I've been reading Mermin's book on Quantum Computer Science, and in the section in which he discusses the construction of a QFT using 1-Qbit and 2-Qbit gates, he makes reference to some expressions involving linear operators that I'm not familiar with (at least if I've seen them before I've forgotten). I'm wondering if someone can help understand/interpret these expressions.(adsbygoogle = window.adsbygoogle || []).push({});

In particular, he defines the following linear operator,

[itex]\textbf{Z}|y\rangle_{n} = e^{2\pi i / 2^{n}}|y\rangle_{n}[/itex]

Where [itex]|y\rangle_{n}[/itex] represents the vector for the basis state corresponding to the n-qbit number whose binary expansion is the integer y.

From there, the following expression for Z is defined as follows (for a specific case in which n=4):

[tex]\textbf{Z} = exp\big(\frac{i \pi}{8}(8\textbf{n}_3 + 4\textbf{n}_2 + 2\textbf{n}_1 + \textbf{n}_0 ) \big) [/tex]

where [itex]\textbf{n}[/itex] represents the 1-qbit number operator [itex]\textbf{n}|x\rangle = x|x\rangle , x \in \{ 0,1 \}[/itex] (subscript denotes the number operator referring to the ith qbit)

This expression refers to an operator in the argument of an exponential, which I've never seen before, so I'm not sure how to interpret this. I did find a definition of a matrix exponential in terms of an infinite series:

[tex]e^{\textbf{Z}} = \sum_{k=0}^{\infty}\frac{1}{k!}\textbf{Z}^{k}[/tex]

The number operator can be expressed in terms of the matrix [itex]\begin{pmatrix}0 & 0 \\ 0 & 1 \end{pmatrix}[/itex], so I'm not sure if the corresponding operator expression can be understood in a similar way? It's a little hard to tell because Mermin seems to present the definitions and then make derivations from them without a lot of detailed explanation, the implication perhaps being that things follow straightforwardly from the basic definitions (which they might if I understood them properly).

I suspect I don't, because just taking as an example another simple definition of the same kind, it is stated that the following operator identity holds:

[tex]exp(2\pi i \textbf{n}) = \textbf{1}[/tex]

[itex]\textbf{1}[/itex] being the identity operator. I tried to see if I could derive this identity interpreting it in terms of the corresponding matrix exponential, but for this to be valid, it seems to me that the corresponding infinite series had better converge to the identity matrix. But based on the definition of the number operator, and multiplying the matrix representation by the complex scalar [itex]2\pi i[/itex], I get:

[tex]e^{(2\pi i \textbf{n})} = \sum_{k=0}^{\infty}\frac{1}{k!}(2\pi i \textbf{n})^{k}[/tex].

In terms of matrices,

[tex](2\pi i \textbf{n})^{k} = \begin{pmatrix}0 & 0 \\ 0 & (2\pi i)^{k} \end{pmatrix}[/tex]

I don't see how the infinite series could possibly ever converge to the identity matrix, not least of which because there is only one element of any power of this matrix that will ever be nonzero? Or am I way off here?

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# Definition Of Operator Exponential?

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