introduction to quantum computation - asz.ink€¦ · 9/47 postulates of quantum computation 1.a...

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Introduction toQuantum Computation

March, 2018

Alan [email protected]

imec-COSIC, KU Leuven

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Table of Contents

1 Introduction

2 Quantum Computation

3 Shor’s Algorithm

4 Grover’s Algorithm

5 HHL Algorithm

6 Quantum Finance

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Introduction

1 IntroductionApplicationsResources




5 HHL Algorithm

6 Quantum Finance

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Applications of Quantum Computers

• simulate QM for science

• simulate QM for industry• pharmaceuticals• nanomaterials• circuits ⇒ bootstrapping potential?

• break crypto

• prove crypto

• do crypto

• optimization / machine learning

• big data

• finance (?)

• blockchain

• money / identity

• NP-complete problems (depends ...)

• in general: small data + big complexity

}my field

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Resources

• bible • poetry

• zoo• NIST Quantum Algorithm Zoo:https://math.nist.gov/quantum/zoo/

• lecture notes• Preskill: http:

//www.theory.caltech.edu/%7Epreskill/ph219/index.html

• De Wolf: https://homepages.cwi.nl/~rdewolf/qcnotes.pdf

https://math.nist.gov/quantum/zoo/

http://www.theory.caltech.edu/%7Epreskill/ph219/index.html

http://www.theory.caltech.edu/%7Epreskill/ph219/index.html

https://homepages.cwi.nl/~rdewolf/qcnotes.pdf

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Quantum Computation

1 Introduction

2 Quantum ComputationPostulatesExampleModes of ComputationPostulates Probabilistic ComputationInterferenceDensity OperatorEPR Paradox and Bell-CHSH InequalityQuantum Gates



5 HHL Algorithm

6 Quantum Finance

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Postulates of Quantum Computation

1. A quantum system is fully defined by its state |ψ〉 ∈ H where

H ⊂ C2k is a Hilbert space of unit-length vectors, i.e.,‖|ψ〉‖22 = |ψ〉∗T|ψ〉 = 〈ψ|ψ〉 = 1.

2. Any valid computation is a unitary transformationT : H → H : |ψ〉 7→ T |ψ〉 of the state and can be described by a

unitary matrix T ∈ C2k×2k such that T ∗TT = I.

3. The composition of two quantum states |ψ〉 ∈ H1 ⊂ C2k and

|φ〉 ∈ H2 ⊂ C2` is described by their tensor product:

|ψφ〉 = |ψ〉 ⊗ |φ〉 ∈ H1 ⊗H2 ⊂ C2k+`

.

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Tensor Product

Inner product: (a1a2

)T(b1b2

)= a1b1 + a2b2

Outer product: (a1a2

)(b1b2

)T

=

(a1b1 a1b2a2b1 a2b2

)

Tensor product: (a1a2

)⊗(b1b2

)=

a1b1a1b2a2b1a2b2

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Postulates of Quantum Computation

1. A quantum system is fully defined by its state |ψ〉 ∈ H where

H ⊂ C2k is a Hilbert space of unit-length vectors, i.e.,‖|ψ〉‖22 = |ψ〉∗T|ψ〉 = 〈ψ|ψ〉 = 1.

2. Any valid computation is a unitary transformationT : H → H : |ψ〉 7→ T |ψ〉 of the state and can be described by a

unitary matrix T ∈ C2k×2k such that T ∗TT = I.

3. The composition of two quantum states |ψ〉 ∈ H1 ⊂ C2k and

|φ〉 ∈ H2 ⊂ C2` is described by their tensor product:

|ψφ〉 = |ψ〉 ⊗ |φ〉 ∈ H1 ⊗H2 ⊂ C2k+`

.

4. Measurement M happens with respect to an orthogonal basis|b1〉, |b2〉, . . . , |bk〉 ∈ H and fixes the state to one basis vectorM(|ψ〉) = |bi〉 with probability 〈ψ|bi〉〈bi|ψ〉.

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Quantum Computation Example

(4.) Use the basis |0〉, |1〉 for H.

(1.) A qubit |ψ〉 ∈ H is described by a vector (α, β) ∈ C2 such that|ψ〉 = α|0〉+ β|1〉.

(4.) Measuring yields |0〉 with probability α∗α and |1〉 with β∗β.

. Let |ψ〉, |φ〉 ∈ H be two qubits set to zero, i.e., |ψ〉 = |φ〉 = |0〉.(3.) The composite system is described by

|ψφ〉 = |ψ〉 ⊗ |φ〉 = α|00〉+ β|01〉+ γ|10〉+ δ|11〉 with e.g. α = 1and β = γ = δ = 0.

(2.) Apply the unitary transformation

T =

12

√2 0 0 − 1

2

√2

0 1 0 00 0 1 0

12

√2 0 0 1

2

√2

to |ψφ〉 ∼=

αβγδ

.

. Result: T |ψφ〉 = 12

√2|00〉+ 0|01〉+ 0|10〉+ 1

2

√2|11〉.

(4.) Measuring yields |00〉 with probability 12 and |11〉 with 1

2 .

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Modes of Computation

P = 1

deterministic

0.2 0.8

0.30.7 0.6

0.4

P = 0.2× 0.7+ 0.8× 0.6= 0.62

probabilistic

√0.2

√0.8

−√0.6

√0.6 √

0.4√0.4

P =(−√0.2√0.6 +

√0.8√0.6

)2

≈ 0.352 ≈ 0.12

quantum nondeterministic

P = 1

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Postulates of Probabilistic Computation

1. A quantum probabilistic system is fully defined by its state (or

probability distribution) ψ ∈ [0; 1]2k

having `2 `1-norm equal to 1

2. Any valid computation is a transformation T : ψ 7→ Tψ of the state

and can be described by a unitary stochastic matrix T ∈ [0; 1]2k×2k

such that all rows and columns sum to 1.

3. The composition of two states ψ ∈ [0; 1]2k

and φ ∈ [0; 1]2`

is

described by their tensor product: ψ ⊗ φ ∈ [0; 1]2k+`

.

4. Measurement Sampling (denoted by M) happens with respect to an

orthogonal basis b1, b2, . . . , bk ∈ [0; 1]2k

and fixes the state to onebasis vector M(ψ) = bi with probability 〈ψ|bi〉〈bi|ψ〉 bTi ψ.

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Quantum Computation in 2 Easy Steps

0. Start with probabilistic computation.

1. Compute with probability distributions as opposed to samples.• sample afterwards

2. Use continuous transitions as opposed to discrete ones.• the nth root of a computation is always well defined• −→ complex numbers, `2 norm• fixedness of quantum circuits is a minor detail• use unitary local transformations

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Interference

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Interference for Quantum Computers

|0〉 U U |1〉 with U =

(1√2

−1√2

1√2

1√2

)U |0〉 = 1√

2|0〉+ 1√

2|1〉 and U2|0〉 = |1〉

|0〉

observable outcomes:

|0〉

|0〉 |1〉

|1〉

−|0〉 |1〉

• destructive interference is the source of all quantum wierdness

• quantum algorithm design = engineering interference

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Density Operator

• equivalent characterization of quantum computation

• in terms of state matrices instead of state vectors via |ψ〉〈ψ| ∼= |ψ〉• useful for

• probability distributions over quantum states• partial quantum systems (i.e., ignoring certain qubits)

• ultimately not necessary to understand quantum computation

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EPR Paradox

• How to define existence?

• Einstein: something exists iff• it has a state• that has a value• that can be exposed through experimental measurement• while being independent of measurement.

• Einstein: whatever QM describes, is not something that exists.

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EPR Paradox

• Einstein, Podolsky, Rosen:• Take two electrons in state | ↑ ↑〉+ | ↓ ↓〉;• send one to Alice and one to Bob.• If Alice measures | ↑〉, she knows instantly Bob will measure | ↑〉.• If Alice measures | ↓〉, she knows instantly Bob will measure | ↓〉.• But before Alice measures, Bob has 50-50 chance.• So: Alice influences Bob’s probabilities faster than the speed of light.• So: QM is not reconcilable with GR.

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Bell’s Theorem

• Bell identifies assumptions in EPR paradox:• realism: all observable behavior of an object is completely

determined by its state• locality: an object’s state is affected only by its immediate

surroundings• “local realism” / “local hidden variables theory”

• Bell:• local realism is testable!• there are functions (inequalities) of experiment outcomes (variables)

that hold for all probability distributions, but not for QM• Bell inequality holds for all experiments =⇒ local realism is true• some experiments violate Bell inequality =⇒ local realism is false

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Bell-CHSH Inequality

• Alice and Bob receive one qubit each from | − 1,−1〉+ |+ 1,+1〉• Alice flips a coin and measures with Q or R

• Bob flips a coin and measures with S or T

• measurement outcomes: +1 or −1

• If Q,R, S, T measure different aspects of the same state vector, theymust follow a joint probability distribution

• If Q,R, S, T follow a classical probability distribution, then:

E[QS] + E[RS] + E[RT ]− E[QT ] ≤ 2

(Bell-)CHSH Inequality ↑• experiments violate Bell inequalities

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Bell-CHSH Inequality| − 1〉

|+ 1〉Q

S

RT

• Let |ϕ〉 = 1√2|+ 1,+1〉+ 1√

2| − 1,−1〉

• Let L =

(cos 22.5◦ −sin 22.5◦

sin 22.5◦ cos 22.5◦

), i.e., CCW rotation by 22.5◦

• Q = MA, S = MB ◦ (I⊗L), R = MA ◦ (L2⊗ I), T = MB ◦ (I⊗L3)

• E[QS] = Pr[A = B|Q,S]− Pr[A 6= B|Q,S]

= ‖〈+1,+1|(I ⊗ L)|ϕ〉‖2 + ‖〈−1,−1|(I ⊗ L)|ϕ〉‖2 − ‖〈+1,−1|(I ⊗L)|ϕ〉‖2 − ‖〈−1,+1|(I ⊗ L)|ϕ〉‖2 = (cos 22.5◦)2 − (sin 22.5◦)2 =E[RT ] = E[RS]

• E[QT ] = Pr[A = B|Q,T ]− Pr[A 6= B|Q,T ]

= ‖〈+1,+1|(I⊗L3)|ϕ〉‖2 +‖〈−1,−1|(I⊗L3)|ϕ〉‖2−‖〈−1,+1|(I⊗L3)|ϕ〉‖2 − ‖〈+1,−1|(I ⊗ L3)|ϕ〉‖2 = (sin 22.5◦)2 − (cos 22.5◦)2

• E[QS] + E[RS] + E[RT ]− E[QT ] = 4(cos 22.5◦)2 − 4(sin 22.5◦)2

≈ 2.82 > 2⇒ QM violates Bell-CHSH inequality

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Quantum Gates

• CNOT: |x, y〉 7→ |x, x⊕ y〉•⊕

1 0 0 00 1 0 00 0 0 10 0 1 0

• Hadamard: |x〉 7→ (−1)x 1√

2|x〉+ 1√

2|¬x〉 H 1√

2

(1 11 −1

)• T (AKA. π/8) : |x〉 7→ exiπ/4|x〉 T

(1 00 eiπ/4

)• S (AKA. phase) : |x〉 7→ ix|x〉 S

(1 00 i

)• C-U: |x〉 ⊗ |y〉 7→ |x〉 ⊗ Ux|y〉

•U

|0〉〈0| ⊗ I + |1〉〈1| ⊗ U

• Measurement

• { CNOT, H, T } is a universal gate set

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Shor’s Algorithm

1 Introduction


3 Shor’s AlgorithmGenericInteger FactorizationHidden Subgroup ProblemSimon’s Algorithm vs. Crypto


5 HHL Algorithm

6 Quantum Finance

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Shor’s Algorithm: Generic

• first quantum algorithm for real world problem

• breaks RSA ... and DLOG ... and ECC

• reduces factorization of n = pq to order finding

• order ϕ(n)⇔ aϕ(n) = 1 modn

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Quantum Fourier Transform

• Classical Fourier Transform• x = (x0, . . . , xN−1)T

• y = (y0, . . . , yN−1)T with yj =∑N−1k=0 e

−2iπjk/Nxk• complexity: O(N logN)

• Quantum Fourier Transform• |x〉 =

∑N−1j=0 xj |j〉 i.e., log2N qubits

• |y〉 =∑N−1j=0 yj |j〉 with yj =

∑N−1k=0 e

−2iπjk/Nxk• complexity: O(logN log logN)

time domain

p

Fourier domain

1/p

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DFT Example

• factorize n = 35

• x = 2, b = 32 −→ order r = 12

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

|a〉 |QFT(a)〉

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Hidden Subgroup Problem

Definition

Let G,+ be a group with subgroup H. Let f : G→ {0, 1}∗ be afunction that produces the same image iff its inputs are from the samecoset of H, i.e.,

∀g1, g2 ∈ G . g1 − g2 ∈ H ⇔ f(g1) = f(g2) .

The Hidden Subgroup Problem (HSP) is to find a generating set of Hgiven oracle access to f .

• factorization: G = Z,+ and H = ϕ(n)Z,+

• discrete logarithm: G = Z× Z,+ and H = Z(

1x

),+

• Simon’s problem: G = {0, 1}k,⊕ and H = {0, p},⊕

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Simon vs. Even-Mansour

• Even-Mansour construction:

Pm c

k k

• given quantum access to the circuit C|m〉 = |c〉 and given P ; find k

• solution1:

P

k k

P

• f(x) = P (x⊕ k)⊕ k ⊕ P (x)

• f(x⊕ k) = P (x)⊕ k ⊕ P (x⊕ k)

• period is k!1H. Kuwakado, M. Morii. “Security on the Quantum-type Even-Mansour Cipher”

IEICE 2012

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Grover’s Algorithm

1 Introduction



4 Grover’s AlgorithmGenericExact DescriptionComplexityAmplitude AmplificationDrawbacks

5 HHL Algorithm

6 Quantum Finance

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Grover’s Algorithm: Generic

• Let F : {0, 1}k → {0, 1} with 1 input a satisfying F (a) = 1

(and F (b) = 0 for all b 6= a)

• task: given oracle access to F , find a

• captures generic search, optimization, preimage-search, NP-completeproblems, ...

• best classical algorithm: exhaustive search — O(2k)

• quantum oracle access: F|a, c〉 7→ |a, c⊕ F (a)〉• best quantum algorithm: Grover — O(2k/2)

• polynomial speedup — nice, but no holy grail

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Grover’s Algorithm: Elements

• A — uniform sampler

• A|0k〉 = 1√2k

∑2k−1i=0 |i〉 = |Ψ〉

• Apply a Hadamard gate to each qubit

• working plane• the vectors |a〉 = |Ψ1〉 and |Ψ〉 span a plane• let |Ψ0〉 lie in this plane, perpendicular to |a〉

• S0 — mirror about |0k〉

• S0|x〉 =

{−|x〉 ifx 6= 0|x〉 else

• compute |x, q〉 7→ |x, q ⊕ (x 6= 0)〉 and keep |q〉 = 1/√

2(|0〉 − |1〉)• 1/√

2|x〉(|0〉 − |1〉) 7→ 1/√

2|x〉(|0〉 − |1〉) (if x = 0)• 1/√

2|x〉(|0〉 − |1〉) 7→ 1/√

2|x〉(|1〉 − |0〉) = −1/√

2|x〉(|0〉 − |1〉)(otherwise)

• Sχ — mirror about |Ψ0〉

• Sχ|x〉 =

{−|x〉 ifF (x)|x〉 if¬F (x)

• compute |x, q〉 7→ |x, q ⊕ F (x)〉 and keep |q〉 = 1/√

2(|0〉 − |1〉)

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Grover’s Algorithm: Complexity

|Ψ1〉

|Ψ0〉

|Φ〉

Sχ|Φ〉

|Ψ〉

Q|Φ〉

δ

θ

sin δ = 〈Ψ1|Ψ〉

δ ≈ (· · · 0 1 0 · · · )(· · · 1/√

2k · · · )T = 1/√

2k

#itr = π/2−θinit2δ = π/2−δ

2δ ≈ π4 2k/2

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Amplitude Amplification

|Ψ1〉

|Ψ0〉

|Φ〉

Sχ|Φ〉

|Ψ〉

Q|Φ〉

δ

θ

Q: what if #A = #{x |F (x) = 1} > 1 ?

Q: what if A is better than uniform?

A: no problem

for uniform A and #A > 1 holds:

#itr ∼ O(√

2k/#A)

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Drawbacks of Grover’s Algorithm

• quantum oracle access to F• expensive when F depends on big data

• no parallelism• partition search space• classically: 2× faster• quantumly:

√2× faster

• no meet-in-the-middle• requires lots of quantum memory

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HHL Algorithm

1 Introduction




5 HHL AlgorithmGenericHamiltonian SimulationPhase EstimationMechanics

6 Quantum Finance

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HHL: Generic Description

• Harrow, Hassadim, Lloyd

• exponential speed-up

• solve large + sparse linear system

• sledgehammer waiting for killer application

• lots of fine print

• Given A ∈ CN×N ,b = (b1, . . . , bN )T ∈ CN ,find x ∈ CN s.t. Ax = b• A is Hermitean: A∗T = A and invertible• A is sparse: ≤ s nonzero entries per row• b is constructable as |b〉 =

∑Ni=1 bi|i〉

• superdense ↑ requires O(logN) qubits

• output |x〉 =∑Ni=1 xi|i〉

• measure via 〈x|M |x〉 for some M

• complexity HHL: O(log(N)s2κ2/ε)• condition number κ = max eigA

min eigA• success probability ε

• classical complexity: O(Ns√κ log(1/ε)) (for positive definite A)

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HHL: Mechanics• construct superdense vector |0〉 7→ |b〉• Hamiltonian simulation |b〉 7→

∑k e

iA2k |b〉• write in eigenbasis |b〉 =

∑N−1`=0 β`|u`〉

• phase est. + Hamiltonian sim.: |0,b〉 7→∑N−1`=0 β`|φ`, u`〉

• φ` are eigenvalues of A• invert |φ`〉 7→ φ−1` |φ`〉

• add ancilla qubit |φ`〉 ⊗ |0〉• rotate by C/φ` radians|φ`〉 ⊗ |0〉 7→ |φ`〉 ⊗ (sin(C/φ`) |1〉+ (cos(C/φ`) |0〉)

≈ |φ`〉 ⊗ ( Cφ`|1〉+

√1− C2

φ2`|0〉)

• post-select for ancilla |1〉• hide normalizing constant• nonzero probability of failure

• undo phase estimation

N−1∑`=0

β`φ−1` |φ`, u`〉 7→

N−1∑`=0

β` φ−1` |0, u`〉 = |0〉 ⊗A−1|b〉 = |0〉 ⊗ |x〉

• measure aspect M of |x〉 via 〈x|M |x〉

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Quantum Finance

1 Introduction




5 HHL Algorithm

6 Quantum FinancePath Integral / Quantum Walk / Black-ScholesClustering

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Applications of Quantum Computers toQuantum Finance

• big data

• generic optimization

• small data + complex problem

• interference

• superdense operations + delayed sampling

• in particular:

× dynamic portfolio optimization? scenario analysisX option pricingX clustering

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Random and Quantum Walks

• Black-Scholes equation ∂V∂t + 1

2σ2S2 ∂2V

∂S2 + rS ∂V∂S − rV = 0• models price of (European style) options

• equivalent to heat equation ∂u∂τ = 1

2σ2 ∂2u∂x2

• special case of diffusion equation

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Quantum Walk

• path integral formulation

• faster spreading

• interference

f( )

• function-of-distribution (avoids sampling)

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Clustering

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x1

-1.5

-1

-0.5

0

0.5

1

1.5

x2

• via Kernel Principal Component Analysis:• data: (~p0, . . . , ~pm−1) ∈ Rn×m• kernel matrix K ∈ Rm×m with ki,j =

(ϕ(~pi) · ϕ(~pj)

)• principal eigenvector ~v0• color pi red if ~vT0K[:,i] < 0; and blue otherwise

• kernel trick: ϕ : Rn → R` with `≫ nbut ki,j =

(ϕ(~pi) · ϕ(~pj)

)is easy to compute

e.g. ki,j = exp(− ‖~pi−~pj‖

2

σ2

)• quantum speedup?

• how about |ϕ(~pi)〉? ⇒ more allowable kernel functions

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Last Slide

1 Introduction




5 HHL Algorithm

6 Quantum Finance

?

[email protected]

introduction to quantum computation - asz.ink€¦ · 9/47 postulates of quantum computation 1.a...

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