A qubit is a physical object (for instance, elementary particle) with two “orthogonal” states, typically denoted by |0 > and |1 >. Like the spin (spin up is |0 > and spin down is |1 >), or low and high electron orbits in a atom.
For quantum computing it is convenient to abstract from a particular physical realization of qubit and use a mathematical definition for it.
Def. A pure quantum state of qubit is a unit norm vector , i.e.
Measurementof|𝑣>(withrespectto|0>and 1>
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Quantum output
Classical output
Measurement (special case) During the measurement of the quantum state |𝑣 >
it collapses either to |0 > or |1 >. The Classical Output shows us to which state ( |0 >
or |1 >) |𝑣 > collapsed. Q. mechanics predicts the probabilities of collapsing to |0 > and |1 >:
Cl. Output
|𝑗 >=|0> |𝑗 >=|1>
Probability
Let us assume that we have n qubits
Postulate 1 A pure state of n qubits is a unit vector
An overall phase rotation does not change the state, so the vectors and , define the same quantum state
Arithmetic Rules for Dirac notations
Multiplication:
We can combine like terms:
We can factor out:
If we have two qubits:
Then their joint state is obtained as the product:
Qubits may have a joint state that is not representable as a product of individual states, for example
In this case we say that these qubits are entangled, or that this is an entangled state.
Example 1 A generic quantum state of 3 qubits has the form
If, for example, all the coefficients are we have the state
Measurement of n Qubits.
The joint state of n qubits before the measurement is
Postulate 2 Upon the measurement each qubit collapses either to |0 > or |1 >. The probabilities of the classical and quantum outcomes are
Cl. Output Q. Output Probability
Example 2 Let 2 qubits have the state Note that
The possible results of the measurement with the corresponding probabilities are
Cl. Output Q. Output Probability
Using Postulate 2, we can find the probability whether the m-th qubit will collapse to |0 > (or to |1 > ). We simply sum up the probabilities of all the outputs in which the m-th qubit collapses to |0 > (or to |1 > ).
Cl. Output Q. Output
Probability
this is summation over all possible binary values of , but always with
Probability
In Example 2 we have
Cl. Output Q. Output
Postulate 2 (continued) What if we measure only m of n qubits? We consider the case of measuring 2 of 3 qubits: ( you will be able to generalize it to any other case)
is the joint quantum state of all 3 qubits
after the measurement
Each of the first 2 qubits collapses (either to |0 > or |1 > ). The 3-rd qubit does not
collapse. Classical and quantum outcomes, and the corresponding probabilities are
Cl. Output Q. Output Probability
Here is a normalization factor to insure that (see Postulate 1). Note that the 3-rd qubit is in the state:
Cl. Output Q. Output Probability
Def. A linear operator (matrix) U is unitary iff . Example 3
Postulate 3 The evolution of a closed quantum system is described by a unitary operator. Time Quantum State
Note that U depends on
Quantum Circuits . Classical Not Gate:
Quantum Not Gate:
The corresponding unitary operator:
Quantum CNOT (control not) Gate (analog of classical XOR)
The joint state of 2 qubits before the gate is
CNOT flips 2-nd qubit if 1-st qubit is 1, that is in each it flips b if a = 1.
The state after the gate is
Unitary operator of CNOT gate is 12
Indeed, in linear algebra notation we have
Switching from Dirac’s notations to linear algebra notations and back, we get
Hadamard Gate:
Toffoli Gate:
In each Toffoli gate flips c if a = 1 and b = 1. Hence,
Theorem Classical AND and NOT gates form a universal set, i.e., they allow one to implement any Boolean function.
Let us assume that we have a quantum circuit with input and output :
Quantum Circuit
We will say that this circuit approximate a unitary operation U with error e if for all (or almost all) states we have
Theorem H, S, CNOT, and Toffoli gates form a universal set, in the sense that for any given unitary operator U these gates allow one to construct a quantum circuit that approximates U with arbitrary small error e.
Note that there are many other universal sets of quantum gates.
• For making the approximation error e smaller and smaller, one ,typically, should make the circuit larger and larger (in terms of used quantum gates).
• Let U be a unitary operator that acts on n qubits, that is U is a 2𝑛 𝑥 2𝑛 unitary matrix. Can this U be approximated with small error e and polynomial number of gates, i.e. 𝑂(𝑛𝑡) gates (t is a positive constant)?
• Unfortunately, NOT. There are infinitely many 2𝑛 𝑥 2𝑛 unitary operators U, and for most of them we need exponentially many, i.e., 2𝛼𝑛 (𝛼 is a positive constant), gates.
• Only some “nice” Us can be implemented with small size quantum circuits.
• Good news is that among these nice Us there are unitary transformations that are very useful for solving certain computational problems.
Einstein, Podolsky, Rosen (EPR) pair is a pair of qubits in the state 12
This state happened to be very surprising and useful for many quantum protocols. Note that this is an entangled state (see page 4). Below we consider Quantum Teleportation.
Quantum Teleportation
Let us pre-share an EPR pair between Alice (A) and Bob (B). Thus, A has in her possession
qubit 1 and Bob has qubit
Let A have another qubit 0 in the state . Alice does not know and
Goal: create in Bob’s possession a qubit in the state using only classical communication; sending qubits to Bob is not allowed.
Actions of Alice: 1. She applies CNOT to qubits 0 and 1 ; 2. she applies H gate to 0 ; 3. she measures 0 and 1 , and sends classical outputs to Bob.
On the next page we analyze how the q. state of all 3 qubits evolves during these steps
Classical Channel
Appling CNOT for 0-th and 1-st qubits, and next factoring out |0> and |1>, we obtain :
After applying H gate to 0-th qubit, making expansion, and factoring out |00>,|01>,|10>, and |11>, we obtain
Using Measurement Postulate (Postulate 2, page 8) we obtain :
Cl. Output Q. Output Probability
To get these results you have to apply carefully the measurement postulate to ; do not forget about normalization factor , which is 2 in this case.
Note that the state of qubit 0 collapsed (either to |0> or |1>).
• Bob gets bits and depending on their values he applies a particular
unitary operator to qubit
2 , as it is shown in the following table. Bob’s Unitary Operator for qubit
• In particular, if state
, then 2 is already in exactly the same
as Alice’s qubit 0 was originally (see measurement results
on the previous page), and so Bob does not have to do anything more
• If , then qubit 2 is in the state , and
therefore Bob has to apply X to move it to the needed state
• Cases of other values of are similar
Thus, we transferred (teleported) the unknown state to Bob without sending any qubit. Amazing!
Quantum teleportation finds many applications. In particular, it is very important for quantum internet and for quantum data exchange inside of a quantum computer.
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