COL863: Quantum Computation and Information
Transcript of COL863: Quantum Computation and Information
COL863: Quantum Computation and Information
Ragesh Jaiswal, CSE, IIT Delhi
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Computation: Phase estimation
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
We will use the assumption that there are black-boxes that:
prepare the state |u〉, and
perform the controlled-U2j operation.
We will describe a phase estimation procedure that uses tworegisters:
A t-qubit register initially in state |0...0〉 (the value of t to bedecided later), anda register that begins in the state |u〉.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
Claim 1: The final state of the first register in the circuit below isgiven by:
1
2t/2
(|0〉 + e(2πi)2t−1ϕ |1〉
)(|0〉 + e(2πi)2t−2ϕ |1〉
)...
(|0〉 + e(2πi)20ϕ |1〉
)= 1
2t/2
∑2t−1k=0
e(2πi)ϕk |k〉
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
Claim 1: The final state of the first register in the circuit below isgiven by:
1
2t/2
(|0〉 + e(2πi)2t−1ϕ |1〉
)(|0〉 + e(2πi)2t−2ϕ |1〉
)...
(|0〉 + e(2πi)20ϕ |1〉
)= 1
2t/2
∑2t−1k=0
e(2πi)ϕk |k〉
Question: Suppose ϕ may be expressed exactly asϕ = [0 · ϕ1ϕ2...ϕt ]. Suggest a way to retrieve the value of ϕ?
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
Claim 1: The final state of the first register in the circuit below isgiven by:
1
2t/2
(|0〉 + e(2πi)2t−1ϕ |1〉
)(|0〉 + e(2πi)2t−2ϕ |1〉
)...
(|0〉 + e(2πi)20ϕ |1〉
)= 1
2t/2
∑2t−1k=0
e(2πi)ϕk |k〉
Question: Suppose ϕ may be expressed exactly asϕ = [0 · ϕ1ϕ2...ϕt ]. Suggest a way to retrieve the value of ϕ?
Taking the inverse-fourier transform and measuring the value ofthe first register in the computational basis gives ϕ.
In general, we will show that the inverse Fourier transform has thefollowing behaviour:
12t/2
∑2t−1j=0 e(2πi)ϕj |j〉 |u〉 → |ϕ̃〉 |u〉
where |ϕ̃〉 denotes a state that is a good estimator for ϕ whenmeasured.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
We will use the assumption that there are black-boxes that:
prepare the state |u〉, and
perform the controlled-U2j operation.
We will describe a phase estimation procedure that uses tworegisters:
A t-qubit register initially in state |0...0〉 (the value of t to bedecided later), anda register that begins in the state |u〉.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
Claim 1: The final state of the first register in the circuit below isgiven by:
1
2t/2
(|0〉 + e(2πi)2t−1ϕ |1〉
)(|0〉 + e(2πi)2t−2ϕ |1〉
)...
(|0〉 + e(2πi)20ϕ |1〉
)= 1
2t/2
∑2t−1k=0
e(2πi)ϕk |k〉
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
Claim 1: The final state of the first register in the circuit below isgiven by:
1
2t/2
(|0〉 + e(2πi)2t−1ϕ |1〉
)(|0〉 + e(2πi)2t−2ϕ |1〉
)...
(|0〉 + e(2πi)20ϕ |1〉
)= 1
2t/2
∑2t−1k=0
e(2πi)ϕk |k〉
Question: Suppose ϕ may be expressed exactly asϕ = [0 · ϕ1ϕ2...ϕt ]. Suggest a way to retrieve the value of ϕ?
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
Claim 1: The final state of the first register in the circuit below isgiven by:
1
2t/2
(|0〉 + e(2πi)2t−1ϕ |1〉
)(|0〉 + e(2πi)2t−2ϕ |1〉
)...
(|0〉 + e(2πi)20ϕ |1〉
)= 1
2t/2
∑2t−1k=0
e(2πi)ϕk |k〉
Question: Suppose ϕ may be expressed exactly asϕ = [0 · ϕ1ϕ2...ϕt ]. Suggest a way to retrieve the value of ϕ?
Taking the inverse-fourier transform and measuring the value ofthe first register in the computational basis gives ϕ.
In general, we will show that the inverse Fourier transform has thefollowing behaviour:
12t/2
∑2t−1j=0 e(2πi)ϕj |j〉 |u〉 → |ϕ̃〉 |u〉
where |ϕ̃〉 denotes a state that is a good estimator for ϕ whenmeasured.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
In general, we will show that the inverse Fourier transform has thefollowing behaviour:
12t/2
∑2t−1j=0 e(2πi)ϕj |j〉 |u〉 → |ϕ̃〉 |u〉
where |ϕ̃〉 denotes a state that is a good estimator for ϕ whenmeasured.
Claim 2
It is sufficient to run the phase estimation technique witht = n + log
(2 + 1
2ε
)in order to obtain ϕ accurate to n bits with
probability at least (1− ε).
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Claim 2
It is sufficient to run the phase estimation technique witht = n + dlog
(2 + 1
2ε
)e in order to obtain ϕ accurate to n bits with
probability at least (1− ε).
Proof sketch
Let 0 ≤ b ≤ 2t − 1 be an integer such that b2t = [0 · b1...bt ] is the
best t bit approximation to ϕ that is less than ϕ. Let δ = ϕ− b2t
(which implies 0 ≤ δ ≤ 2−t).Claim 2.1: Applying the inverse Fourier transform on the firstregister in state 1
2t/2
∑2t−1k=0 e(2πi)ϕk |k〉 ends in the following state:
12t
∑2t−1k,l=0 e
−(2πi)kl
2t e(2πi)ϕk |l〉 .
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Claim 2
It is sufficient to run the phase estimation technique witht = n + dlog
(2 + 1
2ε
)e in order to obtain ϕ accurate to n bits with
probability at least (1− ε).
Proof sketch
Let 0 ≤ b ≤ 2t − 1 be an integer such that b2t = [0 · b1...bt ] is the
best t bit approximation to ϕ that is less than ϕ. Let δ = ϕ− b2t
(which implies 0 ≤ δ ≤ 2−t).Claim 2.1: Applying the inverse Fourier transform on the firstregister in state 1
2t/2
∑2t−1k=0 e(2πi)ϕk |k〉 ends in the following
state: 12t∑2t−1
k,l=0 e−(2πi)kl
2t e(2πi)ϕk |l〉 .Claim 2.2: Let αl be the amplitude of |(b + l) mod 2t〉. Then
αl = 12t
(1−e(2πi)(2tϕ−(b+l))
1−e(2πi)(ϕ−(b+l)/2t )
)= 1
2t
(1−e(2πi)(2tδ−l)
1−e(2πi)(δ−l/2t )
).
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Claim 2
It is sufficient to run the phase estimation technique witht = n + dlog
(2 + 1
2ε
)e in order to obtain ϕ accurate to n bits with
probability at least (1− ε).
Proof sketch
Let 0 ≤ b ≤ 2t − 1 be an integer such that b2t = [0 · b1...bt ] is the
best t bit approximation to ϕ that is less than ϕ. Let δ = ϕ− b2t
(which implies 0 ≤ δ ≤ 2−t).Claim 2.1: Applying the inverse Fourier transform on the firstregister in state 1
2t/2
∑2t−1k=0 e(2πi)ϕk |k〉 ends in the following
state: 12t∑2t−1
k,l=0 e−(2πi)kl
2t e(2πi)ϕk |l〉 .Claim 2.2: Let αl be the amplitude of |(b + l) mod 2t〉. Then
αl = 12t
(1−e(2πi)(2tϕ−(b+l))
1−e(2πi)(ϕ−(b+l)/2t )
)= 1
2t
(1−e(2πi)(2tδ−l)
1−e(2πi)(δ−l/2t )
).
Claim 2.3: Let e be the error parameter and let m be theoutcome of the measurement. Then
Pr[|m − b| > e] ≤ 1
2(e − 1).
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Claim 2
It is sufficient to run the phase estimation technique witht = n + dlog
(2 + 1
2ε
)e in order to obtain ϕ accurate to n bits with
probability at least (1− ε).
Proof sketch
Let 0 ≤ b ≤ 2t − 1 be an integer such that b2t = [0 · b1...bt ] is the
best t bit approximation to ϕ that is less than ϕ. Let δ = ϕ− b2t
(which implies 0 ≤ δ ≤ 2−t).Claim 2.1: Applying the inverse Fourier transform on the firstregister in state 1
2t/2
∑2t−1k=0 e(2πi)ϕk |k〉 ends in the following
state: 12t∑2t−1
k,l=0 e−(2πi)kl
2t e(2πi)ϕk |l〉 .Claim 2.2: Let αl be the amplitude of |(b + l) mod 2t〉. Then
αl = 12t
(1−e(2πi)(2tϕ−(b+l))
1−e(2πi)(ϕ−(b+l)/2t )
)= 1
2t
(1−e(2πi)(2tδ−l)
1−e(2πi)(δ−l/2t )
).
Claim 2.3: Let e be the error parameter and let m be theoutcome of the measurement. Then
Pr[|m − b| > e] ≤ 1
2(e − 1).
The claim follows by setting t = n + p and ε = 12(2p−1) .
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
The phase estimation protocol works when the second register isset to the eigenstate |u〉. In general, this may not be feasible.Observation: Any general state |ψ〉 may be written in terms ofthe eigenstates of U as
∑u cu |u〉.
Exercise: The phase estimation procedure takes state(|0〉)(
∑u cu |u〉) to
∑u cu |ϕ̃u〉 |u〉. If t = n + dlog
(2 + 1
2ε
)e,
then the probability of measuring ϕu accurate to n bits at the endof the phase estimation procedure is at least |cu|2(1− ε).
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation
Phase estimation
Suppose a unitary operator U has an eigenvector |u〉 with eigenvaluee2πiϕ. The goal is to estimate ϕ.
Phase estimation enables us to design quantum algorithms for theorder-finding and factoring problems.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum Computation: Order finding
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation → Order-finding
Given integers N > x > 0 such that x and N have no commonfactors, the order of x modulo N is defined to be the leastpositive integer r such that x r = 1 (mod N).Exercise: What is the order of 5 modulo 21?
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation → Order-finding
Given integers N > x > 0 such that x and N have no commonfactors, the order of x modulo N is defined to be the leastpositive integer r such that x r = 1 (mod N).Exercise: What is the order of 5 modulo 21? 6
Order finding
Given co-prime integers N > x > 0, compute the order of x modulo N.
Exercise: Is there an algorithm that computes the order of xmodulo N in time that is polynomial in N?
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation → Order-finding
Given integers N > x > 0 such that x and N have no commonfactors, the order of x modulo N is defined to be the leastpositive integer r such that x r = 1 (mod N).Exercise: What is the order of 5 modulo 21? 6
Order finding
Given co-prime integers N > x > 0, compute the order of x modulo N.
Exercise: Is there an algorithm that computes the order of xmodulo N in time that is polynomial in N? YesExercise: Is it an efficient algorithm?
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation → Order-finding
Given integers N > x > 0 such that x and N have no commonfactors, the order of x modulo N is defined to be the leastpositive integer r such that x r = 1 (mod N).Exercise: What is the order of 5 modulo 21? 6
Order finding
Given co-prime integers N > x > 0, compute the order of x modulo N.
Exercise: Is there an algorithm that computes the order of xmodulo N in time that is polynomial in N? YesExercise: Is it an efficient algorithm?Let L = dlog ne. The number of bits needed to specify theproblem is O(L). So, an efficient algorithm should have runningtime that is polynomial in L.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation → Order-finding
Order finding
Given co-prime integers N > x > 0, compute the order of x modulo N.
Consider the operator U that has the following behaviour:
U |y〉 ≡
{|xy (mod N)〉 if 0 ≤ y ≤ N − 1
|y〉 if N ≤ y ≤ 2L − 1
Exercise: Show that U is unitary.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation → Order-finding
Order finding
Given co-prime integers N > x > 0, compute the order of x modulo N.
Consider the operator U that has the following behaviour:
U |y〉 ≡
{|xy (mod N)〉 if 0 ≤ y ≤ N − 1
|y〉 if N ≤ y ≤ 2L − 1
Exercise: Show that U is unitary.Exercise: Show that the states defined by
|us〉 ≡1√r
r−1∑k=0
e−(2πi)skr
∣∣∣xk (mod N)⟩
are the eigenstates of U. Find the corresponding eigenvalues.
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information
Quantum ComputationPhase estimation → Order-finding
Order finding
Given co-prime integers N > x > 0, compute the order of x modulo N.
Consider the operator U that has the following behaviour:
U |y〉 ≡
{|xy (mod N)〉 if 0 ≤ y ≤ N − 1
|y〉 if N ≤ y ≤ 2L − 1
Exercise summary: Let |us〉 ≡ 1√r
∑r−1k=0 e
−(2πi) skr
∣∣xk (mod N)⟩
be an eigenstate of U. Then U |us〉 = e(2πi)sr |us〉
Ragesh Jaiswal, CSE, IIT Delhi COL863: Quantum Computation and Information