State-injection schemes of quantum computation in Spekkens' toy theory

Spekkens' toy theory is a noncontextual hidden variable model with an epistemic restriction, a constraint on what the observer can know about the reality. In reproducing many features of quantum mechanics in an essentially classical model, it clarified our understanding of what behavior can be truly considered intrinsically quantum. In this work, we show that Spekkens' theory can be also used to help understanding aspects of quantum computation—in particular, an important subroutine in fault-tolerant quantum computation called state injection. State injection promotes fault-tolerant quantum circuits, which are usually limited to the classically efficiently simulatable stabilizer operations, to full universal quantum computation. We show that the limited set of fault-tolerant operations used in standard state-injection circuits can be realized within Spekkens' theory, and that state injection leads to nonclassicality in the form of contextuality. To achieve this, we extend prior work connecting Spekkens' theory and stabilizer quantum mechanics, showing that subtheories of the latter can be represented within Spekkens' theory, in spite of the contextuality in qubit stabilizer quantum mechanics. The work shines light on the relationship between quantum computation and contextuality.

Figure 1

Computational scheme. Schematic representation of the computational scheme of the paper.

Figure 2

Representation of a noncontextual subtheory of qubit stabilizer quantum mechanics in Spekkens' toy model. In the figure above, in order, the allowed pure states, observables, and gates of the noncontextual subtheory of qubit stabilizer quantum mechanics considered in the proof of Theorem t4 are represented in Spekkens' toy model, according to the definitions of epistemic states, observables, and evolutions (gates) defined in Sec. 2a. We have also indicated the probability distributions associated to the epistemic states, where $x,y\in {\mathbb{Z}}_d,$ and how the gates act on them. In the last two figures we have considered the scenarios corresponding to acting with $X,Y$ on $\left|0\right.〉,$ with $Z$ on $\left|+\right.〉$ and with CNOT on $\left|+0\right.〉$ (thus obtaining the Bell state $\frac{\left|00\right.〉+\left|11\right.〉}{\sqrt{2}}).$ More detailed examples can be found in Refs. [2] and [3].

Figure 3

State-injection schemes of quantum computation. The state-injection schemes that we consider in this work are the ones developed by Zhou, Leung, and Chuang in Ref. [12]. The diagonal gate $U$ can be injected in the circuit by using objects that are allowed in the free part of the computation. The injected state is $U\left|+\right.〉,$ which is subjected to a controlled not with the input state $\left|\psi \right.〉.$ Conditioned on the outcome of the measurement of the Pauli $Z$ on the state $\left|\psi \right.〉$ after the CNOT, the correction $UX{U}^{†}$ is applied to the state $U\left|+\right.〉.$ At the end we obtain the gate $U$ applied to the input state $\left|\psi \right.〉.$

Figure 4

cz injection. The first injection of our scheme is the injection of the $cz\left|++\right.〉$ state, needed to perform the correction in the injection scheme for the CCZ gate and to produce the Hadamard gate. In the figure above the correction is $\left(cz\right)\left(X^a{X}^b\right)\left(cz\right)$ conditioned on obtaining $x,y$ outcomes from the $Z$ measurements, where ${(-1)}^a=x$ and ${(-1)}^b=y$. For example, if the outcomes are $x=1,y=-1,$ the correction is $\left(cz\right)\left(\mathbb{I}X\right)\left(cz\right)=XZ,$ which is an allowed gate in our subtheory.

Figure 5

Hadamard gate via cz. The Hadamard gate can be obtained once the cz gate is available. Obtaining $H$ from cz and $X$ measurements was originally shown in Ref. [31].

Figure 6

State-injection scheme based on CCZ injection. By injecting first the cz state and then the CCZ state we can boost the subtheory of rebit stabilizer quantum mechanics made of observables that are tensors of nonmixing Pauli $\mathbb{I},X,Z,$ and the gates generated by CNOT and the Pauli rotations $X,Z$ to universal quantum computation.

Figure 7

CCZ injection. The second injection of our scheme is the injection of the $\mathrm{CCZ}\left|+++\right.〉$ state. In the figure above the correction is $\left(\mathrm{CCZ}\right)\left(X^a{X}^b{X}^c\right)\left(\mathrm{CCZ}\right)$ conditioned on obtaining $x,y,z$ outcomes from the $Z$ measurements, where ${(-1)}^a=x,{(-1)}^b=y$ and ${(-1)}^c=z$. For example, if the outcomes are $x=-1,y=1,z=1$ the correction is $\left(\mathrm{CCZ}\right)\left(X\mathbb{I}\mathbb{I}\right)\left(\mathrm{CCZ}\right)=\left(X\right)\left(cz\right),$ which is an allowed gate in our subtheory.

Figure 8

Peres-Mermin square via Spekkens' subtheories and cz gates. The above circuit provides a way of implementing all the contexts of the Peres-Mermin square. Each block denoted by cz corresponds to the injection scheme of Fig. 4 endowed also with a swap gate (which is present in our Spekkens' subtheory as it can be made of a series of three alternated cnot gates) to set the output state $cz\left|\psi \right.〉$ as a precise modification of the input state $\left|\psi \right.〉$ (and not of the ancillary resource state $cz\left|++\right.〉$). Each context can be selected according to some combinations of the classical control bits $a,b,c,d,e,\alpha ,\beta ,\gamma$ that can take values in $\{0,1\}.$ The value 0 indicates that the corresponding gate is not performed, while the value 1 that it is performed. At the end of every gray block (labeled by numbers) we assume that we can read the output outcome. The three row contexts of the Peres Mermin square are identified by the variables $(d,e),(a,b,c)$ and $(\alpha ,\beta ,\gamma ,d,e)$ assuming value one, respectively. The three column contexts are identified by $(a,d,\gamma )(b,e,\gamma )$ and $(c,d,e,\gamma ).$ Notice that in the last case where we implement the context $XX,ZZ,YY,$ the measurement of XX is implemented by performing $\mathbb{I}X$ first and then $X\mathbb{I},$ and in this case we consider the outputs related to the blocks labeled by 3,5,6.