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m tained only with some probability. Whereas in the other
(QWC), and it combines the sub-waves in H into a
⊕ twoscenarios,aresulttoapartialmeasurementisalways
single Hilbert space H,
obtained and the wave function should be renormalized.
m The fundamental difference between duality computer
C(p1 ψ1 p ψ )= p ψ . (2)
m m i i andquantumcomputeris that dualitygates neednotbe
| i⊕···⊕ | i | i
Xi unitary. A quantum computer can not perform U1+U2,
Adualitycomputingprocessisdescribedinthefollowing it can only perform U1U2 operation.
Duality Mode in a Quantum Computer—Now
m m m
ψ p ψ p U ψ p U ψ . (3) we give a quantum computer simulation of the duality
i i i i i i i
| i→ Xi ⊕ | i → Xi ⊕ | i → Xi | i computer. To simulate an n-dubit duality computer, we
2
need an (n+1)-qubit quantum computer, andone qubit slits with different phases and so on. By adding more
is used as auxiliary qubit and n qubits are used as work auxiliary qubits, one can simulate a duality computer
qubits. Let’s make the following correspondence with more slits.
Wenowgivethemathematicaldescriptionforsuchdu-
ϕ κ ϕ 0 ,
u ality mode. Some of the result in Ref. [5] can be used
| i| i↔| i| i
ϕ κ ϕ 1 , (7) directly, though the results regarding QWD and QWC
d
| i| i↔| i| i
need modification. Denoting the set of duality gates on
namely, when the auxiliary qubit is in 0 (1 ), it resem-
H as (H), there are four theorems and corollaries re-
| i | i
blesadualitycomputersub-wavefromtheupper(lower) G
garding duality gates in Ref. [5] and we directly copy
slit. We ascribe the initial and finalwavefunction of the
them here. For proof of these results, see Ref. [5].
dualitycomputertothewavefunctionwhentheauxiliary
Theorem 1. The identity I is an extreme point of
H
is in state 0 . Thus, before the QWD, the state of dual-
(H).
| i
itycomputeris ϕ 0 ,aftertheQWD,thewave-function G
Corollary 2. The extreme points of (H) are pre-
| i| i
becomes G
cisely the unitary operators in H.
0 + 1 Thetheoremandcorollarytellusthatthedualitygate
ϕ | i | i, (8)
is unitary only when U0 = U1, and in particular, when
| i √2
U0 =U1 =I H, the duality gate is the identity gate.
namely, the QWD is simulated by a Walsch-Hadamard Let (H)be the set ofbounded linearoperatorsonH
andletBR+
transformation in the auxiliary qubit. Thus the wave- (H)bethepositiveconegeneratedby (H),
G G
function describes the n-bit quantum computer simulta- that is
neouslyintwoslits, 0 and 1 . Differentgateoperations
on different routes | cai n be| sii mulated using conditional R+ (H)= αA:A (H),α 0 . (11)
G { ∈G ≥ }
gates, as shown in Fig. 1.
Then the next theoremtells that any operatoron H can
|φ(cid:1) be simulated.
U0 U1 M Theorem 3. If dim H < , then (H)=R+ (H).
∞ B G
The next corollary is about normal operators.
|0(cid:1) Corollary4. IfdimH < ∞,thenA ∈B(H)isnormal
H H
ifandonlyifA=α p U whereα 0,p >0, p =
i i i ≥ i i i
1 and U are unitaryPoperators that mutually comPmute.
i
FIG. 1: An n-dubit duality computer can be simulated by
We now expound the significance of duality mode in a
an (n+1) qubit quantumcomputer. The Walsch-Hadamard
quantumcomputer. First,an(n+1)-qubitquantumcom-
gatescanbereplacedbyotherunitarygatestosimulatemore
puter can simulate an n-dubit duality computer. This
complicated slits. One can also use more qubits as auxiliary
allows a quantum computer to perform any operation
to simulate more slits.
in the n-qubit Hilbert space. Hence, classicalalgorithms
canbetranslatedintoquantumalgorithmsinthisduality
Then we have mode. This is significant because this is not a direct use
of a quantum computer as a classical computer using a
U0 ϕ 0 +U1 ϕ 1
| i| i | i| i. (9) qubit as a classicalbit, and the qubit resourcein duality
√2
modeisexponentiallysmall. Forinstanceforanunsorted
The quantum combiner operation is again a Walsch- database search problem with N = 2n items, a classical
Hadamardtransformation,afterwhichthewave-function computer needs at least n2n = Nn bits to express the
becomes database. However in a quantum computer running in
dualitymode,itneeds only(n+1)qubits toexpressand
U0+U1 U0 −U1
ϕ 0 + ϕ 1 . (10) manipulate the database. This also provides a natural
2 | i| i 2 | i| i
bridgebetweenclassicalandquantumcomputing. Itwas
We make a measurement on condition that the auxil- suggested that future quantum computer may be used
iary qubit is 0 . The probability to obtain a result is as special-purpose processor to perform task that needs
| i
the square of the norm of the wave function in 0 state, quantum acceleration, and then the result is returned to