherethefirstandthethirdelement,Ux^and

[0212] Ux,correspond tothesametransformation ITx(0 / 4,ni;6 / 2,ni)(seeEq.(41))with assigned axisnx in thexy-pl;ine. On thecontraryweseethatthesecond and fourth term,Ux)'and Ux4\ areequalto TUx(7r,nq;2TF,nq)and ITX(TT,—nq;2TF,—mx).respectivelywithnq alsointhexy-pl;inebutorthogonal toiq (i.e.nq •nx = 0).UsingEq.(39),thecompositionrulegiveninEq.(50)andtheidentities(53)we canhenceconcludethat whichforsakeofsimplicitywewritedroppingthecontribution exp / / |reeT|Oij.Equation (54)implies thatusingthecontrolpulsesofthemodelwecangeneratetransformationsthatactselectivelyonthecrossed elementsofthey-typequbitsinducingarbitraryrotationaroundanyaxisnx inthexy-pl;ine(inparticular wecaninducerotationaroundthexandyaxis).InvokingthestandardEulerrotationtheorem [Sil]wecan generalizethistoanycontrol-unitaryTUxx(6,n)withrespecttoanypossible(nonnecessarilyorthogonalto z)axisn.Indeedrecallthatgiven (6,n)arbitrary,thereexistthreeanglesa,f3,and7,suchthatwecan write

[0213] Usingsequencesoftheform (54)wecanhencetranslatetheaboveidentityatthelevelofourcontrolunitary gates,obtaining

[0214] Noticethattoachievethisgoaloneneedsnomorethan3x4= 12unitarypulsesUx.A similarresultcanbe appliedtothecontrolunitariesthatactselectivelyontheregularelementsofthey-typequbits.Forinstance usingthetransformation(54)wecancompensatetheyxcomponentofthegates(51),i.e. whichshowsthatalsofortheregularelementsofthey-typequbitswecaninducearbitraryrotationaround anyaxisnx inthea?y-plane.From thiswecanthenusetheargumentthatledusto(56)toconcludethat sequencesunitarypulsesUxcanleadstoanyoperationsoftheform ITxr(0,n)witharbitrarynand6,i.e.

[0215] FinallyconcatenatingtheresultsofEqs.(56)and (58)wecaninduceanyarbitrarytransformationsofthe form Wx(6',n'-,6",n"'). III. INFORMATION ENCODING AND QUANTUM COMPUTING

[0216] As discussed in themain text,ourmodelencodestheinformation in oneofthecolumnsofthedevice.

[0217] Specificallyateachstepofthecomputation,weassumetheladdertobeinastateoftheform Inthisexpressiontheket|•••)&'describesthestateofthefe'-thcolumnofthedevice:inparticular\g^>N)k’ (|e0JV)fc / )representsthecasewhereallthequbitsofthefe'-thcolumnareinthe\g)(|e))configuration. isthelogical(possiblyentangled)stateoftheICC,and g! X [')\ describesthestateofthe,\— lcrossed ,1-lvpequbitswhich actasconnectorsbetween adjacentrows(noticethatin writingEq.(7)ofthemain textthepresenceofthelastcontributionwasdroppedforeaseofnotation).Itisalsoworthstressingthat, in ourmodel,thesearetheonlycrossedelementsof,1-lvpe:in particular,alltheA columnsoftheladder arecomposedofonlyregular(non-crossed)elements.

[0218] Statesasdefined in Eq.(59)aresaid tobewell-formed. Thesestatesexhibitno quantum correlations amongthedifferentqubitsofthedevice,exceptforthoselocatedintheICC,which aretheonlyonesthat canshareentanglement.Analternative,morecompactwaytoexpressthem is:

[0219] I’E;k)= |para)® I’I'jj.® jferro), whichemphasisesthattheICC isaninterfacebetweenaparamagneticregionontheleft,where,startingfrom therightmostelementwhichisalwaysin yv).allthecolumnsarepreparedin an alternatingsequenceof |g0JV)and e configurations,andaferromagneticregionontherightwhereallthecolumnsareinsteadin the|g®N)configuration.Inthiscompactnotation,theferromagneticregionalwaysincludesalsothecrossed qubitsoftypeA.Theprocedureforinitializingtheladderinthisspecialtypeofstateswillbediscussedin Sec.IIIE.Here,instead,wediscusshow thewell-formedstateofthemodelevolvesundertheactionofthe system Hamiltonian.

[0220] Remark:Tosimplify thepresentation in thefollowingsection,weshallintroducethespecialnotation “=”toindicatethattwovectorstateofthesystem areequaluptoan(irrelevant,notstatedependent)global phase.Specifically

[0221] A. Preliminaryobservations

[0222] Tobeginwith,letusnoteafew usefulproperties. i)InouranalysiswewillonlyusestatesI’E;fe)with valuesofkthatarelargerthanorequalto3(i.e., thefirstcolumn oftheprocessing areaofthedevice). In otherwords,thelogicalinformation will alwaysbelocatedeitherintheprocessingareaorintheread-outarea,neverintheinitializationarea. Additionally,the|^;kfscanbecategorizedintotwogroups:

[0223] — A-type well-formed states:thesearestatesI’E;fe)wheretheICC index kidentifiesacolumn oftheladderformedby ,1-lvpequbits;

[0224] — B-typewell-formed states:thesearestates|^;k)wheretheICC indexkidentifiesacolumnof theladderformedbyB-typequbits. ii)Thewell-formed states areinvariantundertheaction oftheglobalunitary e~iHfrBBTtotwhich entersinthedecomposition (19).Specifically,

[0225] Thisisduetothespecialarrangementoftheinteractionsintheladder,whereinthestate|^;fe),each ZZ-interactingpair(i,f)hasatleastoneelementthatisinthegroundstate,so(e^Cj|^;k)= 0.Thanks toEq.(62),andrecallingthate commuteswithallthe terms,wecanwrite whichjustifiesdroppinge_sfffreeTtotinEq.(54)andinEq.(5)ofthemaintext. hi)LetI’Jqfe)beaB-typewell-formed state. Theapplication ofthegateZ^ ofEq.(49)inducesthe followingmapping: where representsawell-formedstateofthesystem withtheICC stilllocatedinthefe-th column ofthedevice,butwith theinternalstateofthelogicalqubitsevolved undertheaction ofa contemporaryapplicationof gates.

[0226] ToseewhyEq.(64)holds,recallthatZffl actsselectivelyontheregularA-typequbitsofthemodel, applyingtoeachofthem acontrol-27rrotation,i.e.acontrol-phaseshiftof(—1).Observethatsince theICC column isofB-type,then theonly A-typecolumnsthatcan acquiresuch extraphaseare thoseintheferromagneticarea.Indeed alltheA-typecolumnsontheleft-hand-sideoftheICC have atleastaninteractingneighbouringB elementthatisintheexcited state.Apartform thefirstone, the,1-lypecolumnsintheferromagneticregionarecoupledwithneighbouringB-typesiteswhichare in adefinitegroundstate\g)'.Actingonthem,Z^fwillassigntothestateanirrelevantglobalphase thatisaproductof(—l)’s.Theexactnumber ofsuch termsdependson theactuallocation of ICC column.IntheendtheonlynontrivialeffectthatZ^fcanintroduceinthemodeloccurswhen itactsonthefirst,1-lypecolumnoftheferromagneticarea.Forfuturereferenceweshallidentifysuch columnwiththesymbolA*.DuetothepresenceoftheICC,theactivationofthe(—1)phaseonthe / -throw elementofthiscolumn dependson thestateofthe / -thlogicalqubit(observethattheonly otherB-typecolumnthatisZZcoupledwithA*hasallthequbitsinthe\g)state).Toformalizethis, defineZ^rastherestrictionofZ^1’1onthecolumnATt,i.e.

[0227] From thepreviousanalysis,wecanwrite:

[0228] Noticethat isacontrol-phasegatethatacton thei-th qubitofA*,which isactivated if and onlyifitstwoZZ coupled neighbouringsitesareboth in theground state.Tomakeuseofthis factexpandthelogicalstate|^)oftheICC as where\gi)and |ej)arethegroundandexcitedstateoftheelementpertainingtotheICC columnthat ison thesamerow ofi-th qubitofATt,|^s)and |^e)are(non necessarilynormalized)statesofthe remainingN — 1elementsoftheICC column.Observethenthat fe), (68) whichwhenreplacedinthepreviousEq.(66)gives provingEq.(64). iv)Theproperty (64)alsoholdsforcertainill-formedstatesoftheladder.Specifically,thisoccursifone ormoreoftheB-typecolumnsin theferromagneticphaseofaB-typewell-formed state are replacedbyarbitraryqubitstates.Theonlyrequirementsweneedtofulfilare

[0229] — theremustbeatleastthreeregularferromagneticcolumnsbetween anytwo “deteriorated”B columns;

[0230] — theminimaldistancebetweenadeterioratedB columnandtheICC mustbeatleastthreeregular ferromagneticcolumns;

[0231] — thelastB columnofthedevicemustnotbedeteriorated;

[0232] An ill-formed statethatobeystheaboveconstraintswillbesaid to bewell-behaved. Underthese conditions,eachofthetwoA columnsnearadeterioratedB columnhasanextraB columninitializedin theproperferromagneticconfiguration yv).Thus,whenZ1!'1isappliedtothesystem,adeteriorated B column willactastheuniquecontrollerforitstwoneighbouringA elements,assigningthesame conditional—1phasetoboth,ultimately notmodifyingthestate. Morespecifically,given awell- behaved,ill-formedstate|1F;fe),define{BI,B2,•••,&s}asthesetofitsdeformedcolumns.LetA(lel1' theA columnsontheleft-handsideandright-handside,respectively,ofthef-thelement ofthisset.AsinthecaseofEq.(69)wecanwrite wheretheglobalphase(—l)MfcdependsonthenumberMk'offerromagneticB columnsthatarenot deteriorated.Here indicatesthepartof thatinvolveselementsofthe

[0233] A column.Duetothecommutativityoftheoperators +R;(0,n)® P^J associated with differentsites, ofdifferentcolumnscommutes.FollowingthesamederivationusedinEq.(68),we thencanshow that sothat whichprovesthethesis.

[0234] B. Motion oftheinterface

[0235] Aswehavementionedinthemaintext,byapplyingaspecificsequenceofpulses(39)wecangeneratea unitarytransformationPshiftthatrigidlymovestheinterfacefrom lefttoright,i.e. forall3< k< 4N (recallthatthetotalnumbersofcolumninthemodelis4N + 1).Inparticulargiven thiscanbedonee.g.bytakinig nArnBnAr if1’1';k)isaB-typewell-formedstate; (78) nSILvnB if1’1';k)isaA-typewell-formedstate, whichwewriteomittingtheterm e_sfffreeTtot(|ue|OprOpertyii)ofSec.IllA.

[0236] ToverifythatourchoiceofBahiftiscorrect,considerthecasewhere|^;k)isaB-typewell-formedstate (theanalysisforthecasewhere|^;k)isaA-typewell-formedstateissimilarandwillnotbereportedhere). Tostudytheevolution ofthisstateunderthesequencenppnBnAritisworth tosplitthevector|^;k)in thefollowingfoursectors: sectorIII sectorIV whereweaddthesubscriptA andB tothevarioustermstospecifythetypeofthecolumn.

[0237] •SectorIV:Firstofall,itisclearthatsincethecrossedA-typequbitsarenotaffectedbytheselected controls,theterm insectorIV willbeleftunchangedbytheevolution,i.e.

[0238] Mostimportantly,itisworthstressingthatthecrossed A elementswillremainin the|g)stateatall timesduringtheentirepulsesequence.Thisfactisextremelyimportantasitmeansthatthesequbits playnorolewhat-so-everintheevolutionoftheotherqubitsofthedevice.Inparticularthepossibility thatagiven control-pulsewillbeactivated on oneofthequbitsthatareZZ coupled with acrossed A-typeelement,doesnotdependonthelattersinceitwillbealwaysinthe “go”state.

[0239] •SectorI:AccordingtoEq.(77)thequbitsofthissectorareinitiliazedintheparamagneticphase.As evidentfrom theequation in thiscasetheA columnshaveatleastanearbyB column in the |e0JV) whichpreventsthefirstlip ofthesequencetooperateonthem.Viceversa,whenweapply II / ;.since alltheB columnshaveneighbouringA siteswhichareallin \g),theywillexperience— rotations thatwillmap them into (— Thefinallip operation willhencenotbeblocked,modifying thestatesoftheA-typequbitsinparamagneticareavia— pulses.Theglobaltrajectoryinduced bytheunitaryBahiftonthecomponent(80)of|^;fe)canhencebeexpressedas uptoanirrelevantglobalphaseshiftgivenby(—i)MkwithMkcountingthetotalnumberoftheapplied rotations(anumberthatdependsonlyonfe).

[0240] •SectorIII:Inthiscaseitisclearthatthefirstlip caninduce— rotationonalltheA columns, bringingthem into |e0JV)stateswhich,inturn,willpreventtheactionofflB.Thefinaloperatorlip willstillbeabletoactreturningalltheA columntotheiroriginalconfiguration,i.e. •SectorII:Thestudyofthissectorisslightlymorecomplexsinceinthiscasethevariouspulsesinduce correlationsamonginvolved columns(i.e.theICC column and itsfirstneighbouron theright).A usefulobservation isthatwhen weapplythetwo lip transformationsofthesequence,theaction of theseoperatorsonthe(fe+ l)-thcolumn(whichisofA-type),onlydependsontheinternalstateofthe ICC column,duetothefactthattheB column atposition k+ 2isinthestate|g®N)-seeEq.(81). SimilarlywhenweapplytheHB,itsactionontheICC columnonlydependsonthe(k+ l)-thcolumn, duetothefactthattheA column atposition k— 1isin |g®N)asshown inEq.(80)(recallthatthe crossed ,1-lypequbitsarealwaysin the \g)duringtheentireprocess).Wecan henceconcludethat, despiteourcontrolspulsesaffectmorethan twoqubitsatthetime,thankstotheselected encoding, theneteffectofthesequenceIIpll / ,dIp istoeffectivelyinduceaselectivecouplingbetweencolumnk withcolumnk+ 1.Tostudyexplicitlywhattypeofevolutionthisinducesinourmodel,itisusefulto considerfirstthescenariowheretheladderisformedbyasinglerow.InthiscasethestateofsectorII canbeexplicitlyexpressedas

[0241] ’I'B)*:® |g,4)fc+l= a\gB)k® |g,4)fc+l+ / 3\eB)k® |g,4)fc+l, (82) whereweexpanded I’I'B)*,in thecomputationalbasis.Wecannow easilytracktheevolutionofthis configurationrecallingthatthevariouscontrol-operationsareactivatedeitherbytheinternalstateof fe-thcolumn (inthecaseof11^):orbytheinternalstateofthe(k+ l)-thcolumn (inthecaseofnB). Accordinglyweget: which showsthatattheend ofthesequencethestatesofthek and k+ 1haveswapped (up to an irrelevantglobalphase). To generalizethisresultto thecaseofN rows,simply recallthatin our model,apartfrom thepresenceofthecrossed ,1-lypequbitswhich in thepresentcasearealways in the g)state,therearenotdirectinteractionsamong thevariouselementsofacolumn. Thisin particularimpliesthatthedynamicsinducedby llpll / ,dlp canbeaddressedtreatingthevariousrow independently.Accordinglywecanconcludethat(uptoaglobalphase)onehas,

[0242] TheproofofEq.(73)finallyfollowsbyputtingtogetheralltheidentitiesobtained forthedifferentsectors (again theresultisobtainedup toan irrelevantglobalphasewhich doesnotdependontheinputstateof theICC).

[0243] We finally remark thatan alternative (yetfully equivalent)implementation ofthetransformation (73) couldhavebeenrealizedbyreplacingalltheTFanglesappearinginEq.(74)with—TF’S(withthischoicethe state|rb;fe)willstillmappedinto|^;k+ 1)uptoaglobalphase).Thisinnocentlookingobservationisuseful toclarifiesthatthesameoperatorsgiveninEq.(76)willalsoinducethereverseofthemapping(73),i.e. ;fe). (85)

[0244] IndeedchangingthesignoftheTF’SinEq.(74)meanstakingtheinverseoflip andIIB-Thereforeforthe casewhere fe)isaB-typewell-formedstatewecanwrite whichprovesthethesis(for,1-lypewell-formedstateswecanproceedsimilarly). 1. Explicitpulsesequence

[0245] Noticethat11 / >■induces7T-pulseson alltheB qubits(crossed and regular).A specificinstanceofthe controlparametersthatallowsforsuchoperationisobtainedbyconsideringthefollowingthreestepsequence

[0246] •Ug1isinducedbysetting<j>^ = TF / 2andlettingevolvingthesystem foratimeTi=

[0247] •Ug7isinducedbysetting<(>g = 0andlettingevolvingthesystem foratime

[0248] •Ug7isinducedbysetting<f>g = TF / 2andlettingevolvingthesystem foratimer3= ff^B (same asthefirststep);

[0249] (recallthatbyconventionQB istheRabifrequencyoftheregularB-typequbits).

[0250] On thecontrary lip induces7T-pulsesonlytheregularelementsoftheA qubits.A specificinstanceof thecontrolsparametersthatallowsforsuchoperationisobtainedbyconsideringthefollowingathreestep sequence now

[0251] •UA^isinducedbysetting = TF / 2andlettingevolvingthesystem foratimeTi=

[0252] •UA7isinducedbysetting<j)A= 0andlettingevolvingthesystem foratime = / 4 / 3;

[0253] •UA7isinducedbysetting<j>A= TF / 2andlettingevolvingthesystem foratimer3= )Q| (sameas thefirststep);

[0254] (also istheRabifrequencyoftheregular,1-lypequbits).

[0255] C. Single-qubitgate

[0256] Referringtotheprevioussubsection,weareabletomovetheinterfaceinwhateverdesiredpositionofthe ladder,specificallywhereaB-typecrossed-qubitispositioned.Inordertoperform thesinge-qubitgate(e.g. theHadamardgate),oncetheinterfaceislocatedattheB-typecrossed-qubit,weneedtosendaspecific globalsequenceofpulsesinvolvingA-andB-typequbits.Specificallytheoperationsneededforimplementing thesingle-qubitgatescanberealizedcomposingcontrolpulsesthatinduceevolutionsoftheform

[0257] Z^WBxm -nx^ WBxm nx) (88) with ni orthogonaltozand definedinEq.(49).IndeedthankstothefactthatthecrossedB-type qubitarelocatedoncolumnswhichareatleastthreecolumnsapartfrom other,whenactingon |'■I'; / r)the abovetransformationwilleffectivelycorrespondtoapplythesinge-qubitrotationR(0,nj_)onthecrossed elementoftheICC.Thekeyobservationherearethepropertiesiii)andiv)ofSec.IllA.Noticeinfactthat since|^,fe)isawell-formedstateofB-type,withkbeingassociatedwithacolumnB thatcontainsacrossed term,thenundertheactionoflVBx(0 / 2;nj_)itwillbecomeawell-behaved,ill-formedstate.

[0258] Wg„.(e / 2-n±)|^;fe)= |^;fe):= 0 R;(0 / 2,n±)|^;fe), (89) i£Bx / PARA whereinthelastidentityweemphasizethatallthecrossedB qubitsthatarenotintheparamagneticarea (includingtheoneintheICC)acquirestherotationR,(0 / 2,n_|_).ThereforeunderZA*thisstatewillevolve asinEq.(72)acquiringd(z)-gatestoeachoftheelementoftheICC,i.e.explicitly jeicc TheactionofTFgx(0 / 2;—n jwillbesimilartowhatseeninEq.(89),i.e.

[0259] Sincethelatterisagainwell-behavedwecanthenreplicatetheargumentof(90)towrite

[0260] Now thethesisfollowsby observing thatifj correspondsto a non-crossed elementofthe ICC,itwill experiencetheactionof similarlyificorrespondstoacrossedelementthatisnotintheICC, itwillexperiencetheactionof (0 / 2,—nx)R / (0 / 2,ni)= 1,;onthecontraryifjhappenstobethecrossed elementoftheICC itwillundergothetransformation

[0261] D. Two-qubitgate

[0262] Asmentionedinthemaintext,toentangletwologicalqubitswesimplyneedtobringtheinterfaceatthe positionofthe,1-lypecrossed-qubitthatconnectsthetwologicalqubitswewanttoentangle.Subsequently, wesend aglobalpulseon the ,1-lypequbits,designed toperform a2TFrotation on thecrossed A-qubits, i.e.thegate definedinEq.(49).Thus,suchqubitswillacquirea(—1)phasefactorifandonlyifthe twoconnectedB-qubitsareinthegroundstate,realizingacontrolled-phasegate(CZ).Actually,thepulse isrealizedtoperform a2TFrotationalsoonthenormalB-qubits.However,thissimplyaccountsforaglobal phasefactor,which wediscard.Insummary,thetransformation canberealized through afivestep sequence

[0263] Beforewestartthecomputation,thedeviceisinafullferromagneticphasel^ferro):= )<8>,yV)••• whereallthequbitsareinthe|g)state.Thisisastableconfigurationofthesystem Hamiltonianand,incase the / interactionterm ispositive,itwouldalsocorrespondtothegroundstateofthemodel.However,thisis notawell-formedstateanditwillnotreactwellwhenweapplythesequences(19).Anessentialingredient ofourarchitectureistheabilitytoforcetransitionsfrom l^ferro)tooneofthevectors|’F;fe).Thisiswhere thecontrolpulseV;nit(i)isuseful:itallowsustoselectivelypromotethefirstcolumninFig.3ofthemain textto |e0JV),whichformally,correspondstothewell-formedstate|^o;3)wheretheICC islocated inthe firstB columnoftheprocessunit,andthelogicalstateis|^o):= |g0JV).From thisstate,wecanstartthe computation movingtheICC back and forth on theprocessingareaand then apply anydesired quantum gate. NoticethatVjn;t(i)contributestothedynamicsasanothercontrolpulse,i.e.addingtotheHamiltonian thefollowingtime-dependentterm where<^init(£)and fhnit(i)aretheassociatedcontrolfunctions,andwherethesum isrestricted toonlythe elementsofthefirstcolumnofthedevice(hereindicatedwithBi).Thereforetailoringthephase,theRabi frequency,andthedurationTinitofthecontrollinewecaninducetheevolution whichindeedallow ustorealizetherequiredmapping ll / y l^ferro)= I'E'o!3).

[0264] F. Read-out

[0265] Attheend ofthecomputation,weread outthelogicalstateoftheICC bymovingittotherightmost elementoftheprocessunit(i.e.,theB columnoftheread-outarea)usingthetransformationsUahift-Despite itistheoreticallypossibletomeasurethelogicalstateoftheICC columnwhileitisinsidetheprocessarea, weprefernottodothis.Suchanapproachwouldinevitablyrequirelocaladdressingoftheelementswithin thecolumn,whichcontradictsthefundamentalprinciplesofourarchitecture.Noticealsothatthescheme presentedinFig.3ofthemaintextrepresentstheminimalsettingthatallowsforaseparationbetweenthe processingareaandtheread-outcolumn.Ifweneedtoincreasethisseparation,itcanbeeasilyaccomplished byaddingextra,1-lypeandB-typecolumnsintheread-outarea.

Claims

Claims1. A quantum processing unit comprising: rows comprising first qubits and second qubits, the first qubits and the second qubits being arranged alternatingly within each row, the first qubits and the second qubits forming columns, respectively; 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 two adjacent qubits within each row are subject to a first two-qubit interaction, one in every two columns of first qubits comprises a first special qubit provided between two adjacent second qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits, and one in every two columns of second qubits contains one second qubit, named second special qubit, such that it is driven with at least twice the second Rabi frequency by the second source.

2. A quantum processing unit comprising: rows comprising first qubits, second qubits and third qubits, the first qubits being arranged alternatingly with a set of the second qubits and the third qubits within each row, the set of the second qubits and the thirdqubits alternating between the second qubits and the third qubits, the first qubits, the second qubits and the third qubits forming columns, respectively; 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; a third source configured to drive the third qubits with a third Rabi frequency, wherein two adjacent qubits within each row are subject to a first two-qubit interaction, each column of first qubits comprises a first special qubit provided between two adjacent second or third qubits, the first special qubit being such that it is driven with at least twice the first Rabi frequency by the first source, and the first special qubit being subject to a second two-qubit interaction with respect to each of the two adjacent second qubits or to each of the two adjacent third qubits, each column of second qubits contains one second qubit, named second special qubit, such that it is driven with at least twice the second Rabi frequency by the second source, and each column of third qubits contains one third qubit, named third special qubit, such that it is driven with at least twice the third Rabi frequency by the third source.

3. The quantum processing unit according to claim 1 or 2, whereinthe 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 2, the third source is configured to drive the third qubits using a single signal provided globally to all of the third qubits.

4. The quantum processing unit according to any one of claims 1 to 3, 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 2, the first source is configured to be off if the third source is on, and the third source is configured to be off if the first 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 2, 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 one of the two second qubits subject to the two-qubit interaction with a first special qubit is the second special qubit of the respective column, and when dependent on claim 2, one of the two third qubits subject to the two-qubit interaction with a first special qubit is the third special qubit of the respective column.

5. The quantum processing unit according to any one of claims 1 to 4, wherein the first qubits have a first qubit frequency, and the second qubits have a second qubit frequency, and when dependent on claim 2, the third qubits have a third qubit frequency, wherein optionally the qubit frequency of each second qubit adjacent to the first special qubit is 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, and when dependent on claim 2, the qubit frequency of each third qubit adjacent to the first special qubit is different from the second qubit frequency, preferably equal to the sum of the third qubit frequency and an interaction strength of the second two-qubit interaction.

6. The quantum processing unit according to any one of claims 1 to 5, further comprising: two initialization columns at a first side of the rows; and an initialization source configured to drive a first initialization column, wherein when dependent on claim 2, the first initialization column is a column of third qubits, and the initialization source is the third source, and when not dependent on claim 2, the first initialization column is a column of second qubits, wherein optionally when dependent on claim 2, the qubit frequency of the third qubits of the first initialization column is different from the third qubit frequency, preferably equal to the difference between the third qubit frequency and an interaction strength of the second two-qubit interaction, and when not dependent on claim 2, the qubit frequency of the second qubits of the first initialization column is different from the second qubit frequency, preferably equal to the difference between the second qubit frequency and an interaction strength of the two-qubit interaction.

7. The quantum processing unit according to any one of claims 5 to 6, further comprising: two readout columns at a second side of the rows different to the first side, wherein a second readout column is a column of second qubits, and the qubit frequency of thesecond qubits of the second read-out column is different from the second qubit frequency, preferably equal to the difference between the second qubit frequency and an interaction strength of the two-qubit interaction.

8. The quantum processing unit according to any one of claims 1 to 7, 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 2, each third qubit and special third qubit are connected to the third source by a third wiring; and / or a set of logical qubits is encoded in a column of first qubits, or in a column of second qubits, or, when dependent on claim 2, in a column of third qubits, wherein optionally states of the first qubits, the second qubits and the second special qubits, and, when dependent on claim 2, the third qubits and the third special qubits on one side of the column encoding the set of logical qubits are in a paramagnetic phase and states of the first qubits, the second qubits and second special qubits, and, when dependent on claim 2, the third qubits and the third special qubits on the other side of the column encoding the set of logical qubits are in a ferromagnetic phase.

9. The quantum processing unit according to any one of claims 1 to 8, wherein the first source and the second source, and, when dependent on claim 2, the third source are configured to move a quantum state from one columnwherein optionally the first source and the second source, and, when dependent on claim 2, the third source are configured to move the quantum state of the column encoding the set of logical qubits, while maintaining the paramagnetic phase and the ferromagnetic phase on the two sides of the column encoding the set of logical qubits.

10. The quantum processing unit according to any one of claims 1 to 9, wherein the first source and the second source are configured to perform a single-qubit gate on a second special qubit of a column of second qubits, and when dependent on claim 2, the first source and the third source are configured to perform a single-qubit gate on a third special qubit of a column of third qubits; and / or the first source and the second source are configured to perform a two-qubit gate on the two second qubits adjacent to the first special qubit of the column, and when dependent on claim 2, the first source and the third source are configured to perform a two-qubit gate on two third qubits adjacent to the first special qubit of the column; and / or an interaction strength of the first two-qubit interaction is different from the first Rabi frequency, and different from the second Rabi frequency, and when dependent on claim 2, different from the third Rabi frequency;and / or the quantum processing unit further comprises a plurality of quantum actuators, wherein one quantum actuator is connected to each first qubit, each first special qubit, each second qubit, and each second special qubit, and, when dependent on claim 2, each third qubit and each third special qubit, respectively, via the two-qubit interaction; and / or the first qubits, including the first special qubits, and the second qubits, including the second special qubits, and, when dependent on claim 2, the third qubits and the third special qubits, are superconducting qubits.

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