Quantum Computing

A paradigm shift for information technology? 

The aim of this brief application note is to introduce the reader to some of the fundamental concepts of quantum computing – distinguishing it from a classical machine. In the first part, we will introduce some of the key properties of quantum systems, as well as how these concepts are being used for quantum information processing. The second part consists of an overview of the most common physical platforms used for building quantum processors, as well as comparing their relative performance to date.

Table of contents

• Classic vs. Quantum Bits
• Superposition States and Quantum Measurements
• Decoherence Characterization
• Entanglement and Quantum Interference
• Different Qubit Types
• Conclusion

Classical vs. Quantum Bits 

To explore the underlying mechanisms of a quantum processor, it is natural to first consider how information is represented and processed in today’s “classical” computers. Information in classical computers is stored and processed in binary, meaning that single bits can only represent two values – either “0” or “1”. Therefore, in order to build the powerful computers that are used today, billions of bits are required. It turns out, however, that there are certain classes of problems where classical computers fall short in terms of architecture and processing power. Examples of such problems are simulations of systems with many degrees of freedom, e.g. quantum systems such as atoms and molecules, logistics modeling, and other problems where the number of potential solutions is large. 

About forty years ago, Richard Feynman postulated that quantum systems could be simulated much more efficiently by using computer hardware that operates quantum mechanics. So, what does it mean to operate quantum mechanically? In short, it means that information is stored and processed in a system that is obeying the laws of quantum mechanics and that these laws form the basis for the information processing. To do this, we introduce the concept of a quantum bit (qubit) – a quantum two-level system with some remarkable properties. 

The information in a quantum processor is encoded into the states of its qubits. A qubit is a quantum two-level system that can be physically realized in many different quantum-mechanical platforms, such as trapped ions, superconducting circuits, and silicon spins, to just mention a few.

As opposed to the classical bit, where information is represented by the "on" or "off" states of a transistor, see Fig 1 (a), a qubit is not limited to only being in the states "0" or "1", but it can actually be in both at the same time, see Figure 1(b). This phenomenon is a fundamental concept in quantum physics, known as superposition, see next section. Mathematically, we can represent the qubit state (independent of a particular physical implementation) as a point on a unit sphere, known as the Bloch sphere, see Figure 1b.

Superposition States and Quantum Measurements

The first “non-classical” property of quantum systems is their ability to be in superposition states, which means that quantum systems can be in several different states at the same time, each with a certain assigned probability. Superposition reflects the statistical nature of quantum mechanics and is maintained for as long a time as the system is not observed (or measured). When we perform a measurement of the quantum state, the system collapses into one specific state, see Figure 2. One famous example of superposition is the thought experiment of Schrödinger’s cat, which is both dead and alive at the same time, until it is observed.

In a quantum computer, computations are performed on information encoded into the states of qubits. Each qubit state can be specified with two quantum basis states |0〉 and |1〉, each with an associated complex coefficient 𝑐0 and 𝑐1. The quantum state of the qubit can then be written as a superposition |𝜓〉 =𝑐0|0〉 + 𝑐1|1〉, where the statistical probability of measuring the system in state |0〉 or |1〉 is given by |𝑐0|2 and |𝑐1|2, respectively. Mathematically this can be visualized as a vector, pointing on the surface of a unit sphere, known as the Bloch sphere, see Figure 2.

The probability of measuring |0〉 or |1〉 depends on the “latitude” of the vector on the Bloch sphere – the further north the vector is pointing, the higher the probability of measuring |0〉. In the example illustrated in Figure 2, the vector is closer to the south pole, meaning that it is most probably to measure |1〉 than |0〉. If the state vector is pointing somewhere along the equator, we say that the state is in an equal superposition between |0〉 and |1〉. This means that we have an equal probability of 50% to measure |0〉 and |1〉.  

Decoherence Characterization

Although the idea of a quantum processor has been around for a long time, there is not yet a large-scale realization of a quantum computer. One of the main reasons for this is that the lifetime of the qubit superposition is limited. One of the main challenges for quantum engineers is therefore to build as long-lived quantum systems as possible, without compromising the performance of the processor. 

The loss of quantum coherence is known as decoherence and has a very fundamental underlying reason that we touched upon already in the previous section: quantum superpositions collapse into classical states when they are measured. This happens regardless if it is an intentional measurement by an observer or caused by noise from the environment – the quantum system cannot tell the difference. 

Due to the presence of decoherence, there are several aspects that quantum engineers need to address when building a quantum computer: 

1. The qubit register needs to be both electromagnetically and thermally isolated from its environment, avoiding spurious exchange of energy.
2. The qubit operations, i.e. single and two-qubit gates need to be fast compared with the decoherence time.
3. Qubit measurements (readout) need to be fast and should not alter the quantum state  (non-demolition). 

To measure the qubit coherence time is therefore one of the cornerstones in any quantum lab, as it provides important information about the quality of the qubit itself and its shielding, how it is operated using quantum gates, as well as the characteristics of the qubit readout.

The coherence time of a qubit can be divided into two different timescales, depending on the impact the noise has on its state. The energy relaxation time, 𝑇1, denotes the time after which the qubit state relaxes to its ground state |0〉. This type of decay is irreversible since the energy leaves the system. The dephasing time, 𝑇2, refers to the time scale after which the state loses its phase information around the “equator” of the Bloch-sphere. As opposed to energy relaxation, loss of phase coherence can (under certain circumstances) be reversible, by applying a refocusing gate sequence.  

Entanglement and Quantum Interference

Quantum superposition states can also extend between several quantum systems, such that an operation on one system immediately affects all other. This type of interaction is called entanglement and provides quantum computers with their advantageous scaling in computational power with each added qubit. 

To illustrate this powerful scaling, we can consider the number of complex coefficients that are needed to describe the full quantum register. For instance, as we discussed previously, a single qubit state can be represented by two complex coefficients, 𝑐0 and 𝑐1. In the same way, if we want to represent the states of two entangled qubits, we need four complex coefficients, i.e. 𝑐00, 𝑐01, 𝑐10,𝑐11, and so on. In fact, the number of coefficients scales as 2𝑁, where 𝑁 is the number of qubits. This means that if we have a qubit register with 300 qubits, the number of coefficients is larger than the number of atoms in the known universe.

In a quantum algorithm, performed on a qubit register containing many qubits, the expected outcome contains a statistical distribution between the probabilities to find the qubits in certain states. Therefore, given that the qubits are entangled with each other, the answer to the computational task at hand can then be retrieved by analyzing the quantum interference pattern (or correlations) of its outcome states. This is illustrated for two qubits in Figure 4.