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Quantum Representations of Sound: From Mechanical Waves to Quantum Circuits

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Quantum Computer Music

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Abstract

This chapter discusses methods for the quantum representation of audio signals. Quantum audio is still a very young area of study, even within the quantum signal processing community. Currently, no quantum representation strategy claims to be the best one for audio applications. Each one presents advantages and disadvantages. It can be argued that quantum audio will make use of multiple representations targeting specific applications. The chapter introduces the state of the art in quantum audio. It also discusses how sound synthesis methods based on quantum audio representation may yield new types of sound synthesizers.

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Notes

  1. 1.

    This also appears in the context of the quantum theory, since quantum systems also show wave-like behaviour and thus have wave-like properties such as superposition and interference, among others. But do not be misled. A classical wave and a quantum particle-wave still are fundamentally different, even if they have similar mathematical properties.

  2. 2.

    It would feel logical to induce that the Flexible Representation of Quantum Audio (FRQA) was derived from the Flexible Representation of Quantum Images (FRQI). Unfortunately, this is not the case, since the FRQI is a Coefficient-Based Representation, and the FQRA is a State-Based one, derived from the Novel Enhanced Quantum Representation for digital images (NEQR). The choice to name the FQRA as such is unclear and might have been made for historical reasons. The FRQI was one of the pioneering representations for images (like FRQA) and probably the first to be widely studied in the literature with a variety of applications.

  3. 3.

    \(\quad \cos {(\theta )}^2+ \sin (\theta )^2 = 1\).

  4. 4.

    \(R_y(2\theta )\) has the same form of a 2D rotation matrix, found in many Linear Algebra textbooks. The main difference is that the angle theta rotates twice as fast in a Bloch Sphere compared to a regular Euclidean space.

  5. 5.

    Consider using the cosine term (\(p_{\gamma _i}(\mathinner {|{0}\rangle })\)) in the nominator of Eq. 10.33. What would happen? The complementarity of the trigonometric functions would result in a reconstructed audio with inverted polarity.

  6. 6.

    In his paper about generative quantum images using a coefficient based image representation, James Wootton [11] states that he measured the n-qubit image state \(4^n\) times before considering it was a good approximation for his application. This number can be much higher for reliable retrieval, and it scales exponentially.

  7. 7.

    We can build circuits that use the result of measurements for controlling quantum gates (due to the deferred measurement principle). But only unitary instructions.

  8. 8.

    The \(q-m-1\) was adapted to our notation. In the original notation, the text uses \(n+1\) qubits instead of q.

  9. 9.

    While this limitation may be true, it can be seen as an advantage inside a sensible noise/degraded aesthetic for artistic purposes.

  10. 10.

    Still, in the QSM histogram Fig. 10.26, there is a slight emergence of periodicity on the qubit sequencing. Can it be used artistically?

  11. 11.

    \(\cos a-\cos b = -2\sin {\frac{a+b}{2}}\sin {\frac{a-b}{2}}\); \(\sin a-\sin b = 2\cos {\frac{a+b}{2}}\sin {\frac{a-b}{2}}\); \(\sin ^2{x} = \frac{1-\textrm{cos}2x}{2}\).

  12. 12.

    Many possible connections between quantum and music software are made available in the QuTune Project repository [16].

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Acknowledgements

The authors acknowledge the support of the University of Plymouth and the QuTune Project, funded by the UK Quantum Technology Hub in Computing and Simulation.

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  1. ICCMR, University of Plymouth, Plymouth, UK

    Paulo Vitor Itaboraí & Eduardo Reck Miranda

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  1. Paulo Vitor Itaboraí
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Correspondence to Eduardo Reck Miranda .

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  1. Interdisciplinary Centre for Computer Music Research, University of Plymouth, Plymouth, UK

    Eduardo Reck Miranda

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Itaboraí, P.V., Miranda, E.R. (2022). Quantum Representations of Sound: From Mechanical Waves to Quantum Circuits. In: Miranda, E.R. (eds) Quantum Computer Music. Springer, Cham. https://doi.org/10.1007/978-3-031-13909-3_10

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  • DOI: https://doi.org/10.1007/978-3-031-13909-3_10

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