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Quantum Circuits

  • Chapter
Explorations in Quantum Computing

Part of the book series: Texts in Computer Science ((TCS))

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Abstract

In this chapter we described how to compute the unitary matrix corresponding to a given quantum circuit, and how to compute a quantum circuit that achieves a given unitary matrix. In the forward direction (circuit to matrix) three matrix products turn out to be useful. The direct product (also known as the tensor or Kroenecker product) is used to describe quantum gates that act in parallel. Such gates are drawn vertically aligned over distinct subsets of qubits in quantum circuits. Similarly, the dot product is used to describe quantum gates that act sequentially. Sequential gates are drawn one after the other from left to right in a quantum circuit. When mapping from a sequential quantum circuit to its corresponding unitary matrix remember that if the circuit shows gate A acting first, then gate B, then gate C, the corresponding dot product describing these steps is C⋅B⋅A, where the ordering is reversed. Finally, we introduced the direct sum, which describes controlled quantum gates. These apply a quantum gate to some “target” subset of qubits depending on the qubit values on another set of “control” qubits. The controlling values can be 0 (white circles) or 1 (black circles), and combinations of control values are allowed. We remind you that in controlled quantum gates we do not have to read the control value in order to determine the action. Instead, the controlled quantum gates apply all the control actions consistent with the quantum state of the control qubits.

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Notes

  1. 1.

    Source: Comment made by physicist Bruce Knapp of Columbia University to reporter William J. Broad recounted in “With Stakes High, Race is on for Fastest Computer of All” from the 1st February 1983 issue of the New York Times. In the 1980’s Japan and the U.S. were racing to make faster and faster supercomputers. Knapp was commenting on the capabilities of a new classical supercomputer he and his colleagues were developing. However, the quotation is even more fitting for quantum computers.

  2. 2.

    This data is synthetic and was generated using the piecewise continuous function f(t)=5t 2+3 (for −2≤t<1) and f(t)=−t 3+9 (for 1≤t≤2) and shifted versions thereof.

  3. 3.

    Note that it is purely a matter of convention whether we pick ω=exp (+2πi/N), or ω=exp (−2πi/N) since exp (+2πi/N)N=exp (−2πi/N)N=1. Physicists tend to use the former and electrical engineers the latter. The two versions of the transform are the inverse of one another. It does not matter which version we pick so long as we use it consistently.

  4. 4.

    An orthonormal basis for a vector space is a set of vectors such that the overlap between any pair of distinct vectors is 0, i.e., 〈ψ i |ψ j 〉=0,iffi≠j, and the overlap of a vector with itself is 1, i.e., ∀i,〈ψ i |ψ i 〉=1.

  5. 5.

    N.B. The superscript is always an even number.

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  1. California Institute of Technology, NASA Jet Propulsion Laboratory, Oak Grove Drive 4800, Pasadena, CA, 91109-8099, USA

    Dr. Colin P. Williams

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Correspondence to Colin P. Williams .

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Williams, C.P. (2011). Quantum Circuits. In: Explorations in Quantum Computing. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-84628-887-6_3

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  • DOI: https://doi.org/10.1007/978-1-84628-887-6_3

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