Table Of ContentA Robust Quantum Random Access Memory
Fang-Yu Hong,1 Yang Xiang,2 Zhi-Yan Zhu,1 Li-zhen Jiang,3 and Liang-neng Wu4
1Department of Physics, Center for Optoelectronics Materials and Devices,
Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China
2School of Physics and Electronics, Henan University, Kaifeng, Henan 475004, China
3College of Information and Electronic Engineering,
Zhejiang Gongshang University, Hangzhou, Zhejiang 310018,China
4College of Science, China Jiliang University, Hangzhou, Zhejiang 310018, China
(Dated: January 12, 2012)
n
A “bucket brigade” architecture for a quantum random memory of N = 2 memory cells needs
n(n+5)/2timesofquantummanipulationoncontrolcircuitnodespermemorycall. Herewepropose
2 a scheme, in which only average n/2 times manipulation is required to accomplish a memory call.
1 This scheme may significantly decrease the time spent on a memory call and the average overall
0 error rate per memory call. A physical implementation scheme for storing an arbitrary state in a
2
selected memory cell followed byreading it out is discussed.
n
a PACSnumbers: 03.67.Lx,03.65.Ud,89.20.Ff
J Keywords: quantum randommemory,bucketbrigade,microtoroidalresonator
1
1
A random access memory (RAM) is a fundamental cuit only if a successful quantum manipulation has been
] computing device, in which information (bit) can be performedonthe trits,leadingtothe timesofmanipula-
h stored in any memory cell and be read out at discretion tions on the nodes N =n(n+5)/2 per memory call for
c
p [1,2]. ARAM is made upofaninput addressregister,a 2n memorycells,wherenisthenumberofbitsinthead-
-
t dataregister,anarrayofmemorycells,andacontrolling dress register. Here we present a QRAM scheme, where
n
circuit. A unique address is ascribed to each memory thequantumobjectineverynodehaveonlytwopossible
a
u cell. When the address of a memory cell is loaded into states |lefti and |righti. On average the times of quan-
q the address register, the memory cell is selected and the tum manipulations onthe nodes per memorycallcanbe
[ informationinthedataregistercanbestoredinitorthe reduced to N = n/2, significantly decreasing both the
c
informationof the cellcan be readout to the data regis- decoherence rate and the time spent on a QRAM call.
1
ter. Likeits classiccounterpart,quantumrandomaccess A physical implementation for information storage and
v
0 memory (QRAM) is the building block of large quan- read-out on a QRAM is presented.
5 tum computers. A QRAM is a RAM working in a way The mainidea is shownin Fig.1. The N memorycells
2 with quantum characteristic: the address and data reg-
are positioned at the end of a bifurcation control circuit
2 isters are comprised of qubits instead of bits, and every
with n = log N levels. At each node of the control cir-
. 2
1 node of the controlling circuit is composed of a quan- cuit there is a qubit with two states |lefti and |righti.
0 tum object. When the address state is in superposition,
Thestateofthejthqubitintheaddressregistercontrols
2 P α |x i, the read-out operation gives the output state
i i i which route to follow when a signal arrives at a node in
1
P α |q i in the data register, where |q i is the quan-
: i i i i thejthlevelofthecircuit: ifthenodequbitis|0i,theleft
v tum information stored in the memory cell i associated path is chosen; if it is |1i, the right path is chosen. For
i with the address |x i. Quantum random access memo-
X i example,an addressregister|001imeans that left at the
ries storing classic data can exponentially speed up the
0thlevel,leftatthenext,andrightatthesecond. Illumi-
r
pattern recognition [3–6], discrete logarithm [7, 8], and
a natedbyacontrollaserpulseanodequbitinstate|lefti
quantum Fourier transform, and quantum searching on
willflip to |rightiif the incoming addressqubitis |1i, or
a classical database [9]. A general QRAM is an indis-
remain in |lefti if the address qubit is |0i. Without the
pensable for the performance of many algorithms, such
controlpulse, a node qubit in |lefti (|righti)will deviate
as quantum searching [10], element distinctness [11, 12],
any incoming signal along the left(right) side route.
collisionfinding [13],generalNAND-tree evaluation[14],
First,allthe node qubits areinitialized instate |lefti.
and signal routing [15].
Then the first qubit of the address register is dispatched
In a seminal paper [15, 16], Giovannetti et al. (GLM) through the circuit. At the first node, the address qubit
proposed a promising bucket-brigade architecture for incursaunitarytransformationU onthenodequbitwith
QRAMs, which exponentially reduce the requirements the help of a control pulse Ω(t): U|0i|lefti = |0i|lefti
for a memory call. However, in GLM scheme, n times andU|1i|lefti=|0i|righti. Nextthesecondqubitofthe
of quantum unitary transformations per memory call is address register is dispatched through the circuit, follow
requiredto turnone quantum trit initialized in |waiti in left or right route relying on the state of the first node
eachnodeofthecontrolcircuitinto|leftior|righti,and qubit, and arrives at one of the two nodes on the sec-
all flying qubits including address qubit and bus qubit ond level of the circuit. The node qubit illuminated by
canpassthroughanarbitrarynodeofthecontrollingcir- the controlpulse will makea correspondingstate change
2
according to the state of the second address qubit, and data register 0 0 1
address register
so on. Note that the ith control pulse Ω(t) address all
the nodes of the ith level control circuit simultaneously. left
After all the n qubits of the address register have gone
through the whole circuit, a unique path of n qubits has left left
been singled out from the circuit (see Fig.1). Subse-
quently, a single photon is sent along the selected path right left left left
to single out a memory cell. After that an arbitrary un-
known state in the data register can be transferred to
memory cells
the selected memory cell along the selected path, or the
state of the selected memory cell can be read out to the
data register along the path with black squares in Fig.1.
FIG. 1. (color online). Schematics for a quantum random
Finally, all the node qubits are reset to |lefti for a next
access memory. In each node of the binary control circuit, a
memory address.
qubit in |righti(|lefti) routes the approaching signals right
Becausethe stateofa node qubit |leftiwillnotbe af- (left). A single photon can excite the qubit from |lefti to
fected by the controlpulse illuminating should the qubit |righti with the aid of a classical impedance-matched con-
of the address register be in state |0i, on average there trol field Ω(t). Here the third level memory cell |001i is ad-
are n/2 node qubits will flip to |righti in each memory dressed through the selected path marked with red circles.
The read-out state is transferred to a data register along the
call. This means that on average only n/2 times of con-
path marked with black squares.
trol manipulations are really performed in each memory
call. As a result, the mean comprehensive error rate per
1
memory address is nǫ/2 = log Nǫ with the assumed
2 2
error rate ǫ per node qubit flip event. In contrast, the atom does not affected by the control pulse illuminating
GLM scheme requires n times state flip for a memory and remains in |lefti [19].
call. Inaddition,inthe GLM schemeaphotonmaypass TheswitchfunctionofanodequbitinaQRAMcanbe
through a node only when a control pulse is applied on realized as follows: first, the address qubit is encoded as
the quantum trit, resulting the overalltimes of quantum α|0i +β|1i withFockstate|ni (n=0,1)andarbitrary
p p p
manipulations on the quantum trits per memory call be unknown complex coefficients α and β; the first qubit of
n(n+5)2,which has included 2n times of manipulations the address register is sent out along the control circuit
for asignalphotongoingto a memorycell andbackto a and is coherently stored in the atom in the first node
data register along a same selected path. Here a single- byapplyingthecontrolpulseΩ(t)simultaneouswiththe
photon can pass through a node without any quantum arrival of the address qubit equally split and incident
manipulation, therefore the average times of quantum frombothsidesofthetaperedfibersimultaneously. This
manipulation really performed on node qubits per mem- storingprocesswillincur astateflipofthe nodeatomto
ory call is n/2. Thus this scheme may significantly de- |righti if the address qubit is |1i, or make no change in
creasetheaverageoverallerrorrateandshortenthetime the atom state |lefti if the address qubit is |0i. When
required for a memory call. the second address qubit is sent out and meet the first
Now we discuss a physical implementation. The node node, it will be reflected back and travel along the left
qubitisencodeonanatomwithlevel|lefti, |righti,and path by applying an optical circulator in one side of the
an intermediate state |ei (see Fig.2 A). The transition tapered fiber (see Fig.2 b) if the atom in the first node
between |leftiand |ei is coupled to the evanescentfields is in |lefti, or will transmit the resonator and go along
of modes a and b of of frequency ω of a microtoroidal therightpathifthe atomisin|righti. Whenthe second
c
resonator. State |righti is coupled to |ei by classical addressqubitarriveatoneofthetwonodesonthesecond
control field Ω(t). A tapered fiber and the resonator are level, it will be coherently stored in the node atom and
assumed to be in critical coupling where the input pho- lefttheatomin|leftior|rightidependentonthephoton
tons of frequency ω = ω are all reflected back and the number contained in the address qubit, and so on.
p c
forward flux in the fiber drops to zero when the atom We assume that each quantum memory cell in the
transition(|lefti→|ei)isfardetunedfromtheresonator memory array consists of a memory atom m and an an-
frequencyω [17]. Iftheatomictransition(|righti→|ei) cillaryatoma, whichareconfinedintwoharmonictraps
c
is onresonantwith the resonator,the input photons can and positioned inside a high quality cavity (see Fig.3A).
transmit the resonator and travel forward one by one The ancillary atom has a three-level structure: |gi is
a
[18]. A single-photon can be coherently stored in the coupled to |ei by the field of the cavity mode with
a
atom initialized in |lefti by applying the control pulse strength g ; |si is coupled to |ei by a classic control
a a a
Ω(t) simultaneous with the arriveof the photon which is field Ω1(t), where the subscript a denotes an ancillary
equally divided and incident from both sides of the ta- atom. Afterapathtothememoryarrayissingledout,a
pered fiber simultaneously (see Fig.2) [19]. The photon single-photonissentalongthepathtotheselectedmem-
storage results in a state flip of the atom to |righti. If ory cell. A controllaser pulse Ω1(t) is applied to the an-
no single-photon is contained in the incoming field, the cillaryatomsinitializedinstate|gi atthemomentwhen
a
3
A B two Rydberg atoms [24, 25]. When both of the memory
|eÚ |rightÚ
atom and the ancillary atom are in Rydberg states, the
Ω ( t ) g |leftÚ strong dipole interaction between them will couple their
|leftÚ motion, which is best described in the basis of normal
|rightÚ
modes |ji (j = 0,1,2,...). All the motion modes of the
n
atom optical a atoms are initially cooled to near their ground state by
microtoroidal circulator Raman sideband cooling on them [26].
a b resonator b Third,wedriveaπ pulseontransitionon|gi →|ri
m m
left right on memory atoms to excite the memory atoms from
state |gi to state |ri . Fourth, a π pulse on transi-
m m
tion |gi → |ri |0i and a blue sideband (BSB) pulse
FIG. 2. (Color online). Schematic diagram of a node con- a a n
ontransition|si →|ri |1i areappliedonthe ancillary
sisting of a three-level atom and a microtoroidal resonator. a a n
atoms, leading to state
(A) An address qubit consisting of zero or one photon is
splitequallyincounter-propagatingdirectionsandcoherently |ψi =α|ri |ri |0i +β|ri |ri |1i (1)
1 a m n a m n
stored using an impedance-matched control field Ω(t), lead-
ing to a state flip of the atom conditioned on the photon for the selected memory cell (see Fig. 3bB) and state
number. (B) By employing an optical circulator an photon |ψi =(α |ri +β |si )|ti forothermemorycellswith
2 x m x m a
travels along a tapered fiber being in critical coupling to the their initial states state α |gi + β |si (see Fig.3C).
x m x m
resonator will go along the left (right) path if the atom is in Here we have used the fact that the normal mode of
state |lefti(|righti).
motion is shared by the memory atom and the ancil-
lary atom, both of which are in Rydberg states. Fifth,
a π pulse on the transition |ri →|gi unperturbed by
m m
the photon arrives at the memory cell, resulting a state thestrongdipoleinteractionbetweentwoRydbergatoms
flip of atom a to |sia [20, 21]. To avoid to be involved is applied on the non-selected memory atoms to restore
intothequantumoperationsaimedontheselectedmem- them to their initial states α |ri +β |si .
x m x m
orycell,theancillaryatomsnon-selectedareexcitedtoa Sixth, a π pulse on transition |ri |1i → |r′i |0i
m n m n
stable state |tia by a π pulses on transition |gia → |tia. illuminatesthememoryatoms,resultingastatemapping
Next we can do some quantum manipulations either to
′
saveanarbitraryunknownquantumstateintheselected |ψi1 →|ψi2 =α|ria|rim|0in+β|ria|r im|0in. (2)
memory cell or to read out its content. ′
Seventh,twoπ pulseson|r i |0i →|siand|ri |0i →
m n m n
In the first place, we discuss how to save an arbitrary
|gi , respectively, are applied on the memory atom,
m
unknown quantum state in the memory cell identified
leading to an unitary transformation |ψi → |ψi =
2 3
by the address register. First, an initialization opera-
(α|gi +β|si )|ri ; the unknown state α|0i+β|1i has
m m a
tion on the memory cell array is performed. This can
beenstoredontheselectedmemoryatom. Notethatthis
be realized as follows: atom a in state |si is excited
a two pulses will not affect the states of the non-selected
to a Rydberg state |ri by two-photon stimulated Ra-
a memoryatomssincethepulsesisdetunedfromthetran-
man(TWSR)pulses[22];TWSRpulsesonthetransition ′
sitions |r i → |si and |ri → |gi , which are free
m m m m
|si → |ri (in terms of the perturbed state ) of the
m m of the influence of strong dipole interaction between two
memory atoms are applied, resulting the selected mem-
Rydberg atoms. Eighth, the ancillary atoms is restored
oryatom being excited to |ri andimmediately flipping
m to the ground state |gi by a a π pulse on the transition
a
to the ground state |gi through spontaneous radiation
m |ri → |gi (see Fig.3E), and the non-selected ancillary
a a
(see Fig.3B). In this way only the memory atom in the
atoms are restored to the ground state |gi for the next
selected memory cell is reset in state |gi , leaving the
m memory call by flipping to a intermediate level with a
contentofthe othersunchanged,becausethe states|ri
m pulse and then falling to |gi through spontaneous radi-
of the non-selected memory atoms are off resonant with
ation. In this way an arbitrary unknown state can be
the TWSR pulses due to the absence of the strong Ryd-
transferred to the selected memory atom, leaving the
berg dipole interactions [23]. The ancillary atom in Ry-
states of other memory atoms unchanged.
dgerg|ri isbroughttothegroundstate|gi byapplying
a a Now we discuss how to read out the content of the se-
a π pulse on its transition |ri →|gi .
a a lected memory atom αx|gim +βx|sim. First, a π pulse
Second,anarbitraryunknownstateα|0i +β|1i with on transition |si → |ri is employed to excite the se-
p p a a
Fockbases|ni (n=0,1)istransferredalongtheselected lected ancillary atom to state |ri . Second, we em-
p a
path to the memory cells. On the arrival of the signal ploy two π pulses on transition |gi → |ri |0i and
m m n
a classic control pulse Ω2(t) is applied on the ancillary |si → |ri |1i , respectively (see Fig.3F), driving the
m m n
atoms initialized in the ground state |gi , leading to a system of the selected atoms m and a into state
a
state map (α|0i +β|1i )|gi →|0i (α|gi +β|si ) [20,
p p a p a a |ψi =α |ri |ri |0i +β |ri |ri |1i . (3)
21]. 4 x m a n x m a n
′
State α|gi +β|si is then transferred to a memory Third, a π pulse on transition |ri |1i → |r i |0i
a a a n a n
atombyemployingthestrongdipoleinteractionbetween drive the selected system into state |ri (α |ri +
m x a
4
(cid:36) m |eÚ not influence the non-selected atoms a and m because
a
a Ω1 ( t ) ga tchlaesysichaalviemnpoedsatnrocne-gmdaitpcohleedicnotnertaroctlifioenl.dΩB(yt)a,papmlyainttgera-
|sÚ |gÚ
a a photon mapping (α |gi +β |si )|0i → |gi (α |0i +
x a x a p a x p
β |1i ) can be accomplished [20, 21], transferring the
(cid:37) (cid:38) (cid:39) (cid:40) (cid:41) (cid:42) (cid:43) x p
state of the selected memory cell to the flying qubit and
|rÚ |1Ú
|0Ún leavingthe ancillaryatominstate |gi . The flying qubit
|r'Ú n a
goesalongthe pathwithblack squaresto the data regis-
|sÚ ||01ÚÚn ter (see Fig.1). Finally, the non-selected ancillary atoms
n
|gÚ ||01ÚÚnn are initialized to state |gia for a next task.
m a m a m a m a m a m a m a In summary, we have presented a scheme for a quan-
tum random access memory. With three-level memory
FIG.3. (color online). Schematicsforwriting toand reading
system been substituted by a qubit in every node of the
out of a memory cell. (A) A single-photon can be coherently
control circuit, this structure may significantly reduce
stored in a memory cell consisting of two atoms m and a
overall error rate per memory address and the memory
positioned inside of a high-Q cavity with a time-dependent
address time. In addition, we have discussed a physi-
control pulse Ω2(t). Diagrams of energy levels and pulses
calimplementationbasedonmicrotoroidalresonatorand
sequences for storing unknown state to the selected memory
cell(BtoE)andforreadingoutofthecontentoftheselected strong-dipoleinteractionbetweentwoRydbergatomsfor
memory cell (F to H). a QRAM writing and read-out. The microtoroidal res-
onator and the tapered fiber may be replaced by a sur-
face plasmon propagating on the surface of a nanowire-
β |r′i )|0i (see Fig.3G). Fourth, the selected memory conductor-dielectric interface [27, 28].
x a n
atom is sent to the ground state by a π pulse on the This workwassupportedbythe NationalNaturalSci-
transition |ri |0i → |gi . Fifth, two pulses on the enceFoundationofChina(11072218,and11005031),by
m n m
′
transition |ri → |gi and |r i → |si , respectively, set ZhejiangProvincialNaturalScienceFoundationofChina
a a a a
the selected ancillary atom in state α |gi +β |si (see (Grant No. Y6110314 and Y6100421), and by Scientific
x a x a
Fig.3H); the contentofthe select memory atomis trans- ResearchFundofZhejiangProvincialEducationDepart-
ferred to the ancillary atom. Note that these pulses do ment (Grant No. Y200909693and Y200906669).
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