The labels used to represent basis states are essentially arbitrary, but different ways of representing states will be useful in different contexts. The most common labellings of basis states for a register of qubits are the binary and decimal representations. To understand how these two are related recall that there are \(N=2^n\) distinct bitstrings \(\omega\in\{0,1\}^n\) of the form \(\omega=b_1b_2...b_n\), where \(b_i\in\{0,1\}\). Each bitstring \(\omega=b_1b_2...b_n\) can be made to correspond to an integer \(k\in\{0,1,2,...,N-1\}\) via the map

\[\displaystyle\sum\limits_{i=1}^{n}b_i2^{n-i}=b_12^{n-1}+b_{2}2^{n-2}+...+b_{n-1}2+b_n=k.\]

In this way the \(N\) distinct bit strings in \(\{0,1\}^n\) can be indexed by the numbers \(0,1,2,...,N-1\), and then used to label the \(N\) basis states of the Hilbert space \(\mathcal{H}=\bigotimes_{i=1}^n\mathcal{H}^2\) corresponding to a register of \(n\) qubits. Thus, an arbitrary normalized state \(\left|\psi\right>\in\mathcal{H}=\bigotimes_{i=1}^n\mathcal{H}^2\) can be expressed as

\[\left|\psi\right>=\displaystyle\sum\limits_{k=0}^{N-1}\alpha_k\left|k\right>,\]

or equivalently as

\[\left|\psi\right>=\displaystyle\sum\limits_{b_1=0}^1\displaystyle\sum\limits_{b_2=0}^1...\displaystyle\sum\limits_{b_n=0}^1\alpha_{b_1b_2...b_n}\left|b_1b_2...b_n\right>,\]

where \(\alpha_{b_1b_2...b_n}=\alpha_k\) if \(b_12^{n-1}+b_{2}2^{n-2}+...+b_{n-1}2+b_n=k\). The former labeling of basis states will be called the

\[\displaystyle\sum\limits_{i=1}^{n}b_i2^{n-i}=b_12^{n-1}+b_{2}2^{n-2}+...+b_{n-1}2+b_n=k.\]

In this way the \(N\) distinct bit strings in \(\{0,1\}^n\) can be indexed by the numbers \(0,1,2,...,N-1\), and then used to label the \(N\) basis states of the Hilbert space \(\mathcal{H}=\bigotimes_{i=1}^n\mathcal{H}^2\) corresponding to a register of \(n\) qubits. Thus, an arbitrary normalized state \(\left|\psi\right>\in\mathcal{H}=\bigotimes_{i=1}^n\mathcal{H}^2\) can be expressed as

\[\left|\psi\right>=\displaystyle\sum\limits_{k=0}^{N-1}\alpha_k\left|k\right>,\]

or equivalently as

\[\left|\psi\right>=\displaystyle\sum\limits_{b_1=0}^1\displaystyle\sum\limits_{b_2=0}^1...\displaystyle\sum\limits_{b_n=0}^1\alpha_{b_1b_2...b_n}\left|b_1b_2...b_n\right>,\]

where \(\alpha_{b_1b_2...b_n}=\alpha_k\) if \(b_12^{n-1}+b_{2}2^{n-2}+...+b_{n-1}2+b_n=k\). The former labeling of basis states will be called the

**decimal representation**and the latter the**binary representation**, and both can be considered as the**standard computational basis**. These two basis representations will be used throughout the presentation of quantum algorithms and it is helpful to be able to use the two interchangeably.
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