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delamonaco
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Need documents to help me understand quantum computing!
Need documents to help me understand quantum computing!
Need resources to help me understand quantum computing please
- Thread starter delamonaco
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What research have you done so far? What have you found?delamonaco said:
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Adrian59
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Below are my notes from a first course in quantum computing. I have more but like the first reply to your request I am not sure what level you are at. I could be well below yours!
Information is physical; storing information requires a physical change.
Maxwell’s daemon
Landauer’s Principle: erasure of one bit of information costs kB T ln(2) energy, and consequent increase in entropy.
Weight hard drive with information < Weight hard drive with random noise.
Quantifying Information:
1) The amount of information depends on probability;
2) Information is a continuous function of probability, ie less P(x) more IX
3) Information is additive – I(P(x), P(y)) = I(P(x)) + I(P(y))
Shannon Entropy (H(x)) = - Σ P(x) log2(P(x))
NB the base of the logarithm is dependent on circumstance; ie base 2 for binary, base e for thermodynamics.
Entropy is a measure of how compressible the data is since completely random string is not compressible. Compression ratio (CR) = H(set)
H(max.)
Extending this to two data sets, gives:
H(x,y) = - Σi,j P(i,j) log2(P(i,j))
And,H(x) = - Σi,j P(i,j) log2(Σj P(i,j))
Note that,H(x,y) < H(x) + H(y)
Conditional entropy (Hx(y)) is the entropy of y, given x which is:
Hx(y)) = - Σi,j P(i,j) log2(Pi(j)) that is how certain of y are we knowing x.
Classical Logic:
Truth tables:
Basic Quantum Computing:
The basic structure of quantum theory is made up of specialized vectors called a bra and a ket, written < A | and | A > respectively. The vectors represent the states of a system. These vectors can be changed by an operator which can be represented by a matrix when the vectors are written in row (for a bra) or column (for a ket) form.
| P > = U| A > or < P| = < A | U where U is an operator.
Since the dot product < A| A > is both real and invariant, U has to be unitary.
In quantum computing the vector, which is a ket vector, represents the input (I) and output (O), and the operators represent the gates (G).
Written in schematic form:G| I > = | O >
The two most important features of quantum theory that have led to the quest for quantum computing are superposition and entanglement.
Superposition is the original driver for a quantum theory when it was discovered that subatomic particles can appear to be in two places at once. The first example of this was the double slit experiment. Another is the Stern-Gerlach Experiment.
Entanglement is the strange phenomenon whereby the properties of distant particles are linked.
Mathematical Development Theory:
An important parameter borrowed from linear algebra is the trace of a matrix. This is the sum of the diagonal elements of a square matrix. The trace is independent of bases. A specialised operator in quantum theory is the density matrix (ρ), written:
ρ = | Ψ > < Ψ |
Extending to two states, gives:
ϬAB = | ΨAB > < ΨAB | = ∑i αi |iA >|iB > ∑j α*j < jA|< jB|
Trace over B (ϬAB) = ∑ij αi α*j |iA > < jA| < jB | iB > = ∑ij αi α*j |iA > < jA| δij
= ∑i |αi|2 |iA > < jA|ie tracing over B
A state is entangled if it cannot be expressed as a product of two states, that is:
| ΨA > x | ΨB >
Bell States:
a)| Φ+ > = 1 (|00> + |11>)
√2
b)| Φ- > = 1 (|00> - |11>)
√2
c)| Ψ+ > = 1 (|01> + |10>)
√2
d)| Ψ- > = 1 (|01> - |10>)
√2
Note that for all the above states, if first qubit is in a known state, the second qubit is given.
Quantum Information Theory:
Classical
Communication from A to B, bits of information in binary will inevitably involve some errors where a sent 0 is received as 1, and vice versa.
Channel capacity is expressed as bits per second; ie Hz.
Decibel is a power law, so 10 dB loss is 1010 loss.
An adopted custom in quantum information theory is for the sender to be called Alice and the receiver Bob.
In quantum information theory, the parameter used is the Shannon entropy (H).
Where H(X/Y) is the entropy of X conditional on Y, and I(A;B) is the mutual information which is the same as the channel capacity.
Fidelity is given as < B | A >, ie the overlap integral which is another way of representing the channel capacity.
An alternative to the Shannon entropy is the von Neumann entropy which is written as:
S(ρA) = - trace ρA log(ρA) = - ∑i λi log(λi)where λi are the eigenvalues of ρA
as, the sum of the eigenvalues = trace of a matrix.
Entropy of a pure state is zero since log(1) = 0.
(aside: entropy of whole universe is zero and is a pure state; the more classical the more entropy)
If Alice and Bob share an entangled state, like one of the Bell states given above, one can trace out Bob in a similar fashion to above:
| Φ+ > = 1 (|0A 0B > + |1A 1B >)
√2
Then, |Φ+ >< Φ+ | = trB {½ (|0A 0B > + |1A 1B >) (< 0A 0B | + < 1A 1B |)}
= {½ (|0A > < 0A| x trB|0B > < 0B|) + |0A > < 1A| x trB|0B > < 1B|)
+ |1A > < 0A| x trB (|1B > < 0B|) + |1A > < 1A| x trB|1A > < 1A|)}
= {½ (|0A > < 0A| + |1A > < 1A|}
A state is entangled when tracing out one, leaves a mixed state.
If trace(ρA 2) < 1A is a mixed state;
if trace(ρA 2) = 1A is a pure state.
No cloning theorem:
If U is a unitary operation of cloning, then:
U | 0 > = | 00 >and U | 1 > = | 11 >
U (|Ψ >) = α |00 > + β |11 > where |Ψ > = α|0 > + β|1 >
But, since U |Ψ > = |Ψ > |Ψ > = α2 |00 > + αβ |01 > + αβ |10 > + β2 |11 >
More complex entanglements have been suggested; eg Greenberger–Horne–Zeilinger state (GHZ state) which is an entanglement of three or more states:
| GHZ > = 1 (|000> + |111>)
√2
Quantum Gates:
Basic notes on nomenclature:
Each input on a quantum circuit can be in two states | 0 > or | 1 > which can be represented in several ways (as can the outputs); gates represent the logic operations:
1) as a circuit diagram with inputs on the left - the number of lines equals the number of input qubits (the first is the top line), and outputs on the right, connections are horizontal lines with vertical connections at certain gates;
2) mathematical operations on kets, eg | a b c > represents 3 qubits with the first in the left;
3) column vectors represent the inputs and outputs with matrices representing the gates, the number of columns is 2^(number qubits).
NB the size of the matrix, representing a gate depends on how many qubits the gate is operating on. Conditional or controlled gates operate on at least two qubits so are at least a 4x4 matrix.
The basic qubits are:
The standard single qubit gates are the identity (I); the three Pauli rotation matrices (X, Y and Z); S, T where S = √Z and T = √S; and finally the Haddamard gate (H).
There is an additional identity: H = X + Z which implies HZ = XH
√2
An example of a conditional gate is the controlled not (CNOT) gate, seen below in circuit and matrix form:
This gate flips the second qubit if the fist is | 1 > but otherwise leaves things unchanged.
This elementary circuit is extremely useful:
It is instructive to rewrite these in standard ket form - noting the above convention, so:
| 00 > → = 1 (|00 > + |01 >)which shows that these states are a superposition.
√2
Then, 1 (|00 > + |01 >)→ 1 (|00 > + |11 >)
√2 √2
The final ket can be recognised as | Φ+ > one of the Bell states, so this is an entangled state.
Explanation: at t1 knowing that the first qubit is in state | 0 > means there is a 50:50 split between the second qubit being | 0 > or | 1 >. However, at t2 knowing which state the first qubit is in then gives the state that the second qubit is in.
Principles of Measurement:
1) Implicit measurement – ie all undetermined channels can be considered a measurement;
2) Deferred measurement – ie all measurements can be deferred to the end of the circuit without loss of generality. However, if a classical outcome is used in the circuit, it should be replaced by a quantum controlled gate.
Quantum Circuits:
There are several of these that have become well known: Grover’ search algorithm, Deutsche’s algorithm, teleportation, dense coding and quantum error correction.
Technical Aspects:
Di Vincento Criteria:
1) A scalable physical system, with well characterised qubits;
2) The ability to initialise the system into a fiducial state;
3) Long relevant decoherence times;
4) A universal set of gates, eg Clifford group + one other;
5) Qubit specific measurement capacity (readout);
Plus for networking:
6) Ability to interconnect, stationary and flying bits;
7) Ability to faithfully transmit flying bits.
1) and 2) OK, 3) hard, 4) and 5) possible, 6) and 7) no one knows.
Quantum Information: Science and Technology
Introduction:Information is physical; storing information requires a physical change.
Maxwell’s daemon
Landauer’s Principle: erasure of one bit of information costs kB T ln(2) energy, and consequent increase in entropy.
Weight hard drive with information < Weight hard drive with random noise.
Quantifying Information:
1) The amount of information depends on probability;
2) Information is a continuous function of probability, ie less P(x) more IX
3) Information is additive – I(P(x), P(y)) = I(P(x)) + I(P(y))
Shannon Entropy (H(x)) = - Σ P(x) log2(P(x))
NB the base of the logarithm is dependent on circumstance; ie base 2 for binary, base e for thermodynamics.
Entropy is a measure of how compressible the data is since completely random string is not compressible. Compression ratio (CR) = H(set)
H(max.)
Extending this to two data sets, gives:
H(x,y) = - Σi,j P(i,j) log2(P(i,j))
And,H(x) = - Σi,j P(i,j) log2(Σj P(i,j))
Note that,H(x,y) < H(x) + H(y)
Conditional entropy (Hx(y)) is the entropy of y, given x which is:
Hx(y)) = - Σi,j P(i,j) log2(Pi(j)) that is how certain of y are we knowing x.
Classical Logic:
Truth tables:
Basic Quantum Computing:
The basic structure of quantum theory is made up of specialized vectors called a bra and a ket, written < A | and | A > respectively. The vectors represent the states of a system. These vectors can be changed by an operator which can be represented by a matrix when the vectors are written in row (for a bra) or column (for a ket) form.
| P > = U| A > or < P| = < A | U where U is an operator.
Since the dot product < A| A > is both real and invariant, U has to be unitary.
In quantum computing the vector, which is a ket vector, represents the input (I) and output (O), and the operators represent the gates (G).
Written in schematic form:G| I > = | O >
The two most important features of quantum theory that have led to the quest for quantum computing are superposition and entanglement.
Superposition is the original driver for a quantum theory when it was discovered that subatomic particles can appear to be in two places at once. The first example of this was the double slit experiment. Another is the Stern-Gerlach Experiment.
Entanglement is the strange phenomenon whereby the properties of distant particles are linked.
Mathematical Development Theory:
An important parameter borrowed from linear algebra is the trace of a matrix. This is the sum of the diagonal elements of a square matrix. The trace is independent of bases. A specialised operator in quantum theory is the density matrix (ρ), written:
ρ = | Ψ > < Ψ |
Extending to two states, gives:
ϬAB = | ΨAB > < ΨAB | = ∑i αi |iA >|iB > ∑j α*j < jA|< jB|
Trace over B (ϬAB) = ∑ij αi α*j |iA > < jA| < jB | iB > = ∑ij αi α*j |iA > < jA| δij
= ∑i |αi|2 |iA > < jA|ie tracing over B
A state is entangled if it cannot be expressed as a product of two states, that is:
| ΨA > x | ΨB >
Bell States:
a)| Φ+ > = 1 (|00> + |11>)
√2
b)| Φ- > = 1 (|00> - |11>)
√2
c)| Ψ+ > = 1 (|01> + |10>)
√2
d)| Ψ- > = 1 (|01> - |10>)
√2
Note that for all the above states, if first qubit is in a known state, the second qubit is given.
Quantum Information Theory:
Classical
Communication from A to B, bits of information in binary will inevitably involve some errors where a sent 0 is received as 1, and vice versa.
Channel capacity is expressed as bits per second; ie Hz.
Decibel is a power law, so 10 dB loss is 1010 loss.
An adopted custom in quantum information theory is for the sender to be called Alice and the receiver Bob.
In quantum information theory, the parameter used is the Shannon entropy (H).
Where H(X/Y) is the entropy of X conditional on Y, and I(A;B) is the mutual information which is the same as the channel capacity.
Fidelity is given as < B | A >, ie the overlap integral which is another way of representing the channel capacity.
An alternative to the Shannon entropy is the von Neumann entropy which is written as:
S(ρA) = - trace ρA log(ρA) = - ∑i λi log(λi)where λi are the eigenvalues of ρA
as, the sum of the eigenvalues = trace of a matrix.
Entropy of a pure state is zero since log(1) = 0.
(aside: entropy of whole universe is zero and is a pure state; the more classical the more entropy)
If Alice and Bob share an entangled state, like one of the Bell states given above, one can trace out Bob in a similar fashion to above:
| Φ+ > = 1 (|0A 0B > + |1A 1B >)
√2
Then, |Φ+ >< Φ+ | = trB {½ (|0A 0B > + |1A 1B >) (< 0A 0B | + < 1A 1B |)}
= {½ (|0A > < 0A| x trB|0B > < 0B|) + |0A > < 1A| x trB|0B > < 1B|)
+ |1A > < 0A| x trB (|1B > < 0B|) + |1A > < 1A| x trB|1A > < 1A|)}
= {½ (|0A > < 0A| + |1A > < 1A|}
A state is entangled when tracing out one, leaves a mixed state.
If trace(ρA 2) < 1A is a mixed state;
if trace(ρA 2) = 1A is a pure state.
No cloning theorem:
If U is a unitary operation of cloning, then:
U | 0 > = | 00 >and U | 1 > = | 11 >
U (|Ψ >) = α |00 > + β |11 > where |Ψ > = α|0 > + β|1 >
But, since U |Ψ > = |Ψ > |Ψ > = α2 |00 > + αβ |01 > + αβ |10 > + β2 |11 >
More complex entanglements have been suggested; eg Greenberger–Horne–Zeilinger state (GHZ state) which is an entanglement of three or more states:
| GHZ > = 1 (|000> + |111>)
√2
Quantum Gates:
Basic notes on nomenclature:
Each input on a quantum circuit can be in two states | 0 > or | 1 > which can be represented in several ways (as can the outputs); gates represent the logic operations:
1) as a circuit diagram with inputs on the left - the number of lines equals the number of input qubits (the first is the top line), and outputs on the right, connections are horizontal lines with vertical connections at certain gates;
2) mathematical operations on kets, eg | a b c > represents 3 qubits with the first in the left;
3) column vectors represent the inputs and outputs with matrices representing the gates, the number of columns is 2^(number qubits).
NB the size of the matrix, representing a gate depends on how many qubits the gate is operating on. Conditional or controlled gates operate on at least two qubits so are at least a 4x4 matrix.
The basic qubits are:
The standard single qubit gates are the identity (I); the three Pauli rotation matrices (X, Y and Z); S, T where S = √Z and T = √S; and finally the Haddamard gate (H).
There is an additional identity: H = X + Z which implies HZ = XH
√2
An example of a conditional gate is the controlled not (CNOT) gate, seen below in circuit and matrix form:
This gate flips the second qubit if the fist is | 1 > but otherwise leaves things unchanged.
This elementary circuit is extremely useful:
It is instructive to rewrite these in standard ket form - noting the above convention, so:
| 00 > → = 1 (|00 > + |01 >)which shows that these states are a superposition.
√2
Then, 1 (|00 > + |01 >)→ 1 (|00 > + |11 >)
√2 √2
The final ket can be recognised as | Φ+ > one of the Bell states, so this is an entangled state.
Explanation: at t1 knowing that the first qubit is in state | 0 > means there is a 50:50 split between the second qubit being | 0 > or | 1 >. However, at t2 knowing which state the first qubit is in then gives the state that the second qubit is in.
Principles of Measurement:
1) Implicit measurement – ie all undetermined channels can be considered a measurement;
2) Deferred measurement – ie all measurements can be deferred to the end of the circuit without loss of generality. However, if a classical outcome is used in the circuit, it should be replaced by a quantum controlled gate.
Quantum Circuits:
There are several of these that have become well known: Grover’ search algorithm, Deutsche’s algorithm, teleportation, dense coding and quantum error correction.
Technical Aspects:
Di Vincento Criteria:
1) A scalable physical system, with well characterised qubits;
2) The ability to initialise the system into a fiducial state;
3) Long relevant decoherence times;
4) A universal set of gates, eg Clifford group + one other;
5) Qubit specific measurement capacity (readout);
Plus for networking:
6) Ability to interconnect, stationary and flying bits;
7) Ability to faithfully transmit flying bits.
1) and 2) OK, 3) hard, 4) and 5) possible, 6) and 7) no one knows.
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Adrian59
- 202
- 25
Did you find my notes helpful, or are you asking for something more in depth?delamonaco said:
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