C/CS/Phys C191 5 Criteria for qu comp, Universal gate sets 10/16/08Fall 2008 Lecture 16

1 Alternative 2-qubit gate realizations with spinsIn the last lecture we saw how to realize a CNOT gate with coupled spins, using frequency selective pulses.In practice, such frequency selective pulses require relatively long time to realize which has disadvantages(allowing more bad interactions with the environment that can destroy the quantum coherence). Experimen-talists implementing quantum gates with nuclear spins usually prefer to work with shorter pulses that thenpossess a broader ’bandwidth’, i.e., include many resonance frequencies and address mutliple transitions.See Stolze and Suter, Ch. 10.2.5 for an example of how the CNOT gate can be achieved in this situation bycombining spin rotations with free precession.

2 5 Criteria (DiVincenzo criteria) for achieving scalable quantum compu-tation

see additional pages at end of notes

3 Quantum GatesWe already had some simple examples of unitary transforms, or “quantum gates”. Here are most of thecommon ones you will encounter.

3.1 One-qubit gates:• Hadamard Gate.

H =1√2

(1 11 −1

)

H∣∣0〉 = 1√

2(∣∣0〉 + ∣∣1〉) = ∣∣+〉

H∣∣1〉 = 1√

2(∣∣0〉 − ∣∣1〉) = ∣∣−〉

The Hadamard Gate is one of the most important gates. Note thatH† = H – sinceH is real andsymmetric – andH2 = I .

In the complex planeH can be visualized as a reflection aroundπ/8, or a rotation aroundπ/4 followedby a reflection.

On the Bloch sphereH can also be visualized in several ways. One is a rotation ofπ/2 about they-axis, followed by reflection in the x-y plane (see Nielsen and Chuang, p. ). Another is a rotation ofπ about the axis(1/

√2,0,1/

√2) (Benenti, p. 111).

C/CS/Phys C191, Fall 2008, Lecture 16 1

Note the action ofH on larger number of qubits:

H⊗H∣∣00〉 ≡ H⊗2∣∣00〉 = ∣∣00〉+∣∣01〉+∣∣10〉+∣∣11〉√

22

H⊗n∣∣00.....0n〉 = 1√2n ∑2n−1x=0 ∣∣x〉

ThusH⊗n produces an equal superposition ofall computational basis states.

• Rotation Gate. This rotates in the complex plane byθ .

R=(

cosθ −sinθsinθ cosθ

)• NOT Gate, also known as bit flip gate, or X (Pauli X). This flips a bit from 0 to 1 and vice versa.

NOT =(

0 11 0

)• Phase Flip, also known as Z (Pauli Z).

Z =(

1 00 −1

)The phase flip is a NOT gate acting in the

∣∣+〉 = 1√2(∣∣0〉 + ∣∣1〉), ∣∣−〉 = 1√

2(∣∣0〉− ∣∣1〉) basis. Indeed,

Z∣∣+〉 = ∣∣−〉 andZ∣∣−〉 = ∣∣+〉 .

• General Phase Gate,Rz(δ ).

Rz(δ ) =(

1 00 eiδ

)ClearlyZ = Rz(π). There are several other special phase gates that are commonly used:S= Rz(π/2),T = Rz(π/4). The latter is sometimes referred to as theπ/8 gate.

S=(

1 00 i

)T ≡ π/8 =

(1 00 eiπ/4

)= eiπ/8

(e−iπ/8 0

0 eiπ/8

)• Phaseflips and bitflips are related by conjugation

Conjugation of X by H means premultiplying X by H−1 and postmultiplying it by H. ButH = H−1.

Claim: HXH = Z (Figure 1).

We can prove this by multiplying out the matrices, or by making use of the decomposition of H intoan X and a Z gate:

H =X +Z√

2

Then(X+Z√

2

)X

(X+Z√

2

)=

C/CS/Phys C191, Fall 2008, Lecture 16 2

[X+Z√

2

][X2+XZ√

2

]=[

X+Z√2

][I+XZ√

2

]=

XI+XXZ+ZI+ZXZ2 =

X+Z+Z+−X2 =

2Z2 = Z.

Conversely,HZH = X. Prove this for yourself.

• Any unitary operation on a single qubit can be constructed with various combinations of gates:

H,Rz(δ ), e.g.,

Rz(π/2+φ)HRz(θ)H∣∣0〉 = eiθ/2(cosθ/2∣∣0〉 +eiφ sinθ/2∣∣1〉)

H,X,T = Rz(π/4)X,Y,Z (Euler rotations)

3.2 Two-qubit gates:• Any one-qubit gate can be tensored with itself or another gate to make a two-qubit gate, as done above

for H⊗H. Such tensor products of one-qubit gates have no ability to generate entanglement and arereferred to as ‘local’ gates.

• Controlled Not (CNOT).

CNOT=

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

The first bit of aCNOT gate is the “control bit;” the second is the “target bit.” The control bit neverchanges, while the target bit flips if and only if the control bit is 1.

The CNOT gate is usually drawn as follows, with the control bit on top and the target bit on thebottom:

tdNote that(CNOT)2 = 1, i.e.,CNOT−1 = CNOT.

• SWAP

×

×

SWAP=

1 0 0 00 0 1 00 1 0 00 0 0 1

C/CS/Phys C191, Fall 2008, Lecture 16 3

3.3 n-qubit gates:• local n-qubit gates formed as tensor products of one-qubit gates, e.g.,H⊗n

• Toffoli gate

This is a 3-qubit generalization of the CNOT gate. The third, target, qubit is flipped iff both the firstand second qubits are in state 1 (Figure 2). You can check thatTOFF2 = 1.

ttdThe Toffoli gate can be decomposed into a combination of one-qubit and two-qubit gates. See Figure3.

3.4 Useful gate equivalences• SWAPequals 3 xCNOT (Figure 4).

Suppose we have two qubits in state∣∣y2,y1〉 :[

ab

]⊗

[cd

]= ac

∣∣00〉 +ad∣∣01〉 +bc∣∣10〉 +bd∣∣11〉Apply the firstCNOT: ac

∣∣00〉 +ad∣∣01〉 +bd∣∣10〉 +bc∣∣11〉Apply the secondCNOT:

ac∣∣00〉 +bc∣∣01〉 +bd∣∣10〉 +ad∣∣11〉

Apply the thirdCNot:

ac∣∣00〉 +bc∣∣01〉 +ad∣∣10〉 +bd∣∣11〉

= ca∣∣00〉 +cb∣∣01〉 +da∣∣10〉 +db∣∣11〉

=[

cd

]⊗

[ab

]The resulting state is

∣∣y1,y2〉 , i.e., the states of the two qubits have been swapped.• Control and target ofCNOT can be swapped by conjugating both qubits withH (Figure 5).

Check this yourself.

C/CS/Phys C191, Fall 2008, Lecture 16 4

4 Universality of Gate Sets4.1 ClassicalThe NAND gate is universal for classical computation. TheNAND gate is the result of applyingNOT toaANDb= a∧b = a ↑ b. See Figure 6.For any boolean function{0,1}n−→{0,1}, there is a circuit built ofNANDgates (possibly withFANOUT=copy) for that function. Note that neither of these gates are reversible.

In general, the circuit may require an exponential number 2n of gates. Functions which can be efficientlyevaluated require only a polynomial numbernc gates. Complexity theory categorizes the scaling of theresources, esp. the number of gates, with the number of bitsn. Provided the gate set is universal, thedistinction between functions which require exponentially large circuits and those which can be computedwith polynomial-size circuits does not depend on the chosen set of gates.

4.2 QuantumA setG of quantum gates is called universal if for anyε > 0 and any unitary matrixU on n qubits, there isa sequence of gatesg1, . . . ,gl from G such that‖U −Ugl · · ·Ug2Ug1‖ ≤ ε.HereUg is V ⊗ I , whereV is the unitary transformation onk qubits operated on by the quantum gateg,and I is the identity acting on the remainingn− k qubits. The operator norm is defined by‖U −U ′‖ =max|v〉unit vector‖(U −U ′)|v〉‖. (Recall that for a vectorw, ‖w‖=

√〈w

∣∣w〉.)Examples of universal gate sets include

• CNOT and all single qubit gates (continuous gates)

• CNOT, Hadamard, and suitable phase flips (continuous gates)

• CNOT, Hadamard,X andT (π/8) (discrete gates)

• Toffoli and Hadamard (discrete gates)

5 Approximating Unitary OperatorsLast time we defined a universal set of quantum gates. Now we consider the question of just how manygates are needed to effect an arbitrary quantum operation, or circuit?

An n-qubit gateU (a 2n×2n unitary matrix) has exponentially many parameters. So typically in general weneed exp(n) many gates to even approximateU .

The Solovay-Kitaev theorem says that, as a function ofε, the complexity of an approximation is only log2 1ε .This is rather efficient – the complexity as a function ofn is the problem.

Quantum computation may be regarded as the study of those unitary transformations onn qubits that canbe described by a sequence of polynomial inn quantum gates from a universal family of gates.U is “easy”(implementable) ifU ≈Ugk · · ·Ug1 for k = O(poly(n)). This definition doesn’t depend on our choice of a(finite) universal gate family, since any particular gate in one gate family can be well-approximated witha constant number of gates from another universal gate family. The constant factor does not affect thedistinction between polynomial- and exponential-size circuits.

C/CS/Phys C191, Fall 2008, Lecture 16 5

H HX Z

Figure 1:An X gate conjugated byH gates is aZ gate.

Figure 2:Toffoli gate, a 3-qubit double controlled NOT gate (bit c is flipped iff both a and b are 1).

Figure 3:A Toffoli gate can be decomposed into a circuit of 1- and 2-qubit gates. HereV =[

1 00 i

]=

Rz(π/2).

C/CS/Phys C191, Fall 2008, Lecture 16 6

Figure 4:A SWAPgate is three back to backCNOT gates with control and target qubits alternating.

H

H

H

H

Figure 5:Control and target qubits ofCNOT can be exchanged by conjugating withH on both qubits.

Figure 6:Classical NAND gate and its truth table.

C/CS/Phys C191, Fall 2008, Lecture 16 7

• a physical system providing a scalable collection ofqubits (need > 100 for significant speedup)

• ability to initialize qubits in a known starting state

• long decoherence time, much longer than the gateoperation time (Td/tg~10,000)

• a universal set of quantum gates

• individual qubit measurement capability

• ability to interconvert stationary/flying qubits

• ability to faithfully transmit flying qubits

|1>

|0>

DiVincenzo 1995, 2001

Requirements for Quantum Information Processing

some candidate quantum bits (‘qubits’)

H13CCl3nuclear spins

atoms in optical lattice

electron spins

superconducting flux loops

• photons

• atom/photon cavity QED

• trapped ions

• trapped neutral atoms

• NMR

• e spin in quantum dots

• e/n spins in bulk semiconductors

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Atoms EM modes

Candidate physical implementations (partial list)

• superconductors

(charge, flux, or phase)

QC Roadmap, http:/qist.lanl.govVersion 1.0, December 1 2002Version 2.0, April 2, 2004

• many-particle excitations in

topological QFTs

Visions and Realities

7-qubit molecule used to factor 15 by Shor’s algorithm: I. Chuang et al., Nature 414, 883 (2001)

Proposal for a Si-based quantum computer: B. Kane, Nature, 393 133 (1998)

NMR Quantum Computing:

Experimental realization of an

order-finding algorithm with an NMR QC

L. Vandersuypen et al., PRL 85, 5452 (2000)

A Cat-State Benchmark on

Seven Bit QC

E. Knill et al., quant-ph/9908051

7-qubit molecule used to factor 15 by Shor’s algorithm: I. Chuang et al., Nature 414, 883 (2001)