Contextuality as a resource for measurement-based quantum computation beyond qubits

Computers that exploit quantum phenomena are believed to be more powerful than those obeying classical rules. Many features of quantum theory have been proposed as the origin of this supposed quantum computational power: entanglement, superposition, and exponential scaling of Hilbert spaces to name a few. Recently, contextuality has been investigated in a variety of scenarios as a potential resource for quantum computation [1–14]. Contextuality can be thought of as the impossibility of assigning pre-determined outcomes to all potential measurements of a quantum system, independent of their measurement context [15–18]. This fundamental but peculiar property of quantum systems can exhibit in many ways, most famously in allowing quantum systems to circumvent constraints on classical correlations, leading to strong non-locality (to be defined shortly).

While the majority of recent research into contextuality as a resource for quantum computation sits in the framework of the circuit model, perhaps the most striking results in this direction arise in the measurement-based model of quantum computation (MBQC) [19]. Anders and Browne [20] showed that a simple control computer limited to evaluating linear boolean functions can be boosted in power, to one that can evaluate general (nonlinear) functions, when given the measurement outcomes on a resource state that constitutes a proof of contextuality. Their key example was Mermin's simplified GHZ paradox [16] (a common proof of contextuality), where linear control of the local measurement settings allows for the evaluation of a nonlinear NAND gate. Subsequently, Raussendorf [1] extended these results, proving that the computation of a nonlinear function (from measurement outcomes with linear pre- and post-processing) implies the impossibility of non-contextual assignments to the single qubit observables.

These previous results are quite strongly dependent on qubits (two-dimensional quantum systems) being the elementary measured systems. While qubits are standard in quantum computation, they are unusual from the perspective of contextuality. For example, an individual qubit is unique as a quantum system that cannot be used to prove Kochen–Specker contextuality [15, 17]. On the other hand, Pauli observables on multi-qubit systems (such as in Mermin's simplified Peres and GHZ paradoxes [16]) allow for state-independent proofs of contextuality, whereas the natural generalisation of these thought experiments to higher-dimensional systems are non-contextual [21, 22]. As highlighted in Anders and Browne, deriving a generalisation of their result for measurements of higher-dimensional systems is not straightforward. In particular, using qudits (higher-dimensional quantum systems) has the potential to confuse the role of contextuality within an individual system with the exhibition of contextuality as a non-local (in Bell's sense) correlation between quantum systems. In addition, the Mermin-style proof of strong contextuality which serves as the key nonlinear example in the qubit case does not straightforwardly generalise to qudits (with generalised Pauli observables). The key prior research in this direction is by Hoban et al [23], where the qudit case was considered from the perspective of Bell inequalities for multi-qudit systems. It was shown that evaluation of a sufficiently high order polynomial function on a multi-qudit system constituted a proof of strong contextuality.

In this paper, we characterise the role of contextuality and non-locality within a general framework of both qubit and qudit MBQC. We give examples of non-contextual qudit MBQCs (with local dimension d ≥ 3) that evaluate nonlinear functions, a result that sits in stark contrast to the qubit case; these examples show that a naive generalisation of Raussendorf's result is not possible. We then consider the space of functions that can be evaluated using MBQCs that admit pre-determined local value assignments to each measured observable. For a polynomial function (in several variables) the existence of a value assignment to local observables places a restriction on its combined degree. In particular, we prove that the evaluation of a polynomial of sufficiently high degree gives a proof of strong contextuality, reproducing the result of [23], in a way that emphasises the distinctive role of contextuality in local versus global systems and correlations. A key ingredient in the argument is the notion of local universality, which captures both the computational power of MBQCs that admit local value assignments, and exposes the importance of non-locality—that grouping qubits or qudits together to form larger degrees of freedom can lead to additional computational power. While our results are general (and can apply to post-quantum theories more broadly), we give particular emphasis to the stabilizer-based MBQCs that are most commonly considered in the quantum computing literature.

The paper is organised as follows. In section 2 we present a general framework for measurement-based computations (MBC) following Anders and Browne [20], and focus on the important case of MBQC on both qubits and qudits. In section 3, we highlight some of the problems in generalising the qubit-based results. For instance, we give explicit examples that show how nonlinear functions can be computed even in non-contextual qudit MBQC. In section 4, we review some results on functions in finite fields, and derive a description of subspaces in the space of polynomial functions invariant under linear pre- and post-processing. We also introduce the crucial notion of local universality, and prove that MBQC within the stabilizer formalism is itself locally universal, i.e., it allows to implement arbitrary functions on individual qudits. The classification of function spaces together with local universality eventually leads to a generalisation of Raussendorf's theorem from qubits to qudits, given in section 5.

We start by introducing the notion of computation we consider, called ld-MBC for 'MBC with ${{\mathbb{Z}}}_{d}$-linear classical processing'. This notion fits within the computational framework first introduced in Anders and Browne [20] to study the computational power of correlated resources in a general setting that includes MBQC. For MBQC it is natural to choose the input and output alphabet to be a cyclic group and we restrict our definition of ld-MBC in this way. While we are primarily concerned with generalising the results of Raussendorf [1] to qudits of prime power dimension, many of our results are not restricted to this particular implementation and hold more generally for any resource with non-local correlations. As such we briefly review the Anders and Browne setup before discussing the special case of MBQC.

A general MBC consists of two components: a correlated resource, and a control computer with restricted computational power. The correlated resource consists of N local parties, each of which is allowed to exchange classical information with the control computer once. No communication between parties is allowed during the computation, and the correlations in their output are entirely due to interactions prior to the computation. During the exchange with the control computer, each party receives an element of ${{\mathbb{Z}}}_{k}$ from the control computer (called the measurement setting), and returns an outcome that is an element of ${{\mathbb{Z}}}_{l}$ (called the measurement outcome). The control computer combines the measurement outcomes in a linear way to produce the computational output.

We are interested in the case where inputs and output are of fixed dimension d, such that d = k = l = p^r for p prime, $r\in {\mathbb{N}}$. In this case we refer to the system as an ld-MBC, which is defined more precisely as follows.

Definition 1. A ld-MBC with classical input ${\bf{i}}$ and classical output $o({\boldsymbol{i}})$ consists of N parties, each of which receives an input ${q}_{k}\in {{\mathbb{Z}}}_{d}$ from the control computer, and returns an outcome ${s}_{k}\in {{\mathbb{Z}}}_{d}$, for $k=1,\,\ldots N$. The inputs and computational output satisfy the following conditions:

• 1.  
  The computational output $o({\bf{i}})\in {{\mathbb{Z}}}_{d}$ is a linear function of the local measurement outcomes ${\bf{s}}={({s}_{1},\cdots ,{s}_{N})}^{\top }$,

  $$\begin{eqnarray}&&o({\bf{i}})=Z{\bf{s}}+{s}_{0}\quad \mathrm{mod}\ d,\end{eqnarray} \tag{ 1 }$$

  for ${s}_{0}\in {{\mathbb{Z}}}_{d}$ and $Z\in {\mathrm{Mat}}_{n}({{\mathbb{Z}}}_{d})$.
• 2.  
  The choice of measurements ${\bf{q}}={({q}_{1},\cdots ,{q}_{N})}^{\top }$ is related to the measurement outcomes $s$ and the ${{\mathbb{Z}}}_{d}$-valued classical input ${\bf{i}}={({i}_{1},\cdots ,{i}_{n})}^{\top }$ via

  $$\begin{eqnarray}&&{\bf{q}}=T{\bf{s}}+Q{\bf{i}}\quad \mathrm{mod}\ d,\end{eqnarray} \tag{ 2 }$$

  for some $T,Q\in {\mathrm{Mat}}_{n}({{\mathbb{Z}}}_{d})$.
• 3.  
  For a suitable ordering of the parties $1,\,\cdots ,\,n$ the matrix $T$ in equation (2) is lower triangular with vanishing diagonal. If $T=0$ the ${ld}$-MBC is called temporally flat.

In the above, ${\mathrm{Mat}}_{n}({{\mathbb{Z}}}_{d})$ denotes the space of n × n matrices with entries in ${{\mathbb{Z}}}_{d}$. We will mostly be concerned with a special implementation of this setup known as ld-MBQC, where the correlated resource is given by a quantum state, and the exchanged information with each party are local measurement settings and outcomes. In particular, a general ld-MBQC consists of two components: a correlated quantum resource, and a control computer with restricted computational power. The quantum resource consists of N local parties, each of which contains a quantum system of dimension d (i.e., a qudit) and a measurement device. For each party there is a choice of d measurement settings, each with d measurement outcomes.

Each party exchanges data with the control computer once. Namely, each party receives a measurement setting ${q}_{k}\in {{\mathbb{Z}}}_{d}$ from the control computer to determine the choice of measurement M_k(q_k), and returns the measurement outcome ${m}_{k}({q}_{k})\in {{\mathbb{Z}}}_{d}$, where here and thereafter k ∈ {1, ..., N} labels the party. We assume that the eigenvalues of each M_k(q_k) are of the form ω^z for $\omega ={{\rm{e}}}^{\tfrac{2\pi {\rm{i}}}{d}}$ a dth root of unity and $z\in {{\mathbb{Z}}}_{d}$. Note that such operators are not Hermitian, but we use the terminology 'measurement of M_k' to denote a projective measurement in the eigenbasis of M_k, where we associate the measurement outcome ${m}_{k}({q}_{k})\in {{\mathbb{Z}}}_{d}$ with the eigenvalue ${\omega }^{{m}_{k}({q}_{k})}$ ³ .

The control computer is tasked with evaluating a function $o\,:{{\mathbb{Z}}}_{d}^{n}\to {{\mathbb{Z}}}_{d}$ on some input ${\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}$. It is responsible for the side processing of classical data that determines both the measurement settings and the overall computational output from the measurement outcomes. It has very restricted computational power in that it is only capable of performing processing that is ${{\mathbb{Z}}}_{d}$-linear. The input q_k to each party is determined in a ${{\mathbb{Z}}}_{d}$-linear way from the input ${\bf{i}}$ only. (This type of computation is called temporally flat, in section 5.3 we consider the case where the input is also determined by previous measurement outcomes.)

In particular, for each input ${\bf{i}}$, the control computer evaluates N linear functions ${f}_{k}\,:{{\mathbb{Z}}}_{d}^{n}\longrightarrow {{\mathbb{Z}}}_{d}$ on the input string i, which determine the local measurement settings ${q}_{k}={f}_{k}({\bf{i}})$ for each party. The control computer evaluates the output function o(i) by adding the measurement outcomes m_k(q_k) modulo d. We can say that the quantum resource state is a genuine resource, in that it increases the computational power of the control computer, if this ld-MBQC scheme allows for the evaluation of functions that are not ${{\mathbb{Z}}}_{d}$-linear.

In order to ensure we are exposing only the computational power of the resource we require measurements to be unitarily related,

$$\begin{eqnarray}&&{M}_{k}({q}_{k})={U}_{k}^{{q}_{k}}{M}_{k}(0){U}_{k}^{-{q}_{k}},\end{eqnarray} \tag{ 3 }$$

where the ${U}_{k}({q}_{k})$ form a projective representation of ${{\mathbb{Z}}}_{d}$ for some fiducial measurement choice M_k(0). In doing so, we have removed the possibility of nonlinear functions being introduced through the choice of measurement setting alone, thereby incidentally increasing the power of the control computer separately from the quantum resource state (see appendix A in [7]).

In practice, we will often restrict unitaries to the Clifford group, a projective representation of the group ${{\mathbb{Z}}}_{d}^{2N}\rtimes \mathrm{Sp}({{\mathbb{Z}}}_{d}^{2N})$ (see section 3.1). The fiducial measurements M_k(0) are required to have a spectrum given by the dth roots of unity as described above. However, we do not require the M_k(0)'s for different k to be generalised Pauli operators in the same Pauli frame, and as such our setup is sufficient to allow for universal quantum computation.

In summary, we have the following definition of a temporally flat ld-MBQC (see figure 1).

Definition 2. A temporally flat ${ld}$-MBQC with input string ${\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}$ and output $o({\bf{i}})\in {{\mathbb{Z}}}_{d}$, consists of the following components:

• 1.  
  an $N$ qudit system each of local dimension $d$ where the overall resource state is represented by $| \psi \rangle \in {({{\mathbb{C}}}_{d})}^{\otimes N};$
• 2.  
  a set of measurement settings ${q}_{k}={f}_{k}({\bf{i}})$ for some ${{\mathbb{Z}}}_{d}$-linear functions ${f}_{k}\,:{{\mathbb{Z}}}_{d}^{n}\to {{\mathbb{Z}}}_{d}$, independent of previous measurement outcomes;
• 3.  
  a set of measurements ${M}_{k}$ on each qubit satisfying equation (3), each with $d$ possible eigenvalues ${\omega }^{{m}_{k}}$, where ${m}_{k}\in {{\mathbb{Z}}}_{d}$ is the measurement outcome;
• 4.  
  the computational output is a linear function of the measurement outcomes ${\bf{m}}=\{{m}_{1},\cdots ,{m}_{N}\}\in {{\mathbb{Z}}}_{d}^{N}$,

  $$\begin{eqnarray}&&o({\bf{i}})=Z{\bf{m}}\quad \mathrm{mod}d,\end{eqnarray} \tag{ 4 }$$

  for some $Z\in {\mathrm{Mat}}_{n}({{\mathbb{Z}}}_{d})$.

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We remark that with a suitably chosen resource state, such as (qudit) cluster states, this model is universal for quantum computation [19, 24].

2.1. Structure of ld-MBQCs

We first consider the case of deterministic computations for which the output function has a fixed value for every input i, independent of the local measurement outcomes (which individually might change at different runs), and where the output is the mod-d sum of the measurement outcomes. In this case, the product of all local measurements stabilizes the resource state $| \psi \rangle$, and we can describe the particular output function,

$$\begin{eqnarray*}&&o({\bf{i}})=\displaystyle \sum _{k=1}^{N}{m}_{k},\end{eqnarray*}$$

concisely through a set of eigenvalue equations,

$$\begin{eqnarray}&&\underset{k=1}{\overset{N}{\displaystyle \bigotimes }}{U}_{k}^{{f}_{k}({\bf{i}})}{M}_{k}(0){U}_{k}^{-{f}_{k}({\bf{i}})}| \psi \rangle ={\omega }^{o({\bf{i}})}| \psi \rangle ,\end{eqnarray} \tag{ 5 }$$

for each ${\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}$. For a given input ${\bf{i}}$, we define the 'global observable' $M({\bf{i}})$ as the tensor product of the local measurements. The global observables are important as their eigenvalues encode the computational output according to equation (5). We will restrict our attention to deterministic ld-MBQCs throughout the paper for simplicity, however our main results generalise to the probabilistic case as discussed in appendix C.

2.1.1. Non-contextual and local value assignments

We first define the notion of contextuality we consider, called strong contextuality, before making the connection with ld-MBQCs. We note that we only present the definition of strong contextuality as it pertains to the quantum case, for the definition that applies to general no-signalling models see [25]. Denote the set of observables by ${ \mathcal O }$. A measurement context ${ \mathcal C }\subseteq { \mathcal O }$ consists of a set of mutually commuting observables, and we denote the set of all contexts by ${ \mathcal M }$. A non-contextual value assignment (NCVA) is a function $s:{ \mathcal O }\to {\mathbb{R}}$ with the following properties

• (i)  
  $s(A)\in \mathrm{sp}(A)\,\forall A\in { \mathcal O }$ (that is, s(A) is an eigenvalue of A),
• (ii)  
  $\forall { \mathcal C }\in { \mathcal M },s(A)s(B)=s({AB})$ $\forall A,B,{AB}\in { \mathcal C }$.

The NCVAs s(A) should be thought of as assigning an outcome that is revealed upon measurement of A. Importantly, in the setting of MBQC, the value assignments must be compatible with the resource state in the following sense: we only consider NCVAs that assign a set of outcomes that can jointly occur with nonzero probability upon measurement of the resource state. If no such globally consistent NCVAs exist, then the system is strongly contextual.

In the special case of product observables $A={A}_{1}\otimes {A}_{2},B={B}_{1}\otimes {B}_{2},{AB}={A}_{1}{B}_{1}\otimes {A}_{2}{B}_{2}$ the locally measurable observables A₁ and B₁ (A₂ and B₂) compose individually, hence local value assignments must too (this is a special case of noncontextuality). When no such local assignment exists, we say that the system is strongly non-local.

The connection with ld-MBQCs is as follows. Each input ${\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}$ can be regarded as selecting a context ${ \mathcal C }({\bf{i}})$ (that is, a set of commuting observables) through

$$\begin{eqnarray}&&{ \mathcal C }({\bf{i}})=\{{M}_{1}({q}_{1}={f}_{1}({\bf{i}})),\,\ldots ,\,{M}_{N}({q}_{N}={f}_{N}({\bf{i}})),M({\bf{i}})\}.\end{eqnarray} \tag{ 6 }$$

We have included the global observable $M({\bf{i}})={M}_{1}({q}_{1})\otimes {M}_{2}({q}_{2})\,\otimes \cdots \otimes {M}_{N}({q}_{N})$ in each context as its measurement outcome is fixed in the deterministic case, and corresponds to the computational output $o({\bf{i}})$ (that is inferred from outcomes of the local measurements). The task of finding a non-contextual hidden variable model is to find (perhaps many) value assignments to local observables that are consistent with the global value assignment. Since the global value assignment is fixed by the computational output, in general, some computations may result in an obstruction to finding a non-contextual hidden variable model in our setting.

2.2. Anders and Browne qubit example

Here we review the instructive example, due to Anders and Browne [20], which shows that (at least in the qubit d = 2 case) evaluating nonlinear functions deterministically with l2-MBQC is possible with an appropriate quantum resource state.

Following Anders and Browne, we consider an l2-MBQC with a three-qubit GHZ state, $| {\psi }_{\mathrm{GHZ}}\rangle =(| 001\rangle -| 110\rangle )/\sqrt{2}$, on which local measurements of Pauli observables X or Y on each qubit enable the deterministic computation of the nonlinear NAND gate. The control computer receives two bits ${\bf{i}}=({i}_{1},{i}_{2})\in {{\mathbb{Z}}}_{2}^{2}$ as input. The classical pre-processing to determine the measurement settings on each qubit amounts to evaluating the linear functions ${f}_{1}({\bf{i}})={i}_{1},{f}_{2}({\bf{i}})={i}_{2}$ and ${f}_{3}({\bf{i}})={i}_{1}\oplus {i}_{2}$. The bits ${q}_{k}={f}_{k}({\bf{i}})$ determine the measurement setting on each qubit according to M_k(0) = X and M_k(1) = Y, for k ∈ {1, 2, 3}. These measurement settings are related by the unitary transformation $U=\tfrac{1}{\sqrt{2}}(X+Y)$. If the eigenvalue +1 (−1) is observed, then the outcome m_k(q_k) = 0 (m_k(q_k) = 1) is recorded.

These measurement settings define the global observables,

$$\begin{eqnarray}&&M(0,0)=X\otimes X\otimes X,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&M(1,0)=Y\otimes X\otimes Y,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&M(0,1)=X\otimes Y\otimes Y,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\,M(1,1)=Y\otimes Y\otimes X,\end{eqnarray} \tag{ 10 }$$

with the state $| {\psi }_{\mathrm{GHZ}}\rangle$ an eigenvector of each observable, and with corresponding eigenvalues given by

$$\begin{eqnarray}&&{(-1)}^{o({i}_{1},{i}_{2})}={(-1)}^{{i}_{1}{i}_{2}+1}={(-1)}^{\mathrm{NAND}({i}_{1},{i}_{2})}.\end{eqnarray} \tag{ 11 }$$

In other words, the measurement outcomes of the local X and Y measurements can be linearly processed to compute the output function $o({\bf{i}})={\sum }_{k=1}^{3}{m}_{k}({\bf{i}})$, which by equation (11) gives $o({i}_{1},{i}_{2})=\mathrm{NAND}({i}_{1},{i}_{2})$. Thus, the l2-MBQC evaluates a nonlinear function of the input that could not be evaluated by the control computer alone.

2.3. Central questions

In this section, we illustrate some of the subtleties involved in the qudit case. We focus on a particularly interesting class of ld-MBQCs based on the qudit stabilizer formalism. Unlike the qubit case, such ld-MBQCs are non-contextual. (An explicit non-contextual hidden variable model for the qubit stabilizer theory is given by the discrete Wigner function [21, 22]. There has been a considerable amount of recent research investigating the differences between the qubit and qudit stabilizer subtheories from the perspective of contextuality; see for example [3–6, 10, 21, 26, 27].)

Contrary to what one might naively expect, we will see that qudit stabilizer ld-MBQCs possess a computational power that exceeds ${{\mathbb{Z}}}_{d}$-linear processing. That is, nonlinear functions can be evaluated using an MBQC that is entirely non-contextual, in stark contrast to the qubit case. This demonstrates that the relationship between strong contextuality and nonlinearity in the qubit case is not the end of the story, and for qudits with d ≥ 3 we need a finer functional constraint.

In this section we will be restricting to the case where the measurements M_k belong to the qudit Pauli group, and the unitaries U_k that relate these measurements as in equation (3) belong to the Clifford group. Important to our considerations is that when d is odd, this subtheory admits a non-contextual description [21]. We outline a very restricted case under which linearity in the output can be recovered, along with two examples within this non-contextual framework that result in nonlinear output functions.

3.1. Symplectic structure of qudit stabilizer formalism

Our results will make extensive use of the symplectic structure of the qudit Clifford group, and so we briefly review this formalism here. For more details, see [28].

Recall that the Pauli group ${{ \mathcal P }}_{d}^{\otimes N}$ over ${{\mathbb{Z}}}_{d}$ is the group generated by N-fold tensor products of individual elements from $\langle {X}_{k},{Z}_{k},\omega {{\mathbb{1}}}_{k}\rangle ,k\in \{1,\,\cdots ,\,N\}$ with $X| z\rangle =| z+1\rangle ,Z| z\rangle ={\omega }^{z}| z\rangle$, and $\omega ={{\rm{e}}}^{\tfrac{2\pi {\rm{i}}}{d}}$ a dth root of unity.

These qudit Pauli operators can be conveniently represented (up to phase) by Weyl operators. A Weyl operator W_v for ${\bf{v}}={({\bf{a}},{\bf{b}})}^{\top }\in {{\mathbb{Z}}}_{d}^{2N}$ is defined as the N-fold tensor product of local operators ${W}_{a,b}={\tau }^{-{ab}}{Z}^{a}{X}^{b}$ where ${\tau }^{2}=\omega$ and $a,b\in {{\mathbb{Z}}}_{d}$. Note that Weyl operators are generalised Pauli operators with a particular choice of phase.

The Clifford group ${{ \mathcal C }}_{N}(d)\subset { \mathcal U }({{ \mathcal H }}_{d}^{\otimes N})$ of ${{ \mathcal P }}_{d}^{\otimes N}$ is the group of unitary operators such that ${{VPV}}^{\dagger }\in {{ \mathcal P }}_{d}^{\otimes N}$ for all $P\in {{ \mathcal P }}_{d}^{\otimes N},V\in {{ \mathcal C }}_{N}(d)$. All (N-qudit) Clifford operators $V\in {{ \mathcal C }}_{N}(d)$ factorise,

$$\begin{eqnarray*}&&V={{UW}}_{{\bf{x}}},\quad {\bf{x}}\in {{\mathbb{Z}}}_{d}^{2N},\end{eqnarray*}$$

into a Weyl operator W_x and an element of the group of symplectic Clifford operators $U\in \sigma {{ \mathcal C }}_{N}(d)$. These are defined as automorphisms on the set of Weyl operators, i.e. for all ${\bf{v}}\in {{\mathbb{Z}}}_{d}^{2N}$ it holds that ${{UW}}_{{\bf{v}}}{U}^{\dagger }={W}_{{\bf{w}}}$ for some ${\bf{w}}\in {{\mathbb{Z}}}_{d}^{2N}$, in fact, they preserve the underlying symplectic structure,

$$\begin{eqnarray*}&&{{UW}}_{{\bf{v}}}{U}^{-1}={W}_{{C}_{U}{\bf{v}}},\mathrm{for}\ \mathrm{some}\ {C}_{U}\in {\mathrm{Sp}}_{2N}({{\mathbb{Z}}}_{d}).\end{eqnarray*}$$

The group ${\mathrm{Sp}}_{2N}({{\mathbb{Z}}}_{d})$ denotes the group of symplectic transformations, i.e. of linear transformations $C:{{\mathbb{Z}}}_{d}^{2N}\longrightarrow {{\mathbb{Z}}}_{d}^{2N}$ such that ${C}^{\top }{\sigma }_{2N}C={\sigma }_{2N}$ where the symplectic matrix is given as ${\sigma }_{2N}=\left[\begin{array}{cc}{0}_{N} & {{\mathbb{1}}}_{N}\\ -{{\mathbb{1}}}_{N} & {0}_{N}\end{array}\right]$. Moreover, the symplectic inner product is given by $[{\bf{v}},{\bf{w}}]={{\bf{v}}}^{\top }{\sigma }_{2n}{\bf{w}}$, for ${\bf{v}},{\bf{w}}\in {{\mathbb{Z}}}_{d}^{2N}$.

3.2. Computational output and (non)linearity

Let us now relate the transformation properties of Pauli observables under Clifford operations to the computational output of the ld-MBQC, using the symplectic formalism. From the defining commutation relation of Weyl operators,

$$\begin{eqnarray}&&{W}_{{\bf{v}}}{W}_{{\bf{w}}}={\omega }^{[{\bf{v}},{\bf{w}}]}{W}_{{\bf{w}}}{W}_{{\bf{v}}},\end{eqnarray} \tag{ 12 }$$

and the fact that symplectic operators preserve the symplectic inner product, we obtain the following relation for individual qudits (for clarity we omit the subscript k labeling different qudit sites),

$$\begin{eqnarray}&&\begin{array}{l}{V}^{f({\bf{i}})}\ {W}_{{\bf{v}}}\ {V}^{-f({\bf{i}})}={({{UW}}_{{\bf{x}}})}^{f({\bf{i}})}\ {W}_{{\bf{v}}}\ {({{UW}}_{{\bf{x}}})}^{-f({\bf{i}})}\\ \quad =\,{({{UW}}_{{\bf{x}}})}^{f({\bf{i}})-1}\ {W}_{{C}_{U}{\bf{x}}}\ {W}_{{C}_{U}{\bf{v}}}\ {W}_{-{C}_{U}{\bf{x}}}\ {({W}_{-{\bf{x}}}{U}^{-1})}^{f({\bf{i}})-1}\\ \quad =\,({W}_{{C}_{U}{\bf{x}}}\ \cdots \ {W}_{{C}_{U}^{f({\bf{i}})}{\bf{x}}})\ {W}_{{C}_{U}^{f({\bf{i}})}{\bf{v}}}\ ({W}_{-{C}_{U}^{f({\bf{i}})}{\bf{x}}}\cdots \ {W}_{-{C}_{U}{\bf{x}}})\\ \quad =\,{\omega }^{[{C}_{U}^{}{\bf{x}},{C}_{U}^{f({\bf{i}})}{\bf{v}}]+[{C}_{U}^{2}{\bf{x}},{C}_{U}^{f({\bf{i}})}{\bf{v}}]+\cdots +[{C}_{U}^{f({\bf{i}})}{\bf{x}},{C}_{U}^{f({\bf{i}})}{\bf{v}}]}{W}_{{C}_{U}^{f({\bf{i}})}{\bf{v}}}\\ \quad =\,{\omega }^{\displaystyle \sum _{k=0}^{f({\bf{i}})-1}[{\bf{x}},{C}_{U}^{k}{\bf{v}}]}{W}_{{C}_{U}^{f({\bf{i}})}{\bf{v}}},\end{array}\end{eqnarray} \tag{ 13 }$$

for any Clifford unitary $V\in {{ \mathcal C }}_{N}(d)$. Note that the phase in equation (13) is state-independent, it only depends on the Weyl commutation relations.

3.2.1. Example 1: linear output

As a first example, we examine the very restrictive case, where the controlled unitary operators in definition 2 are Pauli operators. Using only controlled Pauli operators in equation (13), i.e., $V={W}_{{\bf{x}}},{\mathbb{1}}=U\in \sigma {{ \mathcal C }}_{1}(d)$, the phase depends linearly on the input functions f(i). In fact, we simply obtain a variant of equation (12),

$$\begin{eqnarray}&&{W}_{{\bf{x}}}^{f({\bf{i}})}{W}_{{\bf{v}}}{W}_{{\bf{x}}}^{-f({\bf{i}})}={\omega }^{f({\bf{i}})[{\bf{x}},{\bf{v}}]}{W}_{{\bf{v}}}.\end{eqnarray} \tag{ 14 }$$

From equation (14), we infer that conjugation of a Pauli operator by Pauli operators results in multiplication of a phase, yet does not change the context. That is $M({\bf{i}})\propto M({\bf{0}})$ and ${ \mathcal C }({\bf{i}})\propto { \mathcal C }({\bf{0}})$ for all inputs, meaning the output $o({\bf{i}})$ is linearly related to $o({\bf{0}})$. As a result, we are trivially restricted to be non-contextual.

3.2.2. Example 2: quadratic output

In the next two examples we remain within the stabilizer subtheory but extend to arbitrary Clifford unitaries. First, we show how using control unitaries that are symplectic Cliffords (i.e., Cliffords that are not Paulis) allows us to compute quadratic functions from the symplectic inner product. In the following, we assume d ≥ 3.

The generalised phase gate S is an element of the symplectic Clifford group,

$$\begin{eqnarray*}&&S=\displaystyle \sum _{z=0}^{d-1}{\tau }^{{z}^{2}}| z\rangle \langle z| \ \in \sigma {{ \mathcal C }}_{1}(d),\end{eqnarray*}$$

which up to phase acts on the generalised Pauli $X={W}_{{\bf{v}}},{\bf{v}}={(0,1)}^{\top }$ by multiplication with Pauli $Z,{S}^{k}{{XS}}^{-k}={\tau }^{-k}{Z}^{k}X$. Consider the following state of $N=2d$ qudits,

$$\begin{eqnarray*}&&| \psi \rangle =\displaystyle \frac{1}{\sqrt{d}}\displaystyle \sum _{z=0}^{d-1}{| z\rangle }^{\otimes 2}{| z+1\rangle }^{\otimes 2}\cdots {| z+d-1\rangle }^{\otimes 2}.\end{eqnarray*}$$

We fix all of the linear functions ${f}_{k}({\bf{i}})=f({\bf{i}})$ to be the same, and note that we have the following stabilizer relations,

$$\begin{eqnarray}&&\underset{k=1}{\overset{2d}{\displaystyle \bigotimes }}{({S}^{f({\bf{i}})}{W}_{{\bf{v}}}{S}^{-f({\bf{i}})})}_{k}| \psi \rangle =| \psi \rangle ,\quad {\bf{v}}={(0,1)}^{\top },\end{eqnarray} \tag{ 15 }$$

where the parentheses ${(\cdot )}_{k}$ denote the subsystem on which the operator acts.

We can use this setup together with the symplectic structure of Weyl operators to implement quadratic output functions through accumulated symplectic products. Without loss of generality, we choose the first qudit and take ${V}_{1}={({{SW}}_{{\bf{x}}})}_{1}$ for ${\bf{x}}={(0,-1)}^{\top }$ in equation (15) such that $[{\bf{x}},{C}_{S}^{k}{\bf{v}}]=k$, while leaving V_k = S_k for $k\geqslant 2$. Hence,

$$\begin{eqnarray*}&&o({\bf{i}})=\displaystyle \sum _{k=0}^{f({\bf{i}})-1}[{\bf{x}},{C}_{S}^{k}{\bf{v}}]=\displaystyle \frac{f({\bf{i}})(f({\bf{i}})-1)}{2}.\end{eqnarray*}$$

Despite C_S being a linear map, it leads to quadratic output functions due to the symplectic structure of the Weyl group.

3.2.3. Example 3: general nonlinear output

With our final example, we show that one can even go beyond quadratic functions. Another symplectic Clifford operator is given by,

$$\begin{eqnarray}&&{M}_{u}:= \displaystyle \sum _{k=0}^{d-1}| {uk}\rangle \langle k| ,\quad u\in {{\mathbb{Z}}}_{d}^{\times },\end{eqnarray} \tag{ 16 }$$

where R^×denotes the multiplicative group of units within a ring R.

Now let $| \psi \rangle =| 1\rangle \in {{\mathbb{C}}}_{d},{\bf{v}}={(1,0)}^{\top }\in {{\mathbb{Z}}}_{d}^{2}$ and consider the symplectic Clifford unitary $U={M}_{u}^{f({\bf{i}})}\in \sigma {{ \mathcal C }}_{1}(d)$ for some $u\in {{\mathbb{Z}}}_{d}^{\times }$ acting on $Z={W}_{{\bf{v}}}$,

$$\begin{eqnarray}&&{{UW}}_{{\bf{v}}}{U}^{-1}| \psi \rangle ={M}_{u}^{f({\bf{i}})}{W}_{{\bf{v}}}{M}_{u}^{-f({\bf{i}})}| \psi \rangle ={W}_{{C}_{{M}_{u}}^{f({\bf{i}})}{\bf{v}}}| \psi \rangle ={\omega }^{{u}^{-f({\bf{i}})}}| \psi \rangle .\end{eqnarray} \tag{ 17 }$$

The output function $o({\bf{i}})={u}^{-f({\bf{i}})}$ is again nonlinear, however, the underlying system is part of the qudit stabilizer formalism and can be given a non-contextual hidden variable model with local value assignments m_k(i) such that $o({\bf{i}})={\sum }_{k=1}^{N}{m}_{k}({\bf{i}})$ [21, 28].

Examples 2 and 3 are therefore in clear contrast to Raussendorf's result in the qubit case, as they allow for the evaluation of nonlinear functions with MBQCs that are non-contextual. It turns out that linearity is a special property of boolean functions which does not translate to the qudit case.

Given these conceptual differences between the qubit and qudit cases as illustrated by the examples in the previous section, it will be helpful to take a more abstract view on the problem. We begin by reviewing some properties of functions on finite fields.

4.1. Functions on finite fields

Let ${{\mathbb{F}}}_{d}$ be the finite field with d = p^r (p prime and $r\in {\mathbb{N}}$) elements and ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}:= {{\mathbb{F}}}_{d}[{x}_{1},\,\ldots ,\,{x}_{n}]$ the polynomial ring in n variables, ${x}_{1},\,\ldots ,\,{x}_{n}\in {{\mathbb{F}}}_{d}$.

For infinite fields there are many non-polynomial functions, but not so for finite fields.

Theorem (Functions on finite fields). Let ${{\mathbb{F}}}_{d}$ be the finite field with $d$ elements, and $n\in {\mathbb{N}}$. Then every function $g\,:{{\mathbb{F}}}_{d}^{n}\longrightarrow {{\mathbb{F}}}_{d}$ is given by a polynomial, $g\in {{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}$ of partial degree less or equal to $d-1$ in each variable ${x}_{i}$.

Proof. See appendix A. □

Here, the partial degree of a variable within a product is the exponent of that variable.

For qubits, Raussendorf proved that computability of a nonlinear function using an l2-MBC implies strong contextuality of the underlying system [1]. We note that this result depends critically on the application of the above theorem to d = 2, where it states that all functions $g\,:{{\mathbb{Z}}}_{2}\to {{\mathbb{Z}}}_{2}$ are linear. In a non-contextual hidden variable model, the measurement outcomes m_k are determined by a local value assignment that can depend on the choice of measurement setting q_k = f_k(i); these local value assignments m_k(i) are necessarily linear by the above theorem. In the same way, the final output function o(i) on an l2-MBQC on N qubits is a linear combination of the measurement outcomes at each site,

$$\begin{eqnarray}&&o({\bf{i}})=\displaystyle \sum _{k=1}^{N}{m}_{k}({\bf{i}}).\end{eqnarray} \tag{ 18 }$$

If o(i) is nonlinear, then the assumption of non-contextuality must be incorrect. Nonlinearity in this case can only arise from products of at least two different inputs, as in Anders and Browne's NAND gate illustrated in section 2.2.

In generalising to the qudit case, however, a local value assignment is not required to be a linear function of the choice of measurement. Here, this simple connection between non-contextuality and linearity is lost. Nevertheless, we now show how to use the above theorem for functions on finite fields to build a new, general connection.

4.2. Linearly closed subspaces

We would like to characterise the space of all functions of the form $g\,:{{\mathbb{F}}}_{d}^{n}\longrightarrow {{\mathbb{F}}}_{d}$ in terms of non-trivial subspaces under linear pre- and post-processing, which we denote by ⊂_l. First, define the space of linear functions as

$$\begin{eqnarray*}&&{L}_{n}^{{{\mathbb{F}}}_{d}}:= \{l\in {{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}| l(x)={c}_{0}+\displaystyle \sum _{j=1}^{n}{c}_{j}{x}_{j},x\in {{\mathbb{F}}}_{d}^{n},{c}_{j}\in {{\mathbb{F}}}_{d}\},\end{eqnarray*}$$

where x = (x₁, ..., x_n). Then a schematic that represents the linear pre- and post-processing that we consider is given in figure 2. Every measurement outcome ${m}_{k}\,:{{\mathbb{F}}}_{d}^{n}\to {{\mathbb{F}}}_{d}$ is a function from the inputs ${\bf{i}}\in {{\mathbb{F}}}_{d}^{n}$ to measurement outcomes that decomposes into a linear function ${f}_{k}\in {L}_{n}^{{{\mathbb{F}}}_{d}}$ and a contribution ${\phi }_{k}\,:{{\mathbb{F}}}_{d}\to {{\mathbb{F}}}_{d}$,

$$\begin{eqnarray}&&{m}_{k}={\phi }_{k}\,\circ \,{f}_{k},\quad \forall k\in \{1,\,\cdots ,\,N\}.\end{eqnarray} \tag{ 19 }$$

We refer to ${\phi }_{k}{\rm{s}}$ as correlation functions.

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Note first that the space of functions available after pre- and post-processing will at least contain the original function space, as the identity is a linear function. Consider the two trivial cases: the space of all functions ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}};$ and the space of linear functions ${L}_{n}^{{{\mathbb{F}}}_{d}}$. Any linear combination of linear functions results again in a linear function, hence, we find ${L}_{n}^{{{\mathbb{F}}}_{d}}{\subset }_{l}{{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}$.

Aside from these two trivial cases, there also exist non-trivial spaces that are stable under linear pre- and post-processing. Define the following subspaces for 1 ≤ δ ≤ n(d − 1), using the notation $\langle \cdot {\rangle }_{l}$ for linear span,

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta ):= {\left\langle \displaystyle \prod _{j=1}^{n}{x}_{j}^{{a}_{j}}| {a}_{j}\in {{\mathbb{F}}}_{d},\displaystyle \sum _{j=1}^{n}{a}_{j}\leqslant \delta \right\rangle }_{l}.\end{eqnarray} \tag{ 20 }$$

The function spaces ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )$ depend on the field ${{\mathbb{F}}}_{d}$, the number of inputs n and the maximal combined degree of monomials δ. (The combined degree within a product of variables is the sum of their respective exponents as in equation (20).) In other words, ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )$ contains all polynomials $g\,:{{\mathbb{F}}}_{d}^{n}\longrightarrow {{\mathbb{F}}}_{d}$ of degree at most δ.

We prove a lemma, detailing the behaviour of these subspaces under linear pre- and post-processing.

Lemma 1. Let $g\in {{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}$ be a polynomial of degree $| g| =\delta \leqslant n(d-1)$, then the set of all polynomials generated by linear pre- and post-processing coincides with ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )$, i.e.

$$\begin{eqnarray*}&&{L}_{1}^{{{\mathbb{F}}}_{d}}\,\circ \,g\,\circ \,{L}_{n}^{{{\mathbb{F}}}_{d}}={{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta ).\end{eqnarray*}$$

Proof. We focus on the space of functions generated by linear pre-processing first. We find that $g\circ {L}_{n}^{{{\mathbb{F}}}_{d}}{\subseteq }_{l}{{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )$ as evaluating monomials x^j on linear functions results in polynomials of degree at most j. As the definition of equation (20) already captures linear post-processing, we also have

$$\begin{eqnarray*}&&{L}_{1}^{{{\mathbb{F}}}_{d}}\,\circ \,g\,\circ \,{L}_{n}^{{{\mathbb{F}}}_{d}}{\subseteq }_{l}{L}_{1}^{{{\mathbb{F}}}_{d}}\,\circ \,{{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )={{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta ).\end{eqnarray*}$$

On the other hand we can always choose a linear function such that after evaluation on x^j the resulting polynomial and x^j are linearly independent (by producing other terms of less degree). Hence, taking linear combinations we can generate all polynomials of degree at most j,

$$\begin{eqnarray*}&&{L}_{1}^{{{\mathbb{F}}}_{d}}\,\circ \,g\,\circ \,{L}_{n}^{{{\mathbb{F}}}_{d}}{\supseteq }_{l}{{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta ).\end{eqnarray*}$$

□

We conclude that the subspaces closed under linear pre- and post-processing are exactly ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )$ for 1 ≤ δ ≤ n(d − 1), where ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(1)={L}_{n}^{{{\mathbb{F}}}_{d}}$.

4.3. Local universality

In the qubit case, ${{\mathbb{F}}}_{2}={{\mathbb{Z}}}_{2}$, every local output function ${m}_{k}\,:{{\mathbb{Z}}}_{2}^{n}\longrightarrow {{\mathbb{Z}}}_{2}$ is linear. Put another way, with any given function $\phi \in {{\rm{\Omega }}}_{1}^{{{\mathbb{F}}}_{d}}$ together with linear pre- and post-processing, we can realise arbitrary functions ${m}_{k}\,:{{\mathbb{Z}}}_{2}^{n}\longrightarrow {{\mathbb{Z}}}_{2}$. We say (somewhat trivially) that linear functions are universal for local qubit value assignments.

For qudits (d = p^r, p prime, $r\in {\mathbb{N}}$ and corresponding finite field ${{\mathbb{F}}}_{d}$), local measurement outcomes are still maps of the form, ${\phi }_{k}:{{\mathbb{F}}}_{d}\longrightarrow {{\mathbb{F}}}_{d}$. However, it is no longer true that all functions of this type are linear; any monomial of degree higher than one is clearly not.

Hence, whereas linear functions turned out to be universal (in the sense of being able to evaluate any function ${\phi }_{k}:{{\mathbb{Z}}}_{2}\longrightarrow {{\mathbb{Z}}}_{2}$), this is not the case for higher-dimensional systems where there can be more than two measurement outcomes. Therefore, we define a new property capturing this universality for both cases.

(Local universality).

Definition 3 A ld-MBC is called locally universal if it implements all functions $\phi \,:{{\mathbb{F}}}_{d}\longrightarrow {{\mathbb{F}}}_{d}\in {{\rm{\Omega }}}_{1}^{{{\mathbb{F}}}_{d}}$.

Note that the system size crucially enters this definition by means of the finite field ${{\mathbb{F}}}_{d}$, where d denotes the qudit dimension.

With this definition, we are now able to identify the generalised connection between computational power and contextuality for qudit systems. In a non-contextual hidden variable model, we again have local output functions ${m}_{k}\,:{{\mathbb{F}}}_{d}^{n}\longrightarrow {{\mathbb{F}}}_{d};$ these are not required to be linear in the qudit case. Nonetheless, as we now show, this does not allow for the evaluation of any nonlinear function. For qubits, nonlinear functions arose (by necessity) from crossterms, i.e., terms with combined degree greater than 1, such as the term ${i}_{1}{i}_{2}$ in the NAND gate in the example of section 2.2. For qudits, local universality allows us to implement some crossterms for finite fields with more than two elements; however, by lemma 1 we find that we can only implement a strict subset within the space of all polynomials,

$$\begin{eqnarray*}&&{{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )\ \subsetneq \ {{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}\quad \forall \delta \in \{1,\,\ldots ,\,d-1\}.\end{eqnarray*}$$

Local universality is a restriction on the space of (global) functions $o\,:{{\mathbb{F}}}_{d}^{n}\longrightarrow {{\mathbb{F}}}_{d}$, but a necessary requirement to maximise the computational power of correlations in ${\phi }_{k}:{{\mathbb{F}}}_{d}\longrightarrow {{\mathbb{F}}}_{d}$. In short, locally universal models are in the class ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(d-1)$ and,

$$\begin{eqnarray*}&&{{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(\delta )\ \subseteq \ {{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(d-1)\ \subsetneq \ {{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}},\quad \delta \in \{1,\,\ldots ,\,d-1\}.\end{eqnarray*}$$

The key observation is that for both qubits and qudits non-contextual models are at most locally universal, slightly more general we have:

Theorem 1. Consider a deterministic ${ld}$-MBC of prime power dimension $d={p}^{r},p$ prime, $r\in {\mathbb{N}}$. If the resource state is not strongly non-local then the output polynomial has maximal degree bounded by $d$, i.e.

$$\begin{eqnarray}&&o({\bf{i}})\in {{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(d-1).\end{eqnarray} \tag{ 21 }$$

Proof. The theorem is a direct consequence of lemma 1 and the fact that the correlation functions ${\phi }_{k}$ are restricted to the class ${{\rm{\Omega }}}_{1}^{{{\mathbb{F}}}_{d}}(\delta )$. Here we use strong non-locality in the sense of [25]. □

Note that theorem 1 is independent of the particular physical implementation of these function evaluations. In the next section we will look at the particular case of ld-MBQC.

In this section we connect these results with ld-MBQC as outlined in definition 2. We have the immediate corollary of theorem 1.

Corollary 1. Let M be a ld-MBQC for d prime, which deterministically evaluates a function $o\,:{{\mathbb{Z}}}_{d}^{n}\longrightarrow {{\mathbb{Z}}}_{d}$. If o(i) (when written as a polynomial) involves at least one term of the form ${\prod }_{j=1}^{n}\ {x}_{j}^{{a}_{j}}$, s.t. ${\sum }_{j=1}^{n}{a}_{j}\geqslant d$, then M is strongly contextual, and specifically strongly non-local.

We note that corollary 1 was first reported in [23]. In order to see how tight this bound is, we check whether general ld-MBQC instances, and specifically non-contextual ones, satisfy local universality. For qubits, this is trivially the case. In this section, we prove that local universality rather than linearity also generalises to ld-MBQCs on qudits of prime dimension.

We give a brief note regarding our restriction to prime values of d. Recall that in the setting of ld-MBQC, we are dealing with rings of integers modulo d. For d a prime number, ${{\mathbb{Z}}}_{d}$ is in fact a field, but not all finite fields arise this way. Theorem 1 holds for all finite fields ${{\mathbb{F}}}_{d}$, i.e., also fields of order a prime power d = p^r. However, ${{\mathbb{F}}}_{d}\,{/}\!\!\!\!\!\!{\cong }\,{{\mathbb{Z}}}_{d}$ whenever r ≥ 2, as the latter is a ring but not a field. Functions on unital, commutative rings are not polynomials in general. For these reasons, we will restrict to fields ${{\mathbb{Z}}}_{p}$ in this section, but we note that our results allow for partial inferences in the case of rings ${{\mathbb{Z}}}_{d}$ with d odd as well.

5.1. Local universality in MBQC

5.2. Scaling under composition

5.3. Temporal ordering

In summary, we have placed a bound on the space of output functions of general MBQCs if the underlying system is non-contextual. This generalises Raussendorf's result [1] for qubits. Nontrivial MBQCs on qudits do not directly make use of (local) contextuality (in quantum systems of dimension at least three), but instead harness a type of global contextuality, namely strong non-locality. In fact, theorem 1 can be understood as a deterministic version of Bell's theorem restricted to MBQC: assuming local hidden variables, m_k(x), and local (single qudit) measurements, ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(d-1)$ is already maximal.

Our results highlight strong non-locality as the crucial ingredient to theorem 1, and show how non-local correlations between spatially separated subsystems can boost the computational power of the classical control computer.

We conclude with a number of directions for further research in this area.

A key open question is whether our results (specifically, corollary 1) generalise to qudits with arbitrary non-prime-power dimensions. Most of our proofs rely on the theorem regarding polynomials in finite fields, which says that all functions over finite fields are polynomials. However, this result no longer holds true if $d\ne {p}^{r}$ for p prime and $r\in {\mathbb{N}}$, and so a generalisation would need to consider more general functions.

We have seen that the standard framework for MBQC in quantum theory is locally universal, and due to non-local correlations they can even compute functions outside of ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}(d-1)$. However, it is not clear whether local universality allows us to compute all functions in ${{\rm{\Omega }}}_{n}^{{{\mathbb{F}}}_{d}}$. It remains to determine if MBQCs of this form in quantum theory can saturate the space of functions (perhaps focussing on prime power dimensions). A more detailed analysis of the connections between the computational results and multi-party Bell inequalities as derived in [23] provides a natural starting point.

We focussed on generalised Pauli-like measurements with Clifford control unitaries as our framework, as this is standard for quantum computation, and showed that it is locally universal. With a different choice of control unitaries, it might be possible to exclude local universality. In this case, one may be able to obtain a proof of contextuality with the computation of polynomial with degree lower than d − 1.

Contextuality has recently been related to cohomology [7, 8, 29–31]. It would be a worthwhile goal to pursue such classifications further and bring them in contact with our result on computational complexity. It would be interesting to understand the relationship between polynomial degree in our setting, and the nontriviality of a certain cocycle within the cohomological frameworks of [7, 8].

We thank Robert Spekkens and Joel Wallman for discussions. This work is supported by the Australian Research Council (ARC) via the Centre of Excellence in Engineered Quantum Systems (EQuS) project number CE170100009 and through a studentship in the Centre for Doctoral Training on Controlled Quantum Dynamics at Imperial College London funded by the EPSRC. SR also acknowledges support from the Australian Institute for Nanoscale Science and Technology Postgraduate Scholarship (John Makepeace Bennett Gift).

Let $g\,:{{\mathbb{F}}}_{d}^{n}\to {{\mathbb{F}}}_{d}$. We first show how to obtain the δ-function from a polynomial with partial degree less than or equal to d − 1. For any $y\in {{\mathbb{F}}}_{d}^{n}$ consider the delta function defined as

$$\begin{eqnarray}&&\delta (x-y)=\left\{\begin{array}{l}1\quad \mathrm{if}\ x=y\\ 0\quad \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ A1 }$$

Then it holds that

$$\begin{eqnarray}&&\delta (x-y)=\displaystyle \prod _{i=1}^{n}(1-{({x}_{i}-{y}_{i})}^{d-1}),\end{eqnarray} \tag{ A2 }$$

which follows from Fermat's little theorem for d prime and for general finite fields as every element in the multiplicative group ${{\mathbb{F}}}_{d}^{\times }$ has order a divisor of $d-1$. We can express any function $g\,:{{\mathbb{F}}}_{d}^{n}\to {{\mathbb{F}}}_{d}$ as a linear combination of delta functions,

$$\begin{eqnarray}&&g(x)=\displaystyle \sum _{y\in {{\mathbb{F}}}_{d}^{n}}\delta (x-y)\phi (y),\end{eqnarray} \tag{ A3 }$$

which along with equation (A2), completes the proof.

Proof of theorem 2. Lemma 1 says a polynomial ϕ of degree $| \phi | =l$ generates all functions of less or equal degree via linear pre- and post-processing. The higher its degree the more valuable a function becomes when viewed as a resource for computational tasks under the scheme in figure 2. It is thus sufficient to show that we can implement a function of highest degree within the limits of ld-MBQC. Here we will make use of the Clifford control unitary explored in the example of section 3.2.3.

Instead of proving the existence of a function of degree p − 1, p prime we take a slightly different route and consider the δ(x)-function over the finite field ${{\mathbb{Z}}}_{p}$. It is clear

$$\begin{eqnarray*}\begin{array}{ccccc} & 0 & 1 & \cdots & p-1\\ \delta ({\text{}}x) & 1 & 0 & \cdots & 0\end{array}\end{eqnarray*}$$

that with δ(x) as a resource we can generate in particular any polynomial $m\in {{\mathbb{Z}}}_{p}[x]$ via linear combinations of the form,

$$\begin{eqnarray*}&&m(x)=\displaystyle \sum _{j=0}^{p-1}{m}_{j}\delta (x-j),\end{eqnarray*}$$

where ${m}_{j}\in {{\mathbb{Z}}}_{p}$ are constants specific to m.

In all finite fields the sum of elements within a subgroup of the multiplicative group ${{\mathbb{F}}}_{d}^{\times }$ vanishes as,

$$\begin{eqnarray*}&&S:= \displaystyle \sum _{g\in G}g=\displaystyle \sum _{{h}^{-1}g\in G}g=h\displaystyle \sum _{g^{\prime} \in G}g^{\prime} ={hS}.\end{eqnarray*}$$

Observe, that the exponential function u^x maps the additive group ${{\mathbb{Z}}}_{p}$ into its multiplicative group ${{\mathbb{Z}}}_{p}^{\times }$. We assume u to be a primitive element of ${{\mathbb{Z}}}_{p}^{\times }$, then u^x is surjective and we obtain non-trivial subgroups $\langle {u}^{x}\rangle \subseteq {{\mathbb{Z}}}_{p}^{\times }$, for all $x\ne 0,p-1$. One computes,

$$\begin{eqnarray}\displaystyle \sum _{k=1}^{p-1}{({u}^{x})}^{k}=\displaystyle \frac{| {{\mathbb{Z}}}_{p}^{\times }| }{| \langle {u}^{x}\rangle | }\ \displaystyle \sum _{k=0}^{| \langle {u}^{x}\rangle | -1}{({u}^{x})}^{k}=\displaystyle \frac{p-1}{| \langle {u}^{x}\rangle | }\ \displaystyle \sum _{g\in \langle {u}^{x}\rangle }g=\ \left\{\begin{array}{ll}p-1 & \mathrm{if}\ x=0,p-1,\\ 0 & \mathrm{otherwise},\end{array}\right.\end{eqnarray} \tag{ B1 }$$

as the sum either runs over all non-zero group elements in ${{\mathbb{Z}}}_{p}$ in case u^x is a primitive of ${{\mathbb{Z}}}_{p}^{\times }$, or over the subgroup $\langle {u}^{x}\rangle \subseteq {{\mathbb{Z}}}_{p}^{\times }$ for $x\ne 0,p-1$.

As shown in the example of section 3.2.3, we can obtain this exponential function using the symplectic Clifford unitary M_u of equation (16). Using equation (B1) and the exponential function constructed in equation (17) with linear pre- and post-processing on p − 1 qudits we implement the function,

$$\begin{eqnarray}{\sigma }_{p}(x):= {(p-1)}^{-1}\displaystyle \sum _{l=1}^{p-1}\ {({u}^{x})}^{l}=\left\{\begin{array}{ll}1\ & \mathrm{if}\ x=0,p-1,\\ 0\ & \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ B2 }$$

For p = 5 and r = 2, the exponentials and their sum in equation (B2) are explicitly given in the table below

$$\begin{eqnarray*}\begin{array}{cccccc}{x} & 0 & 1 & 2 & 3 & 4\\ {2}^{x} & 1 & 2 & 4 & 3 & 1\\ {2}^{2x} & 1 & 4 & 1 & 4 & 1\\ {2}^{3x} & 1 & 3 & 4 & 2 & 1\\ {2}^{4x} & 1 & 1 & 1 & 1 & 1\\ {\sigma }_{5}(x) & 1 & 0 & 0 & 0 & 1 & .\end{array}\end{eqnarray*}$$

Finally, we use another linear operation to recover $\delta (x)$ from σ_p(x),

$$\begin{eqnarray}&&\delta (x)=1+\displaystyle \frac{p-1}{2}(1+\displaystyle \sum _{k=1}^{p-1}{\sigma }_{p}({kx})).\end{eqnarray} \tag{ B3 }$$

In summary, we construct any function $m\,:{{\mathbb{Z}}}_{p}\longrightarrow {{\mathbb{Z}}}_{p}$ using the Clifford unitary M_u on $p{(p-1)}^{2}$ qudits under linear pre- and post-processing (without temporal ordering),

$$\begin{eqnarray*}\begin{array}{rcl}m(x) & = & \displaystyle \sum _{j=0}^{p-1}{m}_{j}\left(1+\displaystyle \frac{p-1}{2}\left(1+\displaystyle \sum _{k=1}^{p-1}\left({(p-1)}^{-1}\displaystyle \sum _{l=1}^{p-1}\ {u}^{{lk}(x-j)}\right)\right)\right)\\ & = & \displaystyle \frac{1}{2}\displaystyle \sum _{j=0}^{p-1}{m}_{j}\left(1+\displaystyle \sum _{k=1}^{p-1}\displaystyle \sum _{l=1}^{p-1}\ {u}^{{lk}(x-j)}\right).\end{array}\end{eqnarray*}$$

Note that the proof holds for general finite fields as long as we can implement the exponential function, yet equation (17) relies on the explicit ld-MBQC implementation and constrains to ${{\mathbb{Z}}}_{p}$ for p prime.

In the latter case we can improve the number of qudits needed to implement the δ-function and generalise to qudits of arbitrary odd dimension d. Note that in any ring of integers modulo d,

$$\begin{eqnarray*}&&{a}^{2}=1\ \mathrm{mod}\ d\,\Longleftrightarrow \,(a-1)(a+1)=0\ \mathrm{mod}\ d,\end{eqnarray*}$$

has solutions a = ±1. We can use this to construct,

$$\begin{eqnarray*}\delta (x)=\displaystyle \frac{1}{2}\displaystyle \sum _{k=\pm 1}{(d-1)}^{{kx}}=\left\{\begin{array}{ll}1 & \mathrm{if}\ x=0\\ 0 & \mathrm{otherwise}\end{array}\right.\end{eqnarray*}$$

and thus arbitrary functions $m:{{\mathbb{Z}}}_{d}\longrightarrow {{\mathbb{Z}}}_{d}$,

$$\begin{eqnarray*}&&m(x)=\displaystyle \sum _{j=0}^{p-1}{m}_{j}\left(\displaystyle \frac{1}{2}\displaystyle \sum _{k=\pm 1}{(d-1)}^{k(x-j)}\right),\end{eqnarray*}$$

using the Clifford unitary M_d−1 on $2p$ qudits under linear pre- and post-processing (without temporal ordering).□

Corollary 1 holds in the deterministic case only, however, the result remains valid at least in a small neighbourhood around the success probability p_S = 1, where

$$\begin{eqnarray*}&&{p}_{S}:= \mathop{\min }\limits_{{\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}}\ \mathrm{Prob}(\tau ({\bf{i}})=o({\bf{i}})),\end{eqnarray*}$$

τ(i) the factual result of a computation with input string ${\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}$. This is an immediate generalisation of the analogous result for qubits in [1], with some slight adjustments. First, we define Δ as follows,

$$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Delta }}\,:{{\mathbb{Z}}}_{d} & \longrightarrow & {{\mathbb{Z}}}_{\tfrac{(d-1)}{2}}\\ {\rm{\Delta }}(q) & := & \left\{\begin{array}{l}q\ \ \ \mathrm{if}\ q\lt -q\\ -q\ \mathrm{if}\ q\gt -q\end{array}\right..\end{array}\end{eqnarray*}$$

Next, we define the distance ν of a function $o\,:{{\mathbb{Z}}}_{d}^{n}\to {{\mathbb{Z}}}_{d}$ to the closest polynomial function in ${{\rm{\Omega }}}_{n}^{{{\mathbb{Z}}}_{d}}(d-1)$,

$$\begin{eqnarray}&&\nu (o):= \mathop{\min }\limits_{p\in {{\rm{\Omega }}}_{n}^{{{\mathbb{Z}}}_{d}}(d-1)}\ \displaystyle \sum _{{\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}}{\rm{\Delta }}(o({\bf{i}})-p({\bf{i}})),\end{eqnarray} \tag{ C1 }$$

where the minimum is taken over all polynomial functions of degree at most d − 1. Generalising the qubit case [1] we observe strong non-locality for

$$\begin{eqnarray}&&{p}_{S}\gt 1-\displaystyle \frac{1}{{d}^{n}}\displaystyle \frac{2\nu (o)}{(d-1)}.\end{eqnarray} \tag{ C2 }$$

In [32] the authors derive a stronger result based on the average success probability,

$$\begin{eqnarray*}&&{\bar{p}}_{S}:= \displaystyle \frac{1}{{d}^{n}}\displaystyle \sum _{{\bf{i}}\in {{\mathbb{Z}}}_{d}^{n}}\ \mathrm{Prob}(\tau ({\bf{i}})=o({\bf{i}})).\end{eqnarray*}$$

Decomposing the empirical model e (a set of probability distributions for outcomes and experiments corresponding to different contexts, see [33]) underlying the ld-MBQC into a non-contextual and strongly contextual part,

$$\begin{eqnarray*}&&e=\mathrm{NCF}(e){e}^{\mathrm{NC}}+\mathrm{CF}(e){e}^{\mathrm{SC}},\end{eqnarray*}$$

where $\mathrm{NCF}(e)\in [0,1],\mathrm{CF}(e)=1-\mathrm{NCF}(e)$, the average probability of failure, ${\bar{p}}_{F}=1-{\bar{p}}_{S}$, satisfies the relation,

$$\begin{eqnarray*}&&{\bar{p}}_{F}\geqslant \mathrm{NFC}(e)\nu (o).\end{eqnarray*}$$

A violation of this inequality yields a proof of non-locality and thus generalises the result achieved in [32] where only linear functions were considered. The analogous derivation holds under the appropriate change in equation (C1).

Note that the success probability depends on the non-contextual fraction $\mathrm{NFC}(e)$, which measures the amount of non-contextuality inherent in the probabilistic case. On the other hand it also depends on ν(o), the minimal distance of the output function to a function realisable in the non-contextual case. We could thus achieve a better success probability if we restricted to functions that all have a certain minimal distance from the set of possible output functions. However, as the space of output functions under non-contextuality, ${{\rm{\Omega }}}_{n}^{{{\mathbb{Z}}}_{d}}(p-1)$, is nonlinear, we cannot use bent functions as in the qubit case.