Quantum States vs Classical States

In the last post of this series, we discussed how supercharging quantum computing with Quantum Mechanics’ principles allows high computational power. To come to terms with this, we must first delve into the math behind quantum memory.

A computer needs memory. It stores input and output data as a transitional place to operate on data. We care about this functionality because data encodes states.

In the classical sense, a state refers to the particular arrangement that something is in at a specific moment. Examples of a classical state are the position of a door: either open or closed; the color of a marker: red, blue, yellow, etc.; the value of a bit: 0 or 1 / false or true; and so on.

As you know from daily life, these states are discrete; that is, there is only one particular arrangement a system can be in. However, a quantum state is continuous. Given a set of basis states, a quantum state may be a combination of those basis states. Translated to concrete terms, a quantum state of the colored marker could be 50% red, 25% blue, and 25% yellow. Another well-known example is Schrodinger’s Cat thought experiment; there is a 50-50 chance that the cat is either dead or alive.

Schrodinger’s Cat – Credit: Mother Jones

Intuitively, this does not make sense for how we perceive things to behave in the real world. A bit, by definition, is either 0 or 1; it cannot be in a 0.5 position or some other state because you know what state it is in by seeing the state. The critical point in our perception is seeing, or rather more generally, measuring. Measurement is what differentiates them. When you measure a classical state, you expect the same state to always be the same; a 1 will always be a 1. On the other hand, a measurement of a quantum state need not always be the same. While it is true that a state must be found in some defined arrangement, a quantum state allows you to find a different defined arrangement each time.

By nature, this measurement process on a quantum state is random, and the specific quantum state defines its probability distribution.

Mathematical Foundation

This state hand-wavy explanation becomes challenging to track fast, so to understand, we use math to describe this. To represent a state $\psi$ we use the notation $|{\psi}\rangle$ . For example, the state of the color red and the logical state 0 can be represented as $|\text{red}\rangle$ and $|0\rangle$ , respectively. This is known as the bra-ket notation or Dirac notation in quantum mechanics.

To say that the system $\psi$ is in one of these states, we write $|\psi\rangle = |\text{red}\rangle$ or $|\psi\rangle = |0\rangle$ . For now, this syntax works perfectly for defining the classical state of a system. However, to expand the usefulness of this nomenclature for our needs, we must think of $|\psi\rangle$ as a vector in some space, with each basis state being an axis. For a mixed state, we could write something like $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$ for a bit.

Superposed Quantum State – QOQM Andrew Daley

Considering that a state $|\psi\rangle = 1.3|\text{red}\rangle$ or $|\psi\rangle = 0.8\ |\text{red}\rangle$ would not make much sense because a scaled classical state should not change being in that state. Therefore, we can restrict ourselves to having $|\psi\rangle$ as a unit vector. This forces our state to be a point on the unit circle, in which we know $\alpha^2 + \beta^2=1$ . This is interesting because the probability is always in the $[0,1]$ range, which applies to both $\alpha^2$ and $\beta^2$ , and the sum of the probabilities of all possible states should equal 1, another rule these two follow. Then, we can argue that the square of the state coefficient is the probability of being in that state. Thus, we call these coefficients probability amplitudes.

We can verify this argument with the example of a quantum state being 50% in $|0\rangle$ and $|1\rangle$ . Mathematically, we could write that if $\alpha$ and $\beta$ refer in some sense to the probability or proportion, then we would want $\alpha = \beta$ in this case. For this to be true, and for the vector to be normalized, then we must have $\alpha=\beta=\frac{1}{\sqrt{2}}$ which means that $\alpha^2=\beta^2=\frac{1}{2}$ or 50-50 chance for $|0\rangle$ and $|1\rangle$ .

What about negative numbers, though, for probability amplitudes? It turns out that Quantum Mechanics allows not only negative numbers but also complex numbers. The theory must work!

The Qubit

We have expanded the two possible states of a bit from this quantum mechanical model of states to a whole space of possible linear combinations. We call this quantum mechanical bit a qubit. The state of a qubit is represented in the form $|q\rangle = \alpha |0\rangle + \beta |1\rangle$ where $\alpha$ and $\beta$ are complex numbers. The space with all the possibilities of its state is shown geometrically by the Bloch Sphere.

The Block Sphere is not a geometric representation of the vector form of the state (as that would take four dimensions: two real and two imaginary axes), but rather it is a mapping of every possible state of a single qubit. The basis states are $|0\rangle$ for the positive z-axis and $|1\rangle$ for the negative z-axis. Any state with real probability amplitudes lies along the $xz$ plane. For example, the equal probability state: $|\psi\rangle = \frac{1}{\sqrt{2}} |0\rangle + \frac{1}{\sqrt{2}} |1\rangle$ is the rightmost point of the circle: $x=1,\;y=0,\;z=0$ . Any point with a $y$ component will have a complex probability amplitude.

In general, any state can be mapped on the Bloch Sphere using spherical coordinates:

$|\psi\rangle = \cos{\frac{\theta}{2}} |0\rangle + e^{i\phi}\sin{\frac{\theta}{2}} |1\rangle$

Properties

From the behavior implanted in this mathematical foundation, two key properties arise from quantum mechanics: superposition and entanglement.

We discussed before the ability of a quantum state to combine multiple basis states simultaneously. This is called superposition. When the state of a system is a mixture of basis states, we say that it is a superposed state. The remarkable aspect of superposed states is that measurement is probabilistic by nature. While classical computation finds this property problematic, we will see later that it is this property that provides the quantum computer with its incredible parallel power.

For a group of interacting quantum states, like in a quantum computer, another helpful property arises entanglement. It connects multiple quantum states such that the measurement of one quantum state also obtains information about the other states. For example, if I prepared two qubits with specific states independent of each other and measured their states, I would obtain results only dependent on the initial state I set them in. However, if I made the qubits interact in some predictable way such that I flipped the value of one quantum state depending on the value of the other state, then from the measurement of either qubit, I could know the state of the other qubit without having measured it.

Schrodinger’s Cat also gives us an analogy: knowing the state of the cat or the detonator gives you information about the other. An alive cat means an untriggered detector and vice versa.

As we will find, entanglement enables us to exploit the parallel power from superposition.

With this crash course on qubits, mathematical state representation, measurement, and some quantum properties, we can now tackle how a quantum computer works, which we will discuss in the next post.

References

Nielsen, M., & Chuang, I. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511976667

A Modern Approach to Quantum Mechanics, John S. Townsend

Why are complex numbers needed in quantum mechanics? Some answers for the introductory level, American Journal of Physics 88, 39 (2020); https://doi.org/10.1119/10.0000258

Dirac, P. (1939). A new notation for quantum mechanics. Mathematical Proceedings of the Cambridge Philosophical Society, 35(3), 416-418. doi:10.1017/S0305004100021162

Superposed State Image. Andrew Daley. Quantum Optics and Quantum Many-Body Systems. https://qoqms.phys.strath.ac.uk/research_qc.html

Bloch Sphere Image. By Smite-Meister – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=5829358

Schrodinger’s Cat Image. Mother Jones. https://www.motherjones.com/kevin-drum/2018/09/schrodingers-cat-is-alive-one-twelfth-of-the-time/

ArrayFire Quantum Simulator. https://github.com/arrayfire/afQuantumSim

Mathematical Foundation

The Qubit

Properties

References

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