Noise in Quantum Computers (part 2)
In this lecture, Zlatko continues on with the topic of noise in quantum computation, focusing on sampling, SPAM and incoherent errors. Zlatko discusses quantum measurement theory through an example of von-Neumann measurement and the effects of sampling measurement outcomes. He relates these to how measurement, more specifically measurement and sampling errors affect algorithmic performance on quantum computers. He then continues onwards to SPAM (State preparation and measurement) errors and discuss how they affect circuit performance. Finally, he finishes off the lecture with a brief treatment of incoherent errors.
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FAQ
What exactly do the gates manipulate on the qubit when applying the gates? In the slides we represent that manipulation as the arrow moving, but in the physical world, what does that translate to?
The black arrow that Zlatko showed in some of the pictures (See, slides: 16-23) represents the direction along which the gate is acting, in other words the normal to the plan of rotation—it’s the vector associated with the rotation that is caused. The colored arrows that he showed depicting the trajectory of the state factor along the surface of the blocks fear or depictions of the evolution of the wavefunction as a function of time under the application of the gate, considered not as an instantaneous, but it’s a real finite in time evolution. We idealize the gates to be instantaneous in a quantum algorithm, but in practice they always have some finite time, and the state doesn’t just jump, it must take a continuous pass through Hilbert space.
Is it the coherent noise caused by the constant error in X gate? For example, If there are both positive and negative errors in X gate, errors cancel out the noise in total. So What causes the error in each X gate?
We assumed here that the noise was deterministic, so that each time you apply in the gate, you get the noise to be the same, not different. This means that because of the coherent nature of the noise, each application of the noise builds up. To first order quadratic in the worst case error. If the noise fluctuated, then you would get a slower build up, because each time you played the gate you would sometimes move the left sometimes to the right, you would still drift on average, but slower.
If the error is average +2°, for example: can we do an R_x(180°-2°) like correction, the application of a rotation gate would have an associated error with itself. In the lecture Zlatko said “a measurement of error would allow us to correct it”. Is the correction perhaps done in some manner other than the application of another gate?
Right so the gate you would apply is the noisy \tilde{X} = R_x(182°), you then dial back the amplitude on the pulse by roughly a fraction 2/182 = 0.011, and check the gate with this error amplifying sequences. Then your gate should be roughly \tilde{X} = R_x(182° * 180/182) = R_x(180° )
"The measurement correlation in the Bell State is stronger then could ever be in a classical system" - what does 'stronger' correlation mean? Can we quantify correlation?
Check out bell violation inequalities.
Are all projective measurements, von Neumann measurements? Across von Neumann measurements mostly when reading a bit on open systems, Is there any key points or assumptions with both?.
Von Neumann measurements are the most restrictive class of measurements. Usually projective and von Neumann mean the same thing. However, more subtly some of us are more careful to distinguish the term ‘von Neumann measurement’ from the the term ‘projective measurements’, because von Neumann got it wrong for rank > 1 projectors. Imagine an operator with degenerate eigenvalues.
In the lecture Zlatko showed “a model of a noisy gate with a bit-flit error” (see image below). Is this a gate that is a type of “mixture” of two smaller circuits that randomly outputs zeros or ones?
That's right. It's a mixture of different circuits in the following way: When you run the circuit you don't know which for the two you will actually run. The top line (with I) or the bottom line (with X). You can't tell the difference. So you have to average over them. It's a lack of control and knowledge.
Under which conditions, is calculated the expectation of an operator using Trace?
When are using density matrices.
Other resources
Read about Postulates of Quantum Mechanics (including measurement postulate) in Quantum Computation and Quantum Information by Isaac Chuang and Michael Nielsen, Section 2.2 (Quantum measurement specifically - 2.23 - 2.2.6)
Read about Incoherent noise in Quantum Computation and Quantum Information by Isaac Chuang and Michael Nielsen in Sections 8.1, 8.2, 8.3, 11.1
