Getting Physical

The non-physicist finds it hard to believe that really the ordinary laws of physics, which he regards as the prototype of inviolable precision, should be based on the statistical tendency of matter to go over into disorder

Erwin Schrödinger

It’s time to discuss some considerations about how we go from theoretical mathematics and physics to the applied and experimental.

The qubits we make and use in the lab for research and those we will deploy for commercial applications involve physical hardware devices such as photonic and microwave controllers. They are subject to noise from the environment, their electronic components, and artifacts from manufacturing choices. Hardware improvements decrease the disturbances, but software and system ones can too. The long-term goal is to have fully fault-tolerant, error-corrected quantum computing devices.

This chapter concludes with a discussion of light and photons to illustrate how properties like...

Topics covered in this chapter

1. 11.1. That’s not logical
2. 11.2. What does it take to be a qubit?
3. 11.3. Quantum cores and interconnects
4. 11.4. Decoherence
   1. 11.4.1. T1
   2. 11.4.2. T2 and T2*
   3. 11.4.3. Pure versus mixed states
5. 11.5. Error correction for physical qubits
   1. 11.5.1. Correcting bit flips
   2. 11.5.2. Correcting sign flips
   3. 11.5.3. The 9-qubit Shor code
   4. 11.5.4. Considerations for general fault tolerance
6. 11.6. Quantum benchmarks
   1. 11.6.1. Performance factors
   2. 11.6.2. Performance benchmarks
   3. 11.6.3. Volumetric benchmarks
   4. 11.6.4. Algorithmic and application benchmarks
7. 11.7. The software stack and access
8. 11.8. Simulation
   1. 11...

11.1 That’s not logical

The qubits such as those in the last two chapters are examples of logical qubits. We can use them indefinitely, they never lose state when idle, and we can apply as many gates to them as we wish without errors. qubit$logical logical qubit

When you build a quantum computer, the actual physical implementations of qubits aren’t as perfect as logical qubits. Such a qubit, called a physical qubit, starts to lose its ability to hold onto a state after its coherence time. We also say that the qubit is decohering. qubit$physical qubit$decoherence decoherence qubit$coherence time coherence time

It’s a goal of quantum computing researchers and engineers to delay the decay of a physical qubit’s quantum state as long as possible. Since the decay is inevitable, a goal of fault tolerance and error correction is to handle and fix the effects of the qubits’ decoherence throughout the execution of a circuit.

Is...

11.2 What does it take to be a qubit?

In his 2000 paper “The Physical Implementation of Quantum Computation,” David P. DiVincenzo laid out five “requirements for the implementation of quantum computation.” 71 DiVincenzo, David P.

In his words, they are:

1. A scalable physical system with well characterized qubits
2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000…⟩
3. Long relevant decoherence times, much longer than the gate operation time
4. A “universal” set of quantum gates
5. A qubit-specific measurement capability

Let’s discuss what each of these means, following his lead.

Scalable physical system

In the physical system we manufacture for quantum computing, we need to create a qubit with two clearly delineated states, |0⟩ and |1⟩. Other states may be possible...

11.3 Quantum cores and interconnects

However you build or trap objects that behave like physical qubits, there is a maximum number of qubits that you can put in a device and reliably control. Even if you could put more, there may be other factors that put an upper bound on the qubit capacity. Larger semiconductor chips, for example, are difficult to manufacture, and any flaws could spoil the chips and decrease yield. quantum$core

I call such a unit with a fixed number of qubits a quantum core. These units are analogous to the multiple cores in classical processing units. For example, my desktop computer has an AMD processor with 12 cores. Figure 11.2 is a diagram that helps us visualize a quantum core with 16 qubits.

Figure 11.2: A single quantum core with 16 qubits

I chose 16 as a convenient number of qubits to illustrate the concepts in this section. Industry vendors have demonstrated cores with several dozen to several...

11.4 Decoherence

There are three measurements that quantum computing researchers frequently use to measure coherence time: T₁, T₂, and its cousin T₂^*. They are single physical qubit measurements, so we can use the Bloch sphere to discuss them. Their use goes back to Felix Bloch’s work on nuclear magnetic resonance (NMR) in the 1940s. 19 Bloch, Felix

Let’s begin with T₁.

11.4.1 T₁

T₁ goes under several names, all of them connected to the physics of various underlying quantum processes:

• relaxation time,
• thermal relaxation,
• longitudinal relaxation,
• spontaneous emission time,
• amplitude damping, and
• longitudinal coherence time.

It is related to the energy loss as the quantum state decays from the higher energy |1⟩ state to the |0⟩ ground state. This energy is transmitted to, or leaked into, the environment and lost...

11.5 Error correction for physical qubits

In section 2.1 and section 6.4, we looked at some basic ideas around classical repetition codes. If you want to send information, transmit multiple copies of it. If enough copies get through unscathed, the recipient can determine what you sent. error correction$quantum error correction$repetition code

For the quantum situation, the No-Cloning Theorem (section 9.3.4) says that we can’t copy the state of a qubit, so traditional repetition is not available. No-Cloning Theorem QEC

We can do entanglement, and it turns out that this is powerful enough when combined with aspects of traditional error correction to give us quantum error correction, or QEC.

How can we go from |ψ⟩ = a|0⟩ + b|1⟩ to a|000⟩ + b|111⟩? As you start thinking about such questions, there are two good starting points: “Would applying an H change the situation into something I know how...

11.6 Quantum benchmarks

How powerful is a gate- and circuit-based quantum computer? How much progress are we making with one qubit modality versus another? What does that mean when we say we can find a solution on a “powerful enough” quantum computer? When will we know we have arrived? benchmark

Benchmarks help us compare performance within a particular class of processors or across processors from competitive providers. For example, we can measure the coherence times among superconducting qubits from competitors or compare 2-qubit gate fidelities among all types of quantum computers to which we have access.

While knowing how well a given qubit is doing with coherence times and error rates is undoubtedly useful, it tells you nothing about the overall system and how well the components work together. You may have one or two spectacular, connected, and low-error-rate qubits, but other aspects of your system may make it unusable for executing practical...

11.7 The software stack and access

One way of accessing a quantum computing system looks like this: Jupyter notebook software stack

1. You download and install software development tools such as the Cirq or Qiskit open-source quantum computing framework to your laptop or workstation. 42 181
2. You develop your quantum code in a programmer’s editor or a Jupyter^TM notebook. 174
3. When run, part of your application connects to a quantum simulator on your computer or remotely to a simulator or quantum hardware.
4. The remote connection is via the internet/cloud.
5. Your application invokes one or more processes that run on the quantum hardware or a simulator.
6. Ultimately, your application uses the results of the quantum computation and does something valuable within your use case.

There are at least two other similar scenarios:

• Instead of developing locally, you use a web browser-based...

11.8 Simulation

Is it possible to simulate a quantum computer on a classical computer? If we could do it, “quantum computing” would be only another technique for coding software on our current machines. simulation simulation$state vector

In this section, we look at what you must consider if you want to write a simulator for manipulating logical qubits. In particular, we look at state vector simulation. In section 11.8.6, we look at simulation using density matrices.

Simulation is good for experimentation, education, and debugging part of a quantum circuit. If you have a simulator handy, such as one that Qiskit or Cirq provides, you can use it for small problems.

We look at how you might build a simulator in general terms. I offer no complete code in any specific programming language but more of a list of what you need to consider.

11.8.1 Qubits

When thinking about building a quantum computing circuit simulator...

11.9 Light and photons

Light illuminates the things around us. It can be as dim and small as a faraway star on a clear night or as harsh and bright as the sun or the output of welding equipment. Understanding the nature of light was a central research direction in physics in the nineteenth and early twentieth centuries. light

The answers were far more complicated than anyone imagined, gave birth to quantum mechanics, and involved the electromagnetic spectrum well beyond visible light.

11.9.1 Photons

Does light behave like a wave, with amplitude A (height) and wavelength λ, as in Figure 11.24? λ is the distance between two wave crests or other corresponding points. light$wavelength λ`italic

Figure 11.24: Metrics associated with waves

Or does light behave like a particle with a well-defined shape, shooting off in various directions, as in Figure 11.25? Can particles have...

11.10 Summary

In this chapter, we connected the “logical” idea of qubits and circuits with the “physical” reality of how you might build a quantum computer.

Physical qubits don’t survive forever, and decoherence explains the several ways in which quantum states wander over time. While we do not yet have large enough systems to implement fault tolerance, we examined what error correction might be able to do in the quantum realm.

We need to be able to measure the progress we are making and answer questions such as “How powerful is your quantum computer?”. Our discussion of benchmarks looked at many factors that can affect performance.

We mentioned several technologies that researchers are exploring to implement qubits. According to their proponent scientists and engineers, superconducting transmon qubits, cold atoms, ion traps, and photonic techniques all hold promise.

We looked at the polarization...