Within the last two decades, a second wave of quantum technology (QT) progressed from fundamental research to experiments in university labs. They promise a myriad of applications, products and services that have the potential to transform medicine, information technology, communications and artificial intelligence among others. 

As per DSTL’s definition, we can distinguish two quantum technology revolutions, waves or generations, based on harnessing different quantum phenomena as follows:

Quantum Technology Wave 1.0: These are reliant on quantum effects such as spin, Q-tunnelling, quantised energy. Quantum 1.0 technologies include: lasers, transistors, semiconductor devices and MRI scanners.

Quantum Technology Wave 2.0: These technologies explicitly exploit, create, manipulate and read out quantum states of matter and phenomena such as superposition and correlation. Quantum 2.0 technologies are concerned with imaging, timing, computing, sensors (gravity, magnetic fields), communication and many more.

Quantum 2.0

Introduction to QT 2.0 landscape

In the UK, the QT landscape is evolving fast, due to a series of initiatives in which the public and private sectors came together for the UK National Quantum Technology Programme

The UK’s quantum strategy is focusing on development as well as commercialisation of quantum sciences and technologies, through its quantum hubs, dedicated to four groups of technologies as follows:

More recently, the National Quantum Computing Centre and the Quantum Metrology Institute of the National Physical Laboratory were added to the UK’ quantum research capabilities.    

For more details on quantum landscapes in the UK and EU, including organisations, research activities, funding and development programmes see QT interactive maps:

  • UK Quantum Landscape map, provided by Knowledge Transfer Network.
  • EU Quantum Landscape map, provided by QuantERA which is a network of 32 organisations from 27 countries, coordinated by the National Science Centre, Poland, supporting international research projects in the field of Quantum Technologies (QT).

Why is this of interest to mathematicians and what is their possible role?

New technologies require new mathematics to make them viable. From fundamental research to real world applications, it is a long way in terms of innovation. Mathematics plays a significant role along this journey. This applies to quantum technology as well. For example, we need new algorithms to interact with quantum computers. To be effective, these methods must take into account the qubit physics and their limitations such as noise and coherence.

The same is true for quantum sensors. For example, quantum imaging aims to make ‘the invisible visible’ by operating at previously inaccessible wavelengths, timescales and length-scales. But, the more subtle and sparse the signal acquired (e.g. a single photon), the more difficult it is to interpret it.  As a result, quantum imaging devices require new mathematical methods to invert domains, compose and analyse information. An older but illustrative example is the new mathematics developed around re-purposing compressed sensing, for image composition in single pixel cameras.

For a selection of problems posed by quantum LIDAR and quantum radar that are relevant to the mathematical community, see past seminars and associated materials here

For an overview of the quantum computing landscape in the UK, see materials from our quantum computing seminar here.

There are many opportunities for mathematicians to take part in QT2.0 innovation journey. If you would like to learn more or are curious about using your research to innovate and accelerate the future QT2.0, please get in touch.


Pre-requisite knowledge 

This is a mini-guide and starting library for mathematicians interested in quantum computing (QC), quantum communication and so on. The QC space and pre-requisite knowledge could be partitioned in many ways, using criteria such as applications, software and hardware:

    1.  Quantum computing theory and classes of problems that are currently candidates for simulation on NISQ (noisy, intermediate-scale quantum) computers such as: 
      1. Non-linear differential equations, with applications in fluid dynamics for medical technology, automotive and aerodynamics, molecular simulations for specialty materials and drug discovery,
      2. Linear algebra systems for applications such as risk analysis, DNA sequence classification, market and customer segmentation,
      3. Factorisation, for encryption, codebreaking and other security applications,
      4. Combinatorial optimisation, with applications in supply chain and logistics, airline or ground transport, financial portfolio.
    2. Operating system software and
    3. Technology used to build the quantum hardware: using superconducting circuits (IBM, Google, Rigetti, Alibaba, intel); ion traps (IonQ, Atos, Honeywell, Alpine quantum technologies), Photonics (PsiQuantum), solid-state (Intel), cold atoms (Cold Quanta, Atom computing) and nanowires (Microsoft) and so on.

      Quantum computing theory is at the intersection of mathematics, physics and computer science. For those wishing to enter this field, here is a list of useful background topics:

      • Mathematics: You already have this covered. Whilst all areas of mathematics will overlap with quantum information theory and applications to some degree, special mentions go to linear algebra and probability which are at the heart of quantum mechanics. Further relevant topics are measure theory, group and representation theory and functional analysis.
      • Physics: To be active in QC you do not need in-depth knowledge of quantum physics. If your curiosity gets the better of you, familiarise yourself with quantum mechanics, especially Dirac’s notation, quantum gates and qubits. 
      • Computer Science: Most theory topics are relevant although are less crucial at first. Some relevant areas are information theory, optimisation algorithms, computation complexity, error-correcting codes and machine learning.  Further topics of note are Grover’s search algorithm, quantum Fourier transform and Shor’s algorithm.
    Books, textbooks, courses

    For those who like to develop an intuition before they delve into the full science, the gentle introductory book Q is for Quantum, by Prof. Terry Rudolph explains the concepts of superposition and entanglement and the connection with quantum computers in a simple and easy manner.

    The canonical reference for learning quantum computing is the textbook Quantum computation and quantum information, M. Nielsen and I. Chuang.Cambridge University Press; 10 Anv edition, 2011. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction.

    Some further textbooks

    • A. Yu. Kitaev, A. H. Shen and M. N. Vyalyi. Classical and Quantum Computation, AMS, 2002.
    • P.Kaye, R. Laflamme, M. Mosca. An Introduction to Quantum Computing, Oxford University Press, 2007.
    • M. Wilde. Quantum Information Theory, Cambridge University Press, 2013.

    Some organisations are already making quantum computers (NISQ) available online, together with copious amounts of tutorials and useful documentation to attract interested students, researchers and developers to learn their programming language and interact with their technology. For example see Amazon’s Braket  and Xanadu's Strawberry Fields.

    Events, seminars, workshops


    This year I will host seminars, presentations and workshops on QC, applications, collaborations with industry, entrepreneurship, primer on QC for mathematicians.

    View upcoming seminars here and access the seminar archive here.

    Please get in touch if you are interested in participating as a presenter, educator or industry representative.