Qubit Conspiracy Theories

Chapters 2 and 3 explained measuring individual qubits. You can send a qubit along a wire, measure the qubit, and add this measurement to your tally of results. That’s fine for individual qubits. But in computing and life, things are more interesting when you start combining elements. So, in this chapter, we’ll see what happens when one qubit interacts with another qubit.

We start by examining gates in which the state of one qubit depends on the state of another. It’s like writing the following informal, English-language code:

if q0 is 0    set q1 to 0
else
    set q1 to 1

Since the state of q0 can be somewhere between |0⟩ and |1⟩, the end result for q1 may be between |0⟩ and |1⟩. What’s more, q0 and q1 may become mysteriously linked across time and space. They’re like identical twins who are separated at birth. When one twin stubs her toe, the other...

Multi-qubit gates

In Chapter 1, New Ways to Think about Bits, the classical AND gate has two input bits but only one output bit. When you deal with qubits, this never happens. In quantum computing, each qubit’s wire goes from the beginning to the very end of its circuit. This rule stems from the reversibility requirement that we described in Chapter 3, Math for Qubits and Quantum Gates.

With a quantum gate, the number of outputs must equal the number of inputs. When two qubits pass through such a gate, the two qubits are affected. This chapter explores gates that deal with two or more qubits. In this section, we’ll describe several commonly used multi-qubit gates. Each of these gates is a building block in the construction of quantum computing algorithms.

CNOT and flipped CNOT gates

A controlled NOT (CNOT) gate involves two qubits. We call one qubit the control qubit and the other the target qubit. The control qubit controls whether the target qubit’s...

Magic tricks with multi-qubit gates

Imagine this: You send two people to two different grocery stores. Before doing so, you give each an instruction to buy either meat or fish. You don’t say which of the two products either person should buy. When they return from their respective stores, they both return with meat—not fish.

The next day, you do all this again in exactly the same way. “Buy either meat or fish,” you say. On this second day, they both return with fish—not meat. On the third day, they both return with fish again. On the fourth day, they both return with meat. On a given day, you can’t predict whether they’ll both return with meat or both return with fish. But you know one thing for sure: they’ll always return with the same food. What’s more, if you repeat the experiment 100 times, they’ll return approximately half the time with meat and the other half with fish.

Your first guess is that, before...

Qubits don’t plan ahead

In the 1999 movie Mystery Men, one character has the superpower of making himself invisible, but only when no one is looking. For most people, this raises the question, “What good is that superpower?” For me, it raises an entirely different question: “Since no one can witness this character disappearing, is there a way to find out if the character actually disappears?” Can you verify or disprove the existence of something that, by its very nature, is unobservable? Of course, the knee-jerk answer to this question is, “No, you can’t.”

But, in 1964, physicist John Bell wrote a paper [4] in which he proposed an experiment that could put an end to hidden-variable theories. Then, in 1982, the team of Alain Aspect, Philippe Grangier, and Gérard Roger performed a convincing version of Bell’s experiment [5].

Disclaimer

This section introduces the theory about the nature of entanglement. If this...

Combining probabilities

In the section entitled Qubits don’t plan ahead, we use rules of probability to form conclusions about entangled qubits. If you’re not familiar with these rules, this section is for you.

An outcome is one possible result from a randomly-conducted experiment. For example, you shuffle a standard, 52-card deck of playing cards. Then, you close your eyes and select one of the cards. One outcome of this experiment is that you pick the nine of hearts. Another outcome is that you pick the queen of spades. All in all, this experiment has 52 outcomes.

An event is a set of outcomes in a randomly conducted experiment. Again, pick one card from a shuffled, 52-card deck. Picking a red card is an example of an event because picking a red card means picking a card from the set of all hearts and diamonds. Picking a face card (jack, queen, or king) is another event. Picking an even-numbered card is an event. Picking either a 2 of clubs, a 10 of diamonds,...

Summary

When combined with other gates, a two-qubit CNOT gate can entangle qubits. An entangled pair behaves as a single unit in which the measurement of one qubit determines the outcome of measuring the other. Neither qubit exists independently in a state of its own. The two-qubit system cannot be represented as the tensor product of two single qubits.

The true nature of entanglement remains a mystery for physicists. Experiments indicate that the theory has no hidden variables, so the qubits don’t know if they’ll be 0s or 1s before they’re measured. But at the time of each measurement, the qubits may be separated by many light-years. And yet, news of one qubit’s measurement seems to travel instantaneously to inform the other qubit’s measurement. No one knows exactly why this happens. As physicist Richard Feynman said, the best we can do is to “Shut up and calculate”.

In this chapter, we leveraged the fact that you can’t...

Questions

1. What’s the output of the following circuit?

2. Try to find values of , , , and satisfying . (It can’t be done because the two qubits in question are entangled.)
3. Show that the circuit in Figure 4.22 creates the state by writing the circuit’s matrix representation and calculating the circuit’s output.
4. Modify the code in this chapter’s Working with Qiskit section so that the resulting state is . Output a multi-qubit sphere like the one shown in Figure 4.18.
5. Between 1983 and 1993, there were approximately 105 boys born in the US for every 100 US-born girls. Given this ratio, what’s the probability that a family of three would have exactly two girls and one boy (in no particular order)?
6. In this chapter’s What would happen if there were hidden variables? section, we conclude that the overall probability of measurement disagreement is at least . We say at least because this...

Further readings

[1]. Born, M. (1926). Zur Quantenmechanik der Stossvorgänge. Zeitschrift für Physik, 37, 863-867. Translated as On the quantum mechanics of collisions, in J. A. Wheeler and W. Zurek (eds), Quantum Theory and Measurement, Princeton, NJ: Princeton University Press (1983), pp. 52-55

[2]. A. Einstein; B. Podolsky; N. Rosen (1935-05-15). Can Quantum-Mechanical Description of Physical Reality be Considered Complete? (PDF). Physical Review. 47 (10): 777-780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777

[3]. Bell, J. S. (1964). On the Einstein Podolsky Rosen Paradox (PDF). Physics Physique Физика. 1 (3): 195-200. doi:10.1103/PhysicsPhysiqueFizika.1.195

[4]. Aspect, A., Grangier, P., and Roger, G., 1982. Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell’s inequalities, Physical Review Letters, 49: 91-94