mod04lec24 - Fixing quantum errors with quantum tricks: A brief introduction to QEC - Part 2
Summary
TLDRThis educational segment delves into the framework of quantum error correction, focusing on the three-bit quantum code that mitigates quantum bit flip errors. It explains the concept of redundancy in quantum systems, transitioning from classical bit flip channels to quantum scenarios. The script introduces the tensor product and the no-cloning theorem, emphasizing the encoding of arbitrary superpositions. It outlines the encoding process using a quantum circuit with CNOT gates, creating logical qubits or code words. The discussion then shifts to error detection, highlighting how single qubit errors transform logical states into distinguishable states, crucial for error correction in quantum computing.
Takeaways
- 📚 The script discusses the framework of quantum error correction, focusing on the three-bit quantum code which corrects for the quantum bit flip channel.
- 🧬 Quantum bit flip noise is described, affecting the basis states |0⟩ and |1⟩ with different probabilities for flipping and remaining unchanged.
- 🔗 Redundancy is introduced in quantum systems by encoding a single qubit state into a three-qubit state, expanding from a 2-dimensional space to an 8-dimensional space.
- 🚫 The no-cloning theorem is highlighted, stating that arbitrary superpositions cannot be copied directly, which complicates simple redundancy techniques used in classical systems.
- 🔄 The concept of logical qubits or code words is introduced, representing the encoded states in a higher-dimensional space, distinct from the physical qubits.
- 🛠️ The encoding process is achieved through a quantum circuit using CNOT gates, which transform the input state into an entangled state without violating the no-cloning theorem.
- 🔎 The script explains how the three-qubit code can detect single qubit errors by transforming logical states into distinct, orthogonal states that can be identified.
- 🔄 The effect of noise on the three-qubit system is considered, with the assumption that noise acts independently and identically distributed (i.i.d) on each qubit.
- 📉 The probabilities of different error scenarios are outlined, with single qubit errors being the most likely, followed by two-qubit errors, and three-qubit errors being the least likely.
- 🛡️ The three-qubit code is effective in error detection and correction by identifying the distinct states resulting from single qubit errors, which are mutually orthogonal.
Q & A
What is the main focus of the final segment of the discussion?
-The main focus is on the framework of quantum error correction, using the example of a three-bit quantum code designed to correct errors caused by the quantum bit flip channel.
What is the quantum bit flip channel and how does it affect quantum states?
-The quantum bit flip channel flips the quantum state with a certain probability. For example, the state ket 0 remains ket 0 with probability 1 - p and flips to ket 1 with probability p, and similarly, ket 1 flips to ket 0 with probability p and remains ket 1 with probability 1 - p.
How is redundancy introduced in quantum error correction compared to classical error correction?
-In classical error correction, redundancy is introduced by replicating bits, such as replacing a single bit 0 with three 0s. In quantum error correction, the process involves replacing a single qubit, like ket 0, with a three-qubit state (ket 0 0 0), thus moving from a two-dimensional vector space to an eight-dimensional complex vector space.
Why can’t arbitrary superpositions of quantum states be directly copied using the redundancy method?
-Arbitrary superpositions of quantum states cannot be directly copied due to the no-cloning theorem, which forbids copying an arbitrary superposition of quantum states. This is a key difference from classical information encoding.
What is the correct way to encode quantum information to avoid violating the no-cloning theorem?
-The correct encoding maps an arbitrary superposition, such as alpha ket 0 + beta ket 1, to a coherent superposition in the three-qubit space, resulting in a state like alpha ket 0 0 0 + beta ket 1 1 1.
What is the role of unitary transformations in quantum encoding?
-Unitary transformations allow quantum encoding by ensuring that the transformation from a single qubit to a three-qubit state is done in a reversible and consistent manner, preserving the superposition and quantum information.
What are logical qubits and how are they represented in the three-qubit code?
-Logical qubits represent the encoded qubits in the three-qubit code. They are denoted as ket 0 with a subscript l (for logical), representing the state ket 0 0 0, and ket 1 with a subscript l, representing the state ket 1 1 1.
How does the quantum encoding circuit work to create the three-qubit state?
-The quantum encoding circuit uses CNOT gates to transform the input qubit (which contains the information) and two ancillary qubits (initialized to 0) into a three-qubit state that represents a superposition, such as alpha ket 0 0 0 + beta ket 1 1 1.
What type of errors does the three-qubit quantum code focus on correcting?
-The three-qubit quantum code primarily focuses on correcting single qubit bit flip errors, which are more likely than two-qubit or three-qubit errors.
How does the three-qubit code help in detecting single qubit errors?
-The three-qubit code helps detect single qubit errors because the logical qubits (such as ket 0 0 0 and ket 1 1 1) transform into distinct, mutually orthogonal states when errors occur, making them distinguishable and detectable.
Outlines
🧠 Introduction to Quantum Error Correction
The segment begins with an introduction to quantum error correction, focusing on the three-bit quantum code that addresses quantum bit flip errors. It explains the quantum bit flip channel and its effect on basis states, contrasting it with classical redundancy methods. The concept of encoding a single qubit into a three-qubit state is introduced, emphasizing the need to handle arbitrary superpositions due to the no-cloning theorem. The discussion sets the stage for understanding how redundancy is introduced in quantum systems to protect against errors.
🔄 Correct Encoding of Quantum Information
This paragraph delves into the correct procedure for encoding quantum information. It explains that arbitrary superpositions cannot be simply copied due to the no-cloning theorem, and instead, a coherent superposition must be created in the three-qubit space. The concept of logical qubits or code words is introduced, replacing physical qubits with a three-qubit state. The paragraph also discusses the encoding process using unitary transformations and the significance of entanglement in this context. A quantum circuit for encoding is briefly mentioned as a way to achieve this transformation.
🔗 Quantum Circuit for Encoding
The focus shifts to the practical implementation of the encoding process using a quantum circuit. The circuit involves a data qubit carrying the information and two ancillary qubits initialized to zero. The use of the CNOT gate is highlighted as a crucial element in the circuit, which, when applied, transforms the three-qubit state into the desired superposition. The output state is an entangled state of three qubits, which is essential for error detection and correction. This paragraph provides a detailed step-by-step analysis of the circuit's operation.
🌐 Effects of Noise on Three-Qubit Code
This paragraph discusses the impact of noise on the three-qubit code, considering the independent and identically distributed (iid) assumption. It outlines the set of possible errors that can occur in a three-qubit system, ranging from no error to single, double, and triple bit flip errors. The probabilities of these errors are explained, with a focus on the likelihood of single qubit errors. The paragraph concludes by emphasizing the need to detect and correct single qubit errors, setting the stage for further discussion on error detection.
🔍 Detecting Single Qubit Errors
The final paragraph examines how single qubit errors affect the logical states of the three-qubit code. It describes the transformation of the logical zero and one states under the influence of bit flip errors, resulting in distinct and mutually orthogonal three-qubit states. The importance of these states being distinguishable is highlighted, as it allows for error detection. The paragraph concludes by emphasizing the three-qubit code's ability to detect errors, which is a critical aspect of quantum error correction.
Mindmap
Keywords
💡Quantum Error Correction
💡Quantum Bit Flip Channel
💡Redundancy
💡Tensor Product
💡No Cloning Theorem
💡Unitary Transformations
💡Logical Qubits
💡Code Words
💡Entanglement
💡IID (Independent and Identically Distributed)
💡Error Detection
Highlights
Introduction of quantum error correction using a three-bit quantum code to correct for quantum bit flip channel.
Explanation of the quantum bit flip noise: ket 0 remains ket 0 with probability 1-p and flips to ket 1 with probability p, and vice versa for ket 1.
Redundancy is introduced by replacing ket 0 with three ket 0s and ket 1 with three ket 1s in a three-qubit system, creating an 8-dimensional vector space.
No cloning theorem prohibits naive replication of arbitrary superpositions, preventing direct copying of quantum states.
Correct encoding procedure involves mapping alpha |0⟩ + beta |1⟩ to a coherent superposition: alpha |000⟩ + beta |111⟩.
Logical qubits are introduced, represented as logical 0 (|000⟩) and logical 1 (|111⟩), forming the basis for encoding.
Unitary transformations, particularly the use of CNOT gates, are applied to transform and encode qubits.
Encoding circuit uses CNOT gates to entangle the qubits, creating a coherent superposition state.
Single qubit bit-flip noise model explained: noise acts independently on each qubit, with identity or X errors occurring on one or more qubits.
Eight possible error states are identified, including identity, single qubit, two qubit, and three qubit bit flip errors, with varying probabilities.
Focus on detecting and correcting single qubit errors, as they are the most probable type of error.
Single qubit errors transform logical states (|000⟩ and |111⟩) into distinct and mutually orthogonal three-qubit states.
Orthogonality of error-transformed states enables distinguishability, making error detection possible.
Entangled states play a key role in quantum error correction, highlighting the importance of entanglement in error resilience.
The three-qubit code allows for error detection by transforming quantum states into distinct, orthogonal forms that can be measured and corrected.
Transcripts
[Music]
okay in this final segment we will now
discuss
the framework of quantum error
correction the technique of quantum
error correction using the simple
example of
the so called
three bit
quantum code
which
corrects for
the
quantum bit flip channel
so recall
in the last segment we had introduced
the idea of the quantum uh the quantum
bit flip channel right and
this was the effect of the channel
uh on the basis states that is sched0
and ket one remember that under the
action of the quantum bit flip noise cat
0 would remain kept 0 with probability 1
minus p and k 0 would flip to get 1 with
probability p
similarly get 1 would flip to get 0 with
probability p and it would remain
get 1
with probability 1 minus p
right
so this is the
single
quantum bit flip channel so
recall what we did for the classical
case where we said that we need to that
one way of dealing with big flip errors
would be to introduce redundancy so how
do we introduce redundancy
well this is the simplest thing to do
right which is to take cat 0
right
and
replace it with
let's say three
cad zeros
in the quantum case what does this
entail remember catch 0 is a state which
lives in a 2 dimensional
so this belongs to
c 2 which is a 2 dimensional complex
linear vector space
but now this becomes
a 3 qubit state
right and this is something which
belongs to
three
two dimensional complex linear vector
spaces so the whole thing is actually an
eight dimensional space
it's an eight dimensional complex linear
vector space or two to the three
dimensional complex linear vector space
right
similarly we can now think of
replacing the cat one
by
three cat ones
so when i write these three cat zeros
there is an implied tensor product here
so this
symbol denotes what is called a tensor
product
again we will not get too much into the
details of this but the idea is that
when you're putting together
multiple quantum systems and remember
we've spoken about
um multi-party multi-qubit systems in
week two right so this is essentially a
three qubit system
right and we are replacing a single
qubit system with a three qubit system
right and this
is how we would introduce redundancy or
how we would do the
encoding
right so we
replace the single cat 0 with three cad
zeros we replace the single cat one with
three kit once but now the question is
what happens to an arbitrary
superposition in the classical case this
was sufficient right a single zero
replaced by three zeros a single one
replaced by three ones but now arbitrary
superpositions of zero and one are
allowed in the quantum case so what do
we do with alpha zero plus beta one
can we do this operation where alpha
zero plus beta one
now becomes
we replace this by
alpha zero plus beta one
we replace this by alpha 0 plus beta 1
on 3 qubits
well this is the kind of operation that
is forbidden
by
the no cloning theorem
the point is that this can be an
arbitrary superposition any possible
superposition of 0 and 1 could occur in
our computation so when we encode
quantum information you're not just
encoding the ket 0 and the ket one
rather you have the you need to have the
ability to encode an arbitrary
superposition of get 0 and get 1 not a
fixed superposition if i am always going
to have only a fixed
state of the form 1 over root 2 0 plus 1
over root 2 1 which is the plus state
then of course i can find ways of making
three plus states on three different
qubits
but now we're talking about encoding
arbitrary superpositions
and this cannot happen in this naive way
of making three copies of an arbitrary
superposition arbitrary superpositions
cannot be copied this is forbidden by
the no cloning theorem which was
discussed in week two
so how do we work around this
so here's the way to work around it so
the correct way to encode
so
so let me call this the correct
encoding
procedure
is actually
that alpha 0 plus beta 1
gets mapped to alpha
0 0 0
plus
beta
1 1
1.
so we make a coherent superposition so
this is what i would call a coherent
superposition
right so we make
the same coherent superposition
in the three qubit space
and the claim is that this can somehow
happen
via unitary transformations because
remember what we are allowed to do
the gates that we are allowed to do are
all unitary transformations right so how
do we do this how can we
do this encoding procedure in a unitary
way i'll just show you the circuit but
before that i want to note that what we
are doing is actually replace
our
physical qubits
which are the single qubits
0 and 1
right
and in their place
we are introducing
the idea
of logical qubits
right
which i will now denote as 0 with a
subscript l and a ket 1 with a subscript
l
where
the 0 with the subscript l is simply
the 3 qubit state 0 0 0
and the one with the subscript l is the
three qubit state 1 1 1
right
if you want to quickly recall
what these three qubit states look like
remember
that get 0 suppose was denoted by the
column vector 1 0
so
if get 0 is this
then
this 3 qubit state
is simply
this outer product or tensor product
right
and
this
becomes an 8
dimensional vector
of the form
right because this
this outer product gives me a 4
dimensional vector column vector 1 0 0 0
and
taking the outer product again with
another one zero gives me
this eight dimensional column vector
with just one one and all zeros
similarly you can write down the state
corresponding to the three ones as an
eight dimensional column vector with all
zeros and just one one sitting at the
last entry right so this is what our
encoded
state looks like
okay and the other terminology that i
would like to introduce here is these
logical qubits are also called
code words
okay and just like how the single qubit
state lives in a two dimensional vector
space that is spanned by zero and one
our encoded states or logical states or
logical qubits live inside a two
dimensional space that is spanned by
zero logical and one logical and hence
an arbitrary superposition of the form
alpha zero plus beta one is to be
replaced by the encoded superposition
which is alpha zero zero zero plus beta
one one one
and now we address the question of how
do we actually realize
this encoding so i'm going to show you
the encoding circuit
while we show the encoding simply as
a map from
a single qubit state to
this
three qubit state
the fact that this is unitarily realized
means that this is actually a map or a
transformation from three qubits to
three qubits so it's actually a circuit
with three input qubits and three output
qubits right so the idea is
the first qubit is the qubit that is
carrying the information in the form of
the superposition
the second qubit
and the third qubit are
what are called
ancillary qubits or
ancillar qubits
these are additional qubits which are
initialized to zero
and then unitarily this three qubit
state is transformed to this
superposition
and the way this is done is basically
using a very important gate
namely the c naught gate okay so this is
what the circuit looks like
i have
three registers
the first qubit is
in the state alpha zero
the other two
are the
ancillar qubits
the first qubit is what we often call
the data qubit
because that is the one that contains
the information which is to be encoded
and now how do we proceed with the
circuit
well we simply apply a pair of c naught
gates
and the claim is that this output state
is now of the form
alpha 0 0 0
plus
beta 1 1 1.
so let's quickly check that so what
happens after the action of the first c
naught
so let's do this circuit analysis step
by step so after the first c naught
we have the three qubit state
alpha zero plus beta one
tensor
remember that the c naught gate flips
the target so this is the control and
this is the target and the c naught gate
flips the target whenever the control is
1. so
this becomes
alpha 0 0
plus
beta 1 1.
tensor get zero
right and this is the third ancillar
qubit the second nonselective rather the
third qubit in the circuit
and
after the second c naught
we end up with the state
now this zero gets flipped whenever the
second qubit is set to one the second
qubit has become the control qubit and
whenever the second qubit is one this
qubit has to get flipped so this becomes
alpha
0 0 0
plus
beta
1 1 1
as
desired right so this from just the
state some arbitrary state psi this has
become the logical state side
right so
this is the final state that we want and
this is how we've achieved the encoding
and we've achieved the encoding in a via
a quantum circuit right so this
is the encoding circuit for the three
qubit uh code
right
okay so now how does this help us
of course one final point to note is
that this final state that we have
written out here
this
alpha 0 0 0 plus beta 1 1 1 is actually
an entangled state
since we have introduced the idea of
entanglement briefly in week 2 this is
an entangled state
of
three qubits
right
and this is where entanglement comes to
our rescue
we will now look at what happens to the
state under the action of noise
okay so now that we've gone from a
single qubit to a three qubit scenario
what can we say about the noise so we
now have to look at
the properties of the
three qubit code
first of course we recall the assumption
that we make this was the assumption
that was made even in the classical case
that the noise is iid
right what does this mean this means
that the noise acts independently
so the noise acts independently on each
qubit
and second it is
identically distributed
which means that the error probabilities
are the same for every qubit
so in the single qubit case remember
that
for the single qubit bit flip noise
we could describe it in terms of
two operators right there was either the
x operator which happened with
probability p
and this is what we said you have the x
error with probability p or you have
identity
with probability 1 minus p
okay
but now
you have 3 cubit bit flip noise
so
what are the set of possible errors
now on each qubit you could have either
identity or x
right which means that you have a set of
eight possible errors now so what are
these eight possible errors we can write
them all down
right so you could have identity on all
three qubits and again i'm going to
denote this
by tensor product sometimes we can just
drop the tensor product and write this
as three identities because remember the
noise is acting independently on every
qubit
you could have
a single x error on the third qubit
you could have
a single x error on the second qubit
and you could have a single x error on
the first qubit
now these errors all occur with
probability p
right so these three errors which are
all the
so called so this set is what we would
call the set of
single qubit errors right because this
basically says there was an x error on
the third qubit or the second qubit or
the first qubit but only on one qubit
right so these are the set of single
qubit errors all of which occur
with probability
p
right
but the set of errors doesn't stop here
it is not simply identity or single
qubit errors you can of course have two
bit flip errors so let's complete the
set of errors now
so the set of errors continues we can
now have two bit flip errors so let's
say you could have error
on the second and the third qubit
you could have error on the first
and the second qubit
and you could have errors on the first
and the last two qubits right and this
set of errors is what we would call
the set of
two qubit
errors because now you have big flips
occurring on
two qubits and these occur
with probability order p squared right
and finally we also have the
three qubit error which is the x error
occurring on all three qubits so this is
what we would call the
three qubit error
and this occurs with probability order p
q
and this completes
our set of errors for the three qubit
bit flip channel right this was of
course also the case in the case of the
classical
bit flip error right and like in the
classical case we are now going to argue
that
the single qubit errors are more likely
than the two qubit which is more likely
than the three qubit and so we are only
going to focus our attention on
correcting first of all detecting and
correcting the single qubit with flip
errors right so we're going to focus our
attention only on the single qubit
errors
and like we said there's three of them
right and then of course there's a
possibility that the qubit remains
unaffected
by the error and this occurs with
probability 1 minus p so just to
complete i must say that this occurs
with probability
1 minus p the whole q
right and you can of course check that
the sum of all these probabilities must
um sum to 1
right so i have 3 p squared here i have
1 p cube here and then i have 3 here
right okay
fine
so now how do we detect or how does this
three qubit encoded state help us detect
the single qubit errors so the next
thing to look at is what how do the
single qubit errors affect the so the
what is the effect of
the single qubit
errors
well so let's look at the
let's actually look at just the basis
states right the logical states or the
code words so let me only look at
the logical states
sorry the logical
cubit
states
0l
which is
0 0 0
and
1l
which is 1 1 1.
so what happens to 0 0 0 under the
action of the single bit flip errors
well the third qubit could get flipped
so
this
could become this
i could have the second qubit
getting flipped
so
this would become this
and i could have
the first qubit getting flipped
so these are the possible output states
after the action of the single qubit
error
on
the zero zero zero state or the logical
zero state similarly we can work out
what happens to the logical one state
and again
i could have the third
qubit getting flipped to a zero i could
have the middle qubit getting flipped to
a zero or the first qubit getting
flipped to a 0. note that i am not
explicitly writing the tensor product
here but it is implied everywhere ok
so these are the set of possible states
now what do we notice about these states
like in the case of the classical
three bit
the classical three bit code
the 0 0 0 and the 1 1 1 are getting
mapped to distinct states right so this
state corresponds to a classical bit
string 0 0 1 this corresponds to 0 1 0
and this corresponds to 1 0 0
1 1 0 1 0 1 0 1 1 so all of them are
basically distinct
states
in the
three qubit space
they're all distinct three qubit states
and therefore exactly like in the case
of the classical bit flip code or the
classical repetition code
this set of six states can be
distinguished and detected so
let me just note this point that
the
single qubit errors
transform
the
logical states
that's the logical zero and the logical
one
to
distinct
this very important
three qubit states in fact
this set of three qubit states that i
have marked here are actually all
mutually orthogonal
right because they correspond to
distinct states their distinct bit
strings right so they are mutually
orthogonal which means if i take the
inner product between this three
qubit state with any of the other three
qubit states i get zero
and because they are mutually orthogonal
they are distinguishable
okay
so this is the key property of the three
qubit
code which helps us in error detection
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