Ch. 5.4 (cont.) Irrelevance of Phase

5.4.8: Global Phase and Its Irrelevance

Recall:

If we write $a,b$ in polar form:

Rearranging:

Calculating $|\psi {|}^2$ makes the $e^{i{\theta }_1}$ part go away as it's magnitude is just 1. This factor is the global phase and doesn't affect quantum computation. The quantity $({\theta }_2-{\theta }_1)$ is the relative phase and is observable. To see why, consider:

Applying $X$ we get:

Thus a global phase of $-1$ is added. But this phase is not observable.

Now instead consider the QC:

We begin with 0's at initialization:

Apply $H$-gate to q[0] and an $X$ gate to q[1]

Apply $H$ gate to q[1]:

Here transferring the global phase occurring in a qubit into a relative phase of another qubit using a controlled gate is called phase kickback (or just kickback).

5.4.8.2: Sequence of Gates

We can combine sequences of gates to get a definition of the one, unified gate. For instance, for $HXH$ you can do the math and the following identities:

These identities means we can make composite gates, which transform Pauli operators into other Pauli operators, forming the basis of universality.

5.4.8.3: Universality

In classical computing, we have a fundamental set of building blocks (gates) that compose all possible boolean functions we have. Choosing the two fundamental ones from OR and NOT are not good choices since they alone cannot make something like an XOR. Instead, using NAND and NOR are better as they are complete by themselves and can construct all other logic gates. Hence NAND and NORE are the universal logic gates in the classical world.

We can similarly define universal quantum gates, where any Quantum Computer can reduce these gates to a fundamental combination of ones from this unique set. The problem is that there's so many gates that it may be impossible to actually implement universality using a finite number of them, so we can use approximations. The Solovay-Kitaev Theorem says that approximations to any desired gate can be created using a fairly short sequence of gates from the generating set. In math terms, unitary $U^{\prime }$ simulates the required unitary $U$ to an accuracy within $\epsilon$ if $\Vert U-U^{\prime }\Vert \le \epsilon$.

A common universal set of quantum gates are the Clifford Group and the $T$-gates.

5.4.9: Circuit Optimization

5.4.9.1: Hardware Topology

From Reading Week 5 - Irrelevant Global Phase#5.4.8.2 Sequence of Gates we learned that a sequence of gates reduces to single-qubit operators, so there is some scope for circuit level optimizations. This is required to run some circuits on NISQ processors, due to physical constraints. They have a hardware topology with a limited set of basis gates and qubit connections. Arbitrary Quantum Circuits need to be converted to sequences of these $G$ gates.

Consider constraints to just having access to CNOT and SWAP operations. The imbq_athens has a topology such as:

Notice:

• The lines between the numbers are qubits and which gates are allowed (2 qubit gates only work on one edge)
• Even though physically we can't use qubit gates between unconnected qubits, we can make multi-qubit gates that do essentially what we want out of smaller "legal" operations.

The imbq_athens has a list of basis gates for it:

[’id’, ’rz’, ’sx’, ’x’, ’cx’, ’reset’]

These are hardware gates because they are implemented at the hardware level. On the contrast, gates like $Z$, $T$, ... are not performed by hardware. Instead, these are implemented as virtual gates (also known as frame changes, or free gates) by making them as compositions from the hardware gates.

Running:

qc_basis = qc.decompose()
qc_basis.draw(output='mpl')

Gives us a difference between our original qc and the one that can be made for the imbq_athens:

Ch 4.1: Review of QC Vocab

Ch 4.2: Classical Computers - A Recap

Digital bits are 0 or 1 exclusive. Qubits are a superposition of these two states.

Ch 4.3: Qubits and Usability

Many types of qubit devices are in the works:

• Nuclear Magnetic Resonance devices
• Valency centers in diamonds
• Photonics
• Trapped ion devices
• Superconducting transmons
• Quantum Dots
• Topological qubits

Digital computers are made on semiconductor material. QC's are made differently based on:

• Coherence time (longer)
• High gate fidelity
• Scalability
• Quantum error correction
• Qubit coupling
• Possible to be implemented commercially

Ch 4.4: Noisy Intermediate Scale Quantum Technology

Noisy Intermediate Scale Quantum Technology denotes the current quantum computing era. While some algorithms are able to run, they are sensitive to quantum errors. This limits their usability.

According to Prof. John Preskill:

“Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50–100 qubits may be able to perform tasks which surpass the capabilities of today’s classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away—we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.”

Ch 4.5: Qubit Metrics

4.5.1: Spin or Energy Relaxation Time $T_1$

This measures the characteristic time for the qubit to return to ground state from an exited state due to spin-environment coupling.

To measure:

1. Put the state as $|0⟩$
2. Apply an $X$-transform and wait time $t$ till we measure $|1⟩$

4.5.2: Dephasing Time $T_{\varnothing }$

This is the time for the bloch vector to diffuse to the equator.

4.5.3: Decoherence Time $T_2$

Assume a qubit at $|\curvearrowright ⟩=\frac{|0⟩+|1⟩}{\sqrt{2}}$. When the qubit is not interacting with the environment, it undergoes constant procession at a frequency $\omega$, the qubit frequency at an azimuthal angle $\varnothing$. Via the environment, the Bloch Vector start to decohere:

To get to 4.7.b (total decoherence), is the $T_{\varnothing }$ or the Decoherence Time. Some experiments that help measure the decoherence time are:

• Hann Echo
• Ramsay Experiment
  which we cover below.

4.5.4: Hann Echo $T_2$

1. Prepare the qubit to the $|\curvearrowright ⟩$ state described prior via a Hadamard Gate
2. Wait half the time $\frac{t}{2}$
3. Apply $X$ and wait for $\approx \frac{t}{2}$ again
4. Apply $H$ again and measure the probability of it in state $|0⟩$

The time $t$ here is the time of coherence.

4.5.5: Ramsay Experiment $T_2^{\ast }$

1. Prepare $|\curvearrowright ⟩$ like prior
2. Wait time $t$
3. Apply $H$ again and measure the probability of being in state $|0⟩$

4.5.6: Gate Time

The number of gate operations one can perform before the qubit completely decoheres and loses its usable lifetime. The gate time is:

for leading qubit modalities.