The qubit

In this section you will meet the qubit. You will also see a bit of mathematical notation, including some concepts from linear algebra. A qubit (pronounced “cue-bit” and short for quantum bit) is the physical carrier of quantum information. It is the quantum version of a bit, and its quantum state can be written in terms of two levels, labeled |0\rangle and |1\rangle , which can be represented in the “computational basis” by two-dimensional vectors:

\begin{split}|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} ~~~~ |1\rangle =\begin{pmatrix} 0 \\ 1 \end{pmatrix}\end{split}

What we have described here is the abstract notion of a qubit. The quantum computers you interact with in IBM Quantum Experience use a physical type of qubit called a superconducting transmon qubit, which is made from superconducting materials such as niobium and aluminum, patterned on a silicon substrate. Such systems are not natural qubits, but are instead formed by isolating two energy levels out of many to form our approximate qubit.

The difference in energy between these two energy levels sets the characteristic frequency f of the qubit through Planck’s constant h : \Delta E = h f . These frequencies are nominally targeted to be around 5~\mathrm{GHz} .

For a superconducting qubit to behave as the abstract notion of the qubit, we must have the device at drastically low temperatures T such that k_{\rm B}T \ll hf , where k_{\rm B} is Boltzmann’s constant. In the IBM Quantum lab, we must keep the temperature extremely cold (15 millikelvin in a dilution refrigerator) to minimize ambient noise or heat that could excite the superconducting qubit and increase the error probability. Once a system has cooled to the target temperature, which takes several days, the qubit reaches equilibrium at the ground state |0\rangle .

To get a sense for what “ground state” means, try running the quantum circuit below in simulation mode (or run it on a real device). Here, the qubit is initially prepared in the |0\rangle state, then is followed by the standard measurement. From your execution results, you should find in the simulation mode (and with very high probability for runs using a real system) that the qubit is still in the |0\rangle state. Any errors on the real device are due to imperfect measurements and/or residual heating of the qubit.

First Quantum Circuit

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As you may have guessed, we need to be able to put the qubit in other states. To do this, we require the concept of a quantum gate. A single-qubit quantum gate is a 2 \times 2 unitary matrix (the matrices must be unitary because quantum gates must be reversible and preserve probability amplitudes). Quantum gates are a convenient way in which to describe the evolution of a quantum state. The action of a gate is to transform an initial state |\psi\rangle into a final state |\psi'\rangle=U|\psi\rangle , where U represents the gate. This is simply matrix-vector multiplication.

The simplest gate is the NOT gate, which we denote by X . It takes |0\rangle\rightarrow X|0\rangle=|1\rangle ; in other words, it flips the 0 to a 1 , or vice versa. This gate is classical and its operation is represented by the familiar permutation matrix:

\begin{split}X =\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\end{split}

In Circuit Composer, it is shown as xgate. Try running it. Did you find that, unlike in the example above, the qubit ended up in the |1\rangle state in the simulator (and with high probability if you used the real systems)?

Bit-flip Quantum Circuit

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