MULT90063 Introduction to Quantum Computing
Lecture 13 – Quantum Supremacy
11.1 Boson Sampling
11.2 IQP Problem
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11.3 Google’s pseudorandom circuits
Lecture 14 – Errors
12.1 Quantum errors: unitary and stochastic errors 12.2 Purity
12.3 Tomography
12.4 Randomized Benchmarking
Quantum Supremacy and Errors
MULT90063 Introduction to Quantum Computing
Errors in Quantum Devices
Physics 90045 Lecture 12
MULT90063 Introduction to Quantum Computing
Real quantum devices
MULT90063 Introduction to Quantum Computing
Two types of errors
Quantum computers are extremely fragile, and vulnerable to noise and errors. While errors occur in classical computing too, we’re accustomed to very low error rates – our hard drives rarely forget what they store.
Two types of error:
(1) Systematic unitary errors – eg. Control pulse error
(2) Random noise – eg. Decoherence
MULT90063 Introduction to Quantum Computing
Control Errors
Control of qubits requires high precision, and errors can sneak in. For example:
• Variations in magnetic fields across the sample, or variations in material properties.
• Stray electric fields, charge traps, strain.
• Applying a microwave pulse where the strength of the pulse is slightly too strong or too weak causes a systematic over-rotation or under-rotation.
• Cross-talk between gates.
• Unwanted interaction between qubits.
IBM image, Flickr
MULT90063 Introduction to Quantum Computing
Systematic Errors in the QUI
The QUI is effectively a pristine qubit environment, but we can introduce such effects systematically and investigate how quantum gate errors affect the output of quantum circuits.
We will consider rotation errors around the cartesian axes in the QUI using the R-gate. For example, a Z-rotation error (or just “Z-error”) is a gate dZ defined as:
Z ⌘ ✓ e i✏/2 0 ◆ 0 ei✏/2
where the level of error is governed by the angle e (assumed to be small). Similarly, we could consider small rotations around other axes:
X=RX(✏), Y =RY(✏), Z=RZ(✏)
In the lab, we will consider the effect of these errors on the success of quantum circuits.
MULT90063 Introduction to Quantum Computing
Decoherence
System (the qubit) Environment
In realistic quantum systems there will always be (unwanted) interaction between the qubits and the environment (electrons, spins, phonons, charge traps).
This causes a type of noise on the system (ie. qubits we want to protect) which we call decoherence.
MULT90063 Introduction to Quantum Computing
Modeling Stochastic Errors
Stochastic errors can be modeled in the QUI.
Dephasing noise: Apply Z gate with some probability p.
Depolarizing noise: Apply either X, Y or Z gates, each with with probability p/3.
Perform many “Monte Carlo” simulations, where errors are placed randomly on each run, and average the measurement results.
MULT90063 Introduction to Quantum Computing
Pure and Mixed States
Pure states have no errors, and are perfectly coherent. For example, the state
| i=|0i is a coherent, quantum state.
Mixed states may have errors, and are less coherent.
Imagine we took system in a pure quantum state, and noise – with say 20% probability- flipped the qubit around the X axis. That state would then be a “mixed” state.
MULT90063 Introduction to Quantum Computing
Superposition vs mixed states
Consider the pure state,
and a mixed state which is
|0i + |1i |+i = p2
Is it possible to tell these two states apart in experiment?
Consider what happens if we apply a Hadamard gate, then measure:
MULT90063 Introduction to Quantum Computing
Superposition vs mixed states
For the mixed state if |0iwere prepared : |0i + |1i
50% of the time we will measure 0 50% of the time we will measure 1
H |0i = p2
For the mixed state if |1iwere prepared :
|0i |1i H |1i = p2
So if the mixed state is prepared:
For the pure state: H |+i = |0i
50% of the time we will measure 0 50% of the time we will measure 1
50% of the time we will measure 0 50% of the time we will measure 1
so 100% of the time we will measure 0.
MULT90063 Introduction to Quantum Computing
Purity on the
|0i |1i p2
|0i + i|1i p2
|0i + |1i p2
Pure states lie on the surface of the
|0i i|1i p2
MULT90063 Introduction to Quantum Computing
Mixed states on the
|0i |1i p2
|0i i|1i p2
|0i + i|1i p2
Mixed states lie Sphere.
inside the Bloch
hXi |0i + |1i
The closer to the origin, the more mixed.
MULT90063 Introduction to Quantum Computing
Purity for one qubit
If the distance from the origin to the state is measured to be r, the purity is:
P = 1 + r2 2
Maximum purity of 1 for all pure states.
Minimum purity of 1⁄2 for a completely mixed state.
|0i |1i p2
hYi |0i + i|1i
|0i i|1i p2
|0i + |1i p2
Note: There’s a more technical definition of purity in terms of density matrices, which we won’t cover in this course.
MULT90063 Introduction to Quantum Computing
Reminder: Calculating expectation values
Example of calculating an expectation value for X,
hXi=h |X| i
= (a⇤ h0| + b⇤ h1|)X(a|0i + b|1i)
= a⇤a h0| X |0i + b⇤b h1| X |1i + a⇤b h0| X |1i + b⇤a h1| X |0i = a⇤b + b⇤a
MULT90063 Introduction to Quantum Computing
Example: 20% error
Eg. Consider a state |0i present in a noisy system. 20% of the time, a bit flip X has applied to it.
hXi = 0.80 h0| X |0i + 0.2 h1| X |1i =0+0
hYi=0.80h0|Y |0i+0.2h1|Y |1i =0+0
hZi = 0.80 h0| Z |0i + 0.2 h1| Z |1i = 0.8 0.2
MULT90063 Introduction to Quantum Computing
The purity
The purity of this mixed state is therefore:
P = 1 + r2 2
= 1+0.62 2
MULT90063 Introduction to Quantum Computing
Measuring purity in the QUI
Run many trials – on each trial choosing a different random set of errors. Measure
Measure
Then calculate the purity:
P = 1+hXi2 +hYi2 +hZi2 2
MULT90063 Introduction to Quantum Computing
Quantum State Tomography
“Tomography” is quantum computing jargon for measuring/determining the quantum state, as well as possible. For one qubit, this is just measuring:
hXi, hYi,hZi
For two qubits, we need to accurately measure correlations between the qubits
as well. We measure the 15 parameters:
hXXi, hXYi, hXZi, hXIi hYXi, hYYi, hYZi, hYIi hZXi, hZYi, hZZi, hZIi hIXi, hIY i, hIZi
Because of counting statistics the states can be unphysical (eg. radius greater than 1). However, these measurements can be used to estimate the closest (mixed) quantum state.
MULT90063 Introduction to Quantum Computing
Two qubit example
Measuring
m1 = ±1 m2 = ±1
Find the product of these, m = m1m2
Average over many runs of the experiment, with different locations/errors on each run to determine
MULT90063 Introduction to Quantum Computing
Randomized Benchmarking
How good are our gates individual gate? We want a number for how much error doing each operation is. One way of determining this is to perform randomized benchmarking.
Apply a random sequence of gates
Make a measurement: Should measure 0
Apply the inverse of the whole sequence
MULT90063 Introduction to Quantum Computing
The Clifford Gates (for one qubit)
|0i … Apply a random sequence of gates
Typically this random sequence is chosen from a small gate set. One common choice is called the Clifford gates: These gates only rotate between the states which like along +/- x, y and z axes.
|0i |1i p2
hY |0i + i|1i
|0i i|1i p2
|0i+|1i |0i |1i |0i+i|1i |0i i|1i |0i,|1i, p2 , p2 , p2 , p2
All of the preset gates except T are Clifford: X, Y, Z, S.
MULT90063 Introduction to Quantum Computing
Randomized Benchmarking
Apply the inverse of the
preceding sequence
At the last step we apply the inverse of the preceding sequence. So if we started in the state
We should also end up in that state. If there were no errors, should measure 0, with certainty. However, with errors, the fidelity of the final measurement drops.
MULT90063 Introduction to Quantum Computing
The Clifford Gates (for one qubit)
Repeat sequences of the same length many times. For each length of sequence, average over many runs of the sequences, to work out the probability of measuring the correct result.
We can plot the fidelity (ie. the probability of getting the correct answer) against the length of sequence.
MULT90063 Introduction to Quantum Computing
Example Plot
Fit this curve with:
where Fm is the average fidelity after m steps, and A and B and f are determined by the fit.
Taken from Knill et al, PRA, 2007.
MULT90063 Introduction to Quantum Computing
Randomized Benchmarking Sequence
We have found some number f < 1 which we can then relate to the average fidelity of a gate. If the dimension of the system is d,
f = dFav 1 d 1
In our case d=2, so the average fidelity is
Fav = f + 1 2
MULT90063 Introduction to Quantum Computing
Different Randomized Benchmarking sequences
For two qubit gates, the set can also include gates such as CZ or CNOT. A similar method (with a different d) then can be used to determine the average fidelity of two-qubit gates.
|0i ... |0i ...
MULT90063 Introduction to Quantum Computing
Interleaved Randomized Benchmarking
To determine the fidelity of an individual gate, interleave it throughout the randomized benchmarking. Eg. X
This gives an indication of the error in individual gates.
More advanced schemes also exist, eg. Adaptive versions of randomized benchmarking pinpoint where and what type of errors are occurring, rather than just giving a single number of average error per gate.
MULT90063 Introduction to Quantum Computing
Quantum Process Tomography
Just as we can do tomography to determine a (mixed) quantum state, in principle we can measure what happens in a quantum process. Technically we are determining a completely positive (CP) map.
General strategy for one qubit
For each possible input states:
|0i+|1i |0i |1i |0i+i|1i |0i i|1i |0i,|1i, p2 , p2 , p2 , p2
Act the operation, U, on each input states
Do complete state tomography on each output (ie.
Similar process for multiple qubits – QPT requires many measurements!
MULT90063 Introduction to Quantum Computing
Lecture 13 – Quantum Supremacy
11.1 Boson Sampling
11.2 IQP Problem
11.3 Google’s pseudorandom circuits
Lecture 14 – Errors
12.1 Quantum errors: unitary and stochastic errors 12.2 Purity
12.3 Tomography
12.4 Randomized Benchmarking
Quantum Supremacy and Errors
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