Quantum computing is getting a significant amount of attention these days, especially since IBM put a real 5-qubit machine online in May 2016. Since then, over 100,000 people have themselves directly used the no-charge IBM Q Experience and created software using the open source Qiskit quantum computing software development framework.

**There is a key question though: Why?**

Is quantum computing just a flashy new alternative to the “classical” computers that are our smartphones, laptops, cloud servers, high performance computers, and mainframes? Can they really perform some calculations faster than classical computers can? How do you characterize those areas where they can or potentially can do better? Can you prove it? Ah, proof. Proving something mathematically is not just making a lot of observations and saying, “it seems likely that such and such is the case.”

In 1994 Peter Shor formulated his eponymous algorithm that demonstrated how to factor integers on a quantum computer almost exponentially faster than any known method on a classical computer. This is getting a lot of attention because some people are getting concerned that we may be able to break prime-factor-based encryption like RSA much faster on a quantum computer than the thousands of years it would take using known classical methods. However, people skip several elements of the fine print.

First, we would need millions and millions of extremely high quality qubits with low error rates and long coherence time for this to work. Today we have 50. Second, there’s the bit about “faster than any known method on a classical computer.” Since we do not know an efficient way of factoring arbitrary large numbers on classical computers, this appears to be a hard problem. It’s not proved to be a hard problem. If someone next week comes up with an amazing new approach using a classical computer that factors as fast as Shor’s might, then the conjecture of it being hard is false. We just don’t know.

#### Quantum computers really are faster at some things

Is everything like that? Are we just waiting for people to be more clever on classical computers so that any hoped-for quantum computing advantage might disappear? The answer is no. Quantum computers really are faster at some things. We can prove it. This is important. Let’s set up the problem. The basic computational unit in quantum computing is a *qubit*, short for *quantum bit*. While a classical bit is always 0 or 1, when a qubit is operating it can take on many other additional values. This is increased exponentially, with the potential computational power doubling each time you add an additional qubit through *entanglement*. The qubits together with the operations you apply to them are called a *circuit*.

Today’s qubits are not perfect: they have small error rates and they also only exist for a certain length of time before they become chaotic. This is called the *coherence time*. Because each gate, or operation, you apply to a qubit takes some time, you can only do so many operations before you reach the coherence time limit. We call the number of operations you perform the *depth*. The overall depth of a quantum circuit is the minimum of all the depths per qubit.

There is. It was proved by scientists Drs. Sergey Bravyi of IBM Research, David Gosset of the University of Waterloo’s Institute for Quantum Computing, and previously of IBM Research, and Robert König of the Institute for Advanced Study and Zentrum Mathematik, Technische Universit__ä__t München. It’s now published in *Science* as “Quantum advantage with shallow circuits. The width of a circuit, that is, the number of qubits, can be related to the required depth of the circuit to solve a specific kind of problem. Qubits start out as 0s or 1s; we perform operations on them involving superposition and entanglement; and then we measure them. Once measured, we again have 0s and 1s.

#### Fault tolerant quantum computers will do some things better than classical computers can

What the scientists proved is that there are certain problems that require only a fixed circuit depth when done on a quantum computer even if you increase the number of qubits used for inputs. These same problems require the circuit depth to grow larger when you increase the number of inputs on a classical computer. To make up some illustrative numbers, suppose you needed at most a circuit of depth 10 for a problem on a quantum computer no matter how many 0s and 1s you held in that many qubits for input. In the classical case you might need a circuit of depth 10 for 16 inputs, 20 for 32 inputs, 30 for 64 inputs, and so on for that same problem.

This conclusively shows that fault tolerant quantum computers will do some things better than classical computers can. It also gives us guidance in how to advance our current technology to take advantage of this as quickly as possible. The proof is the first demonstration of unconditional separation between quantum and classical algorithms, albeit in the special case of constant-depth computations. In practice, short-depth circuits are part of the implementations of algorithms, so this result does not specifically say how and where quantum computers might be better for particular business problems. That’s not really the point. As the paper says, “shallow quantum circuits are more powerful than their classical counterparts.” The IBM Q team is now working to show examples of this advantage via Qiskit in a Jupyter notebook.

Quantum computing will advance by the joint scientific research of physicists, material scientists, mathematicians, computer scientists and work in other disciplines and engineering. The mathematics underlying quantum computing is ultimately as important as the shiny cryostats we construct to hold our quantum devices. The scientific advancements at all levels need to be celebrated to show that quantum computing is real, serious, and on the right path to what we hope will be significant advantages in many application areas.

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**Gaining a Quantum Advantage**

Scientists Sergey Bravyi of IBM Research, David Gosset of the University of Waterloo’s Institute for Quantum Computing, and Robert König of the Institute for Advanced Study and Zentrum Mathematik, Technische Universität München, have published in *Science* as “Quantum advantage with shallow circuits.”

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