mod04lec25 - Fixing quantum errors with quantum tricks: A brief introduction to QEC - Part 3
Summary
TLDRThis script delves into quantum error correction, focusing on the three-qubit code's ability to detect and correct single qubit bit flip errors without collapsing the quantum state. It explains the use of ancillary qubits and the concept of syndrome bits for error diagnosis. The script outlines the complete circuit for encoding, error detection, and correction, emphasizing the importance of fidelity in quantum computation. It also touches on the physical resource requirements for fault-tolerant quantum processors, highlighting the progression from noisy intermediate-scale quantum devices to scalable, error-corrected quantum computing.
Takeaways
- 🧠 The discussion focuses on quantum error detection and correction, specifically for single qubit errors on a three-qubit state.
- 🔍 Error detection is crucial for quantum computing as it allows for the identification of errors without collapsing the quantum state.
- 📊 A majority voting or parity check mechanism is used to detect errors, which involves measuring in the 0-1 basis on ancillary qubits.
- 🛠️ The error detection circuit includes three encoded qubits and two ancillary qubits, with the latter initially set to zero.
- 🔄 The CNOT gate is used in the circuit to maintain the superposition of the state, aiding in the error detection process.
- 📉 Syndrome bits, derived from measurements on the ancillary qubits, indicate the presence and location of errors.
- 🔧 Error correction involves applying an X gate to the affected qubit based on the syndrome bits, effectively reversing the error.
- 🔑 The three-qubit code can detect and correct for single qubit bit flip noise, improving the fidelity of quantum states.
- 💾 Quantum error correction requires physical resources, with at least 49 physical qubits needed for one error-corrected logical qubit using a seven-qubit code.
- 📈 The current era of quantum computing involves noisy intermediate-scale quantum devices, with the goal of scaling up to fault-tolerant, error-corrected quantum computation.
Q & A
What is the primary goal of quantum error detection?
-The primary goal of quantum error detection is to identify the presence of errors without collapsing the quantum state, allowing for the extraction of information about the error's presence and location to reverse its action.
Why is it important to know the location of an error in quantum error detection?
-Knowing the location of an error is crucial to reverse the error's action effectively. Without this information, it would be impossible to correct the error accurately.
What role do ancilla qubits play in quantum error detection?
-Ancilla qubits are used in the error detection process to help identify errors without directly measuring the data qubits. They are initially set to zero and are used in conjunction with CNOT gates to detect errors through parity checks.
How are syndrome bits generated in the error detection process?
-Syndrome bits are generated through measurements of the ancilla qubits in the Z basis. These classical bits act as parity check bits, indicating which qubit was affected by a bit flip error.
What is the significance of the syndrome table in quantum error detection?
-The syndrome table is significant as it maps the outcomes of the syndrome bits to specific error patterns, allowing for the identification of the type of error and the qubit it occurred on, which is essential for error correction.
How does the three-qubit code correct for single qubit errors?
-The three-qubit code corrects for single qubit errors by using a syndrome table that corresponds to different error operators. Depending on the syndrome, an X gate is applied to the appropriate qubit to correct the error.
What is the relationship between fidelity and quantum error correction?
-Quantum error correction can improve fidelity, which is a measure of how close the final state is to the initial state. Higher fidelity implies better preservation of the quantum state against noise.
What is the minimum number of physical qubits required for one error-corrected logical qubit?
-At least 49 physical qubits are required for one error-corrected logical qubit when using a seven-qubit error-correcting code, which is more amenable to fault tolerance techniques.
What is the current state of quantum devices in terms of qubit count and error rates?
-As of the script's knowledge, we are in an era of noisy intermediate-scale quantum devices with around a hundred qubits, but with error rates still above the threshold required for error-resilient computation.
What is the significance of the quantum volume metric in the context of quantum computing?
-Quantum volume is a metric that combines the size of the circuit, the connectivity of the qubits, and other factors to indicate the power of a quantum computer. It is used to gauge progress beyond simple qubit count, aiming to reflect the ability to perform complex quantum computations.
Outlines
🛠️ Quantum Error Detection and Three Qubit States
The paragraph discusses the process of detecting quantum errors without collapsing the quantum state. It emphasizes the importance of identifying the location of an error to reverse its effects. The speaker introduces six three-qubit states and explains that error detection involves measuring in the 0-1 basis, or the Z basis, using ancillary qubits. The error detection circuit is described, involving pairs of CNOT gates from data qubits to ancillary qubits, which are initially set to zero. The speaker explains that the CNOT gates do not alter the superposition of the state, and that the measurement outcomes, referred to as syndrome bits, indicate the presence and location of errors.
🔍 Analyzing the Error Detection Circuit
This section delves into the analysis of the error detection circuit. The speaker hypothesizes a bit flip error on the first qubit and explains how this error would affect the parity of the qubits. The role of CNOT gates in detecting such errors is highlighted, showing how they trigger bit flips in ancillary qubits based on the parity of the data qubits. A syndrome chart or table is introduced as a method to map the classical outcomes of measurements to specific errors. The speaker outlines how different syndrome bit values correspond to errors in different qubits and explains the correction process, which involves applying an X gate to the affected qubit.
🔗 Completing the Quantum Error Correction Circuit
The speaker outlines the complete process of quantum error correction, starting from encoding the qubits to detecting and correcting errors. The process involves starting with an arbitrary single qubit state, applying CNOT gates for encoding, and then introducing noise through a bit flip channel. The noisy qubits are then subjected to the error detection and correction process using additional ancillary qubits and CNOT gates. The outcomes of measurements on these ancillary qubits guide the application of X gates for error correction. The paragraph concludes by emphasizing that this process corrects for single qubit errors, improving the fidelity of quantum computations.
📈 Quantum Error Correction and Fidelity
This paragraph discusses the broader implications of quantum error correction, particularly its role in enhancing the fidelity of quantum computations. The speaker notes that the three qubit code can correct for single qubit bit flip noise, improving coherence time provided the noise parameter is small. The discussion extends to different types of errors, such as phase flip and amplitude damping, and how they can be corrected using extended codes like the nine qubit code. The speaker also introduces the concept of fault tolerance and concatenation, which are techniques used to further reduce error rates and achieve a fault-tolerant quantum processor.
🚀 Roadmap to Fault-Tolerant Quantum Computation
The final paragraph provides an overview of the current state and future prospects of quantum computing. The speaker references the Google roadmap from 2018, highlighting the transition from noisy intermediate scale quantum (NISQ) devices to fault-tolerant quantum computation. The speaker discusses the progress in terms of quantum volume, a metric that captures the size and error resilience of quantum devices. The goal is to scale up to thousands of qubits to achieve fault tolerance and robust, scalable quantum devices. The speaker concludes by emphasizing the ongoing efforts to improve quantum volume and the anticipation of achieving higher quantum volumes in the coming years.
Mindmap
Keywords
💡Error Detection
💡Qubit
💡Encoded State
💡Ancillary Qubits
💡Bit Flip Error
💡Parity Check
💡Syndrome Bits
💡Error Correction
💡Fidelity
💡Fault Tolerance
Highlights
Introduction to error detection in quantum computing without collapsing the state
Explanation of the need for error detection to know the location of errors for reversal
Discussion on the use of majority voting or parity check in three qubit states for error detection
Emphasis on measuring in the 0 1 basis for detecting X errors in quantum states
Description of the role of ancillar qubits in error detection circuits
Circuit diagram explanation for error detection using ancillar qubits
Analysis of how the circuit handles different types of bit flip errors
Introduction to syndrome bits as classical outcomes of measurements for error detection
Construction of a syndrome table to map error detection to specific qubit errors
Explanation of how to correct errors by applying an X gate based on syndrome results
Overview of the complete circuit for encoding, detecting, and correcting errors in a three qubit code
Discussion on the limitations of error correction to single qubit errors
Importance of the noise parameter being small for effective error correction
Introduction to the concept of fault tolerance in quantum computing
Exploration of the physical resource requirements for implementing quantum error correction
Discussion on the transition from noisy intermediate scale quantum devices to fault-tolerant quantum computation
Presentation of the Google and IBM roadmaps for scaling up quantum devices
Conclusion on the current state and future prospects of quantum error correction
Transcripts
[Music]
so the next step is to look at error
detection
we have looked at
the action of the
single qubit errors
on the encoded state or the three qubit
state
so the next step is to ask how do we
detect these errors in a way that the
state doesn't collapse but still we are
able to extract information about
the presence of the error the point is
we need to know where the error occurred
in order to be able to
reverse the action of the error right so
how do we know where the error occurred
now look at these six
three qubit states
you notice that once again there is a
majority voting or a parity check
element that comes in right so
but now notice that the measurement that
i need to do as far as the quantum
states are concerned is actually a
measurement in the 0 1 basis so this is
the first thing to note that if i want
to detect x errors
note
that
we must measure
in the so called z basis
or
the 0 1 basis so we must measure measure
each qubit
in the
0 1 basis
and like i said the key point is that
the measurement is not done on the
data qubits itself uh on the encoded on
the three encoded qubits but rather on
two additional qubits called the
ancillar qubits so here is what the
error detection circuit looks like
so we have the three
encoded qubits and then we have two
additional lines
to denote the ancillar qubits
and these ancillar qubits are initially
set to
zero
okay
so here what we have is some arbitrary
superposition of the form
alpha zero zero
plus beta 1 1
1
right so this is what the circuit looks
like it basically entails
pairs of c naughts i'll describe the
circuit first and then we'll see how
it achieves our goal of error detection
so i have a pair of c naughts from the
first two qubits
to the first ancillar qubit and then
i have another pair of c naughts to the
second ancillar qubit coming from the
second and the third
data qubit so essentially the first two
data qubits act as control for the first
two c naughts and the second two data
qubits sorry
act as control
for the
second two c naughts
two c naughts on the first ansel are
qubit
with the first two data qubits as
controlled and two c naughts on the
second ancillar qubit with the second
and third data qubits as control now
notice that the c naught being a two
qubit unitary gate does nothing to the
superposition so this state continues to
remain
as
um actually i should not say 0 0 plus
beta 1 1 1 this could be some noisy
version of 0 0 0 plus beta 1 1 so this
could be a state with say a bit flip
that has occurred in the second qubit or
a big flip that has occurred in the
third qubit and so on right so this
state remains whatever it is
right so let me not write this as an
as an
error free state but some
encoded state so i will say encoded
noisy state
right
and this state continues to remain what
it is so if i call this
maybe psi
logical to denote that this is an
encoded state this continues to remain
psi logic
now what we do with the two ancillar
qubits well we perform a measurement
we perform two measurements both of them
in the z basis
okay
and this leads to
some classical bits right so let me call
this as
s1 and s2
classical outcomes of these two
measurements and these classical
outcomes
are called syndrome bits
because these classical syndrome bits
are essentially the parity check bits
which tell us exactly which error
occurred or rather which qubit was
affected by the bit flip error
so let's analyze the circuit now suppose
there was a bit flip error on the first
qubit right so let me mark this
let me say that there was an error on
the
first qubit right
so then what happens
uh
well let's go back here right
and see
so if there was an error on the first
qubit you notice that the 0 0 0 has
become 1 0 0 and the 1 1 one has become
a zero one one right so this is the
possibility that we are talking about
this one right
so in this case you notice that the
parity of the first two qubits is odd
right but the parity of the second two
qubits is even
so
that's what these c naughts are doing
for us so if either of the first or the
second qubit
has undergone a bit flip
then one of these c naughts gets
triggered one of these two c naughts of
the on the first two qubits gets
triggered and correspondingly the first
ancillar qubit undergoes a bit flip so
this
then undergoes a bit flip
similarly if one of the second two
qubits underwent a bit flip then the
second ancillar qubit will undergo a bit
flip
so either the first ancillar qubit or
the second ancillar qubit undergoes a
bit flip depending on whether the first
two qubits underwent a bit flip or the
second two qubits underwent a bit flame
and so we can actually make a table
right so we can do this where we can
make what is called a syndrome chart or
a syndrome table
right
where
we write down the values of the
classical outcomes s1 and s2 so s1 and
s2 are both
bits right so they cannot be 0 or 1. so
what happens if s1 and s2 are both 0
well this implies that neither of the
two ancillar qubits got flipped right
because both continue to be zero
and this can only happen if
both pairs of qubits the first two
qubits and the second two data qubits
continue to have even parity which means
even number of zeros and ones
right and this can only happen in the
case that there was no noise so if both
syndrome bits
are zero then the diagnosis
is that there are there was no
error right
this is the case where all three of them
simply got acted by the identity
operator
now if the first syndrome bit
has flipped but the second one has not
what it means is that there was an error
only somewhere in the first two qubits
but none in the second pair of qubits
and so the error must be on the first
qubit so this means
that there was a bit flip
on
the first qubit
so this is basically the error operator
x i i
right and once you detect this what do
you do you simply reverse for it by now
acting an x gate on the first
data qubit of the first encoded qubit
similarly if the syndrome was 0 1 then
that implies a bit flip
on the
third qubit
which is corresponding to the error
operator iix
why is this because this means that
there was no error on the first two
qubits there was an error on the second
pair of qubits and that's why the second
ancillar qubit has gotten flipped so
this again means that the second qubit
is unaffected but is the third qubit
that is affected and finally you could
have the possibility that both syndrome
bits
are one which means the outcomes of both
the z measurements at the end on the
ancillars are one in which case there
must have been a bit flip
on
the first pair and the second pair which
then means a bit flip on the second
qubit alone right
and this corresponds to the error
operator i x i
so while i am writing correspondingly
the error operator here note that this
is also the recovery operator in the
sense that if there is no error
then
what is the correction procedure involve
nothing you just do identities do
nothing right so this last column what
i'm writing here is actually in some
sense the
recovery or the correction operation
right
so
if there's a bit flip on the first qubit
you simply apply an x on the first qubit
if there's a big flip on the third qubit
you apply a x on the third qubit and
there's a bit flip on the second qubit
you simply apply an x on the second
cubit right so this syndrome table
essentially captures
the fact that the different single qubit
errors
can be
distinctly identified and they can be
fixed by the simple application of an x
gate on one of the three encoded qubits
or one of the three data qubits
depending on
what the syndrome table tells us right
so
this
summarizes the error detection and the
error correction procedure
or the decoding procedure for the three
qubit code
so finally we can sort of
complete the circuit right we can we can
put together the encoding the detection
and the correction right so we can draw
a complete circuit so let me just
quickly draw that finally so
the three qubit
code
this is the complete
circuit so i first have these three
qubits which are going to become my
encoded qubits
i start with an arbitrary single qubit
state which is alpha 0 plus beta 1.
i do two c naughts
two control not gates
these are two ancillars remember which
were set to zero
right
so i do two c naughts right and i assume
so far
that this process where i have prepared
where where the initial qubit state is
in alpha zero plus beta one i have
prepared the two ancillar qubits in get
zero and i have applied two c naught
gates i assume that this process has
happened somehow in a perfect way we'll
address this
in the very last
few minutes of this lecture
but it's as if no error has happened
until now and at this point we say that
the qubits are now affected by
error
okay
so now we say
you have the
bit flip channel or the bit flip noise
that is affecting the qubits remember we
only
aim to correct detect and correct for
the single qubit bit flip errors so now
we have the noisy qubits right so so far
we have perfect qubits is the assumption
somehow
okay
and now we have the noisy qubit so this
part
you have the noisy encoded qubits
what do we do with them well we now have
to detect and correct for the errors so
we add these two additional ancillar
qubits which are going to help us in
error detection and correction
and now we have all these c naughts c
naughts from the first
and the second
which act on the first ancillar qubit c
naught from the second
and the third
with the second ancillar qubit as target
so c naught on the first ancillar qubit
have
the first two data qubits is controlled
c naught's on the second
answer like cubit have second and third
data qubits is control and then we do
measurement
and depending on the outcome of this
measurement
and i draw two lines here to denote that
this is now classical bits
so depending on the classical outcome
of
these two measurements
we now have to apply simply nx gate
and this
so the double line denotes
classical bits
right and this is the classical bit s1
this is the classical bit s2 and
depending on the values of these two
classical bits we apply an x gate on
these
a single x gate on one of these three
encoded or data qubits and then what
comes out
is
something which is
error corrected
but remember its error corrected only
for the single qubit error so in essence
this is error corrected up to order p
right so this is the complete circuit
and
we simply conclude this discussion by
noting that the three qubit code can
detect and correct for
single qubit bit flip noise so under the
action of a bit quantum bit flip channel
the three qubit code corrects up to
order p errors right and so essentially
this demonstrates that quantum error
correction
can improve
what we call fidelity fidelity is simply
a metric that compares how close the
final state is to the initial state
right so quantum error correction can
improve fidelity
and
this in turn implies that it can improve
the coherence
time of
the physical cubits
provided
this noise parameter is small
because the entire procedure crucially
hinges on the fact that single qubit
errors are the most likely errors and
that is true only if
the noise parameter is small in
particular in this case we require
that the error probability is less than
a half
right because the errors are corrected
only up to order p remember and so we
require that
in order for this entire procedure to
improve the fidelity and correspondingly
effectively improve the coherence time
of the qubits the
error probability or the noise parameter
must be small and in this case in
particular it must be less than a half
so having explained the technique
and the mathematical ideas behind
quantum error correction
we would like to conclude with a brief
comment on what what is the kind of
physical resource requirement in order
to be able to implement quantum error
correction
so as we mentioned earlier
we don't just have big flip errors we
can have a continuum of different errors
which can occur on a single qubit
and
this includes erasure which is an
important error in the classical setting
as well this includes phase flip this
includes amplitude damping it includes
depolarizing noise etc
right
now of course as we said the resolution
is to now
break down your errors in terms of
a
finite set of errors the x error is the
bit flip error that we already discussed
the z error is what we would call a
phase flip error which can again be
corrected in a sort of fashion that is
identical to the bit flip error
and the y error can actually be viewed
as a combination of the x and the z
errors
and together
this set of errors
can be corrected
and if this set of errors if we have an
error correcting code that can correct
for this set of errors then such a code
can indeed correct for an any single
qubit error because any single qubit
error e can be written as
some linear combination
of
x y z and identity this is any
2 by 2 matrix
which is what any single qubit error
would look like so any 2 cross 2 matrix
can be written as a linear combination
of identity x y and z
and therefore if we are able to correct
for x and z errors
and since y is simply a product of x and
z we can also correct for y
then we end up correcting for any single
qubit error
i should note that just as we had a
three qubit code for
just the x errors
if i now
extend the same idea to also correct for
the phase flips then i get
a 9 qubit code
for
x
and z errors
that is a nine qubit code that corrects
for
single qubit x and z errors and this
code
often goes by the name of the short code
okay
so
but the nine qubit code is not the most
optimal code
in fact the shortest code
that is
that can correct for an arbitrary single
qubit error is a so called five qubit
code and this saturates what's called a
quantum hamming bound one can make
certain arguments to say that the
shortest
code that can potentially correct for
any arbitrary single qubit error must be
at least five qubits long and
we indeed have such a code that
a five qubit code that corrects for
arbitrary single qubit errors
but note that so far we have assumed
that
while the qubit itself is susceptible to
noise the gates up have been assumed to
be perfect and so the encoding process
has been assumed to be perfect
the error detection has been assumed to
be perfect the readouts the measurements
have been assumed to be perfect
but in a real circuit in a real quantum
processor all of these are going to be
faulty
and so the next
level is to talk about fault tolerance
the next step is to talk about fault
tolerance
so in order to deal with faulty gates
and circuits we have the theory of
quantum fault tolerance
and finally
we also need additional qubits in order
to bring down the error rate further and
this uses a technique called
concatenation so finally
we have
i'm just trying to now talk about the
kind of physical requirements we need in
order to have a fault tolerant quantum
processor
it turns out
that we need at least 49 physical qubits
in order to have one error corrected
logical qubit and this is due to the
fact that
rather than the five qubit code it is a
seven qubit code which is more amenable
to or more easily amenable to fault
tolerance techniques and so most of the
physical architectures today focus on
implementing the seven qubit error
correcting code
not just at one level but at two levels
of concatenation so that we have a 49
qubits effectively in order to
protect against single qubit errors in a
fault tolerant way right so this is the
kind of resource requirement that one is
talking about if one has to have error
corrected
quantum processors or quantum devices
so just a couple of more slides to talk
about what is the road ahead where are
we today as i mentioned at the very
beginning we are today in an era of
noisy intermediate scale quantum devices
um and this is the google roadmap which
was put out in 2018 as you can see
where we are today um is somewhere here
where we have
we have
we are off the order of let's say a
hundred qubits
but with hundred qubits one can realize
near term applications but these are 100
noisy qubits
right with an error rate which is still
which is still not really below this
threshold right so this is the so-called
error correction threshold which says
that so long as your error rates are
below this
one can do error resilient computation
but today this is where we are at
and
the goal is to then scale up to
say a thousand or ten thousand qubits
which could then actually be error
corrected qubits and this is the regime
where we finally have useful error
corrected quantum computation which is
require something beyond
um 1000 qubits or close to a 10 000
qubits right
there's a similar roadmap that we have
from ibm and this is a very encouraging
map what i have on the left here is the
good old moore's law that many of us are
familiar with which tells us how
it's a log log plot of how the number of
computations per integrated function is
actually
showing a linear rise in a log log plot
with time
what is plotted on the second figure is
an interesting metric which is often
used today called the quantum volume
it's a combined metric which includes
information about
what is the size of the circuit
the connectivity of the qubits and so on
right and the higher the quantum volume
the more
number of qubits the more number of
error resilient qubits that we have
right so
in terms of quantum volume you find that
we in a log log plot we actually doing
better than linear and today ibm is at a
quantum volume of about 128
and the hope is that in the next in the
coming few years we are actually going
to achieve higher and higher quantum
volumes and be able to implement fault
tolerant error resilient quantum
algorithms
and this roadmap talks about the current
state of affairs so we are here where we
have about 60 qubits today on the ibm
processors
and
the hope is to be able to scale to a
thousand qubits in over the next couple
of years and then
we will be in a regime where we can do
quantum error correction hopefully
achieve fault tolerance and
get closer to robust scalable quantum
devices
thank you
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