Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
NowtheUniversalityTheorem,whichrepresentsanamalgamationofseveralresultsthatsuits
theneedsofthisarticle,maybestated. Inparticular,itincorporatestheSolovay–KitaevTheorem,
whichprovidesaboundonthesizeofanapproximatingcircuitasafunctionoftheaccuracy.
Theorem 1 (Universality Theorem). Let Φ be an arbitrary quantum operation from n qubits to m
>
qubits. Then for every ε 0 there exists a quantum circuit Q with n input qubits and m output
qubits such that δ(Φ,Q) < ε. Moreover, for fixed n and m, the circuit Q may be taken to satisfy
size(Q) = poly(log(1/ε)).
9
X Y X Y
1 1 1 1
X Y X Y
2 H 2 2 H 2
X X W
3 Tr 3 1
0 Tr Z 1 W 2
| i
Figure5: Ageneralquantumcircuit(left)anditsunitarypurification(right).
NotethatitisinevitablethatthesizeofQisexponentialinnandmintheworstcase[70]. Further
detailsonthefactscomprisingthistheoremcanbefoundinNielsenandChuang[84]andKitaev,
Shen,andVyalyi[68].
III.3 Unitarypurifications of quantumcircuits
Theconnectionbetweenthegeneralandunitaryquantumcircuitscanbeunderstoodthroughthe
notion of a unitary purification of a general quantum circuit. This may be thought of as a very
specificmanifestationoftheStinespringDilationTheorem[95],whichimpliesthatgeneralquantum
operationscan berepresentedbyunitaryoperationsonlarger systems. Itwasfirstappliedtothe
quantumcircuitmodelbyAharonov,Kitaev,andNisan[10],whogaveseveralargumentsinfavor
of the general quantum circuit model over the unitary model. The term purification is borrowed
from the notionof a purification of a mixed quantumstate, as the processof unitary purification
forcircuitsissimilarinspirit. Theuniversalgatedescribedintheprevioussectionhastheeffectof
makingthenotionofaunitarypurificationofageneralquantumcircuitnearlytrivialatatechnical
level.
Suppose that Q is a quantum circuit taking input qubits (X ,...,X ) and producing output
1 n
qubits(Y ,...,Y ),andassumethereare kancillary gatesand l erasuregatesamongthegatesof
1 m
Q to be labelled in an arbitrary order as G ,...,G and K ,...,K , respectively. A new quantum
1 k 1 l
circuit R may then be formed by removing the gates labelled G ,...,G and K ,...,K ; and to
1 k 1 l
accountfortheremovalofthesegatesthecircuit Rtakeskadditionalinputqubits(Z ,...,Z )and
1 k
producesl additionaloutputqubits(W ,...,W ). Figure5illustratesthisprocess. Thecircuit Ris
1 l
said to be aunitary purification of Q. It is obvious that R is equivalent to Q, providedthequbits
(Z ,...,Z ) are initially set to the 0 state and the qubits (W ,...,W ) are traced-out, or simply
1 k 1 l
| i
ignored,afterthecircuitisrun—forthisispreciselythemeaningoftheremovedgates.
Despite the simplicity of this process, it is often useful to consider the properties of unitary
purificationsofgeneralquantumcircuits.
III.4 Oraclesinthe quantumcircuit model
Oracles play an important, and yet uncertain, role in computational complexity theory; and the
situationisnodifferentinthequantumsetting. Severalinterestingoracle-relatedresults,offering
someinsightintothepowerofquantumcomputation,willbediscussedinthisarticle.
Oraclequeriesarerepresentedinthequantumcircuitmodelbyaninfinitefamily
R : n N
n
{ ∈ }
10
x x
 
| i | i
 
 
 
 H R 6 H 
y  H H  y f(x)
| i | ⊕ i
 
 
H H
 
 1 H Tr 
| i
Figure6: TheBernstein–Vaziranialgorithmallowsamultiple-bitquerytobesimulatedbyasingle-
bit query. In the example pictured, f : Σ3 Σ3 is a given function. To simulate a query to this
→
function, the gate R is taken to be a standard oracle gate implementing the predicate A(x,z) =
6
f(x),z ,for , denotingthemodulo2innerproduct.
h i h· ·i
of quantum gates, one for each possible query length. Each gate R is a unitary gate acting on
n
n+1qubits,withtheeffectoncomputationalbasisstatesgivenby
R x,a = x,a A(x) (1)
n
| i | ⊕ i
for all x Σn and a Σ, where A is some predicate that represents the particular oracle under
∈ ∈
consideration. Whenquantumcomputationsrelativetosuchanoraclearetobestudied,quantum
circuits composedofordinaryquantumgatesaswellastheoraclegates R areconsidered;the
n
{ }
interpretationbeingthateachinstanceofR insuchacircuitrepresentsoneoraclequery.
n
It is critical to many results concerning quantum oracles, as well as most results in the area
of quantum query complexity, that the above definition (1) takes each R to be unitary, thereby
n
allowingthesegatestomakequeries“insuperposition”. Insupportofthisseeminglystrongdef-