What are some introductory textbooks and resources on quantum computing that do not use Dirac-notation, but rather ordinary mathematical notation (i.e. the notation used by mathematicians to represent vectors, dual vectors, norms, tensor products, inner products, operators, etc.)?

  • 1
    $\begingroup$ What do you mean by "ordinary" specifically? $\endgroup$ Commented May 2, 2019 at 14:19
  • $\begingroup$ By 'ordinary mathematical notation' I mean the notation used when students study mathematics or physics at the university. $\endgroup$
    – Fischer
    Commented May 2, 2019 at 14:27
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    $\begingroup$ As far as I know, every physics student uses Dirac notation heavily since the introduction of quantum mechanics $\endgroup$ Commented May 2, 2019 at 19:52
  • $\begingroup$ @user2723984 While the Dirac notation is convenient, it's definitely nothing intrinsic to quantum mechanics or quantum computing; they can be studied perfectly even without it. In fact, the Dirac notation is hardly ever used by pure mathematicians. $\endgroup$ Commented May 2, 2019 at 21:27
  • $\begingroup$ @SanchayanDutta I know, I was just answering to OP suggesting that the "ordinary mathematical notation used when students study physics at a university" is anything other than Dirac notation, which as far as I know is the most common (hence "ordinary") $\endgroup$ Commented May 2, 2019 at 21:30

2 Answers 2


Lance Fortnow has written a brief introduction to quantum complexity theory targeted at computer scientists with no background in quantum physics. It avoids the use of Dirac's bra-ket notation, except in one section that explicitly discusses its absence.

  • $\begingroup$ To be clear, that paper is much less of "an introduction to quantum computing" (it's hardly a few pages) and much more of "an introduction to quantum complexity theory". $\endgroup$ Commented May 4, 2019 at 5:42
  • $\begingroup$ Certainly. Answer updated. $\endgroup$ Commented May 6, 2019 at 7:27

Note: There's some merit in not using the Dirac notation if you're a mathematics student. Mathematics deals with broader classes of spaces than just Hilbert spaces and the properties of Hilbert spaces don't simply carry over cf. this. For instance, in Banach spaces the norm isn't necessarily given by an inner product. In fact, there exist generalizations to quantum theory which specifically employ special Banach spaces rather than the narrower class of Hilbert spaces. It's the additional geometric structure imposed on the Hilbert space that allows for the bra-ket notation. Using this notation naively when you're studying the properties of the Hilbert spaces as an example of a broader class, isn't a great idea. That said, the bra-ket notation has some beautiful properties cf. the discussion on Physics SE and has almost become a tradition in physics. You'll have a hard time following the popular literatures in quantum computing if you do not pick it up over time. Prof. Balakrishnan's has a wonderful lecture on this topic! At the end of the lecture, he remarks that "machinery" of Dirac notation, once established, is almost "automatic" and "self-correcting", which indeed is true.

You may be interested in Quantum Algorithms via Linear Algebra: A Primer (R.J. Lipton & K.W. Regan).

Summary of notation (including correspondence with Dirac notation)

  • Quantum states (vectors in Hilbert space $\Bbb H^N$) $$\text{Ket: }|a\rangle \longrightarrow \boldsymbol{a} = \begin{bmatrix}a_0\\ \vdots \\ a_{N-1}\end{bmatrix}$$ $$\text{Bra: }\langle a| \longrightarrow \boldsymbol{a}^* = \begin{bmatrix}a_0^* \\ \vdots \\ a_{N-1}^*\end{bmatrix}$$ where $a_k \in \Bbb C \space \forall \space k\in\{0,N-1\}$ and * denotes complex conjugate. The $k$-th component of $\boldsymbol{a}$ i.e. $\boldsymbol{a}_k$ is represented as $\boldsymbol{a}(k)$. In the textbook, they don't really distinguish between vectors (kets) and their duals (bras) (i.e. the dual state of $\boldsymbol{a}$ is treated like $\begin{bmatrix}a_0^* & \dots & a_{N-1}^*\end{bmatrix}^T$ rather than $\begin{bmatrix}a_0^* & \dots & a_{N-1}^*\end{bmatrix}$), so be wary. Note that one particular advantage of the bra-ket notation is that it helps to distinguish a vector from its dual.
  • Norm

    $$\langle a|a\rangle^{\frac{1}{2}} \longrightarrow \langle \boldsymbol{a},\boldsymbol{a}\rangle = ||\boldsymbol{a}|| = |\sum_k\boldsymbol{a}(k)^2|^{1/2}$$

  • Tensor Product

    The tensor product of two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ is the vector $\boldsymbol{c} = \boldsymbol{a} \otimes \boldsymbol{b}$ defined by $$|c\rangle=|a\rangle\otimes |b\rangle \longrightarrow \boldsymbol{c}(ij)=\boldsymbol{a}(i)\boldsymbol{b}(j).$$

  • Matrices

    These are linear operators $\boldsymbol{U}$ on Hilbert spaces. The following properties hold: $$U(|a\rangle+|b\rangle) = U|a\rangle + U|b\rangle \longrightarrow\boldsymbol{U(a+b)}=\boldsymbol{U(a)+U(b)}\tag{1}$$ $$U(k|a\rangle) \longrightarrow \boldsymbol{U}(k\boldsymbol{a})=k\boldsymbol{U(a)}\tag{2}$$ where $k$ is a scalar from the underlying field, which is generally $\Bbb C$.

  • Inner product

    The inner product of two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ is given by $$\langle a|b\rangle \longrightarrow \langle \boldsymbol{a},\boldsymbol{b}\rangle = \sum_{k=0}^{m}\boldsymbol{a}^*(k)\boldsymbol{b}(k).$$

From the Preface

This book is an introduction to quantum algorithms unlike any other. It is short, yet it is comprehensive and covers the most important and famous quantum algorithms; it assumes minimal background, yet is mathematically rigorous; it explains quantum algorithms, yet steers clear of the notorious philosophical problems and issues of quantum mechanics.

Almost all summaries, notes, and books on quantum algorithms use a special notation for vectors and matrices. This is the famous Dirac notation that was invented by Paul Dirac — who else. It has many advantages and is the de-facto standard in the study of quantum algorithms. It is a great notation for experts and instrumental to becoming an expert, but we suspect it is a barrier for those starting out who are not experts. Thus, we avoid using it, except for a few places toward the end to givea second view of some complicated states. Our thesis is that we can explain quantum algorithms without a single use of this notation. Essentially this book is a testament to that belief: if you find this book more accessible than others, then we believe it owes to this decision. Our notation follows certain ISO recommendations, including boldface italics for vectors and heavy slant for matrices and operators.

  • $\begingroup$ I think it must be conjugate transpose of $a(k)$ in the inner product formula. $\endgroup$
    – Danylo Y
    Commented May 4, 2019 at 6:40

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