US 2023/0129562 Al

of A, B, or C” means “(A); (B): (C); (A and B); (A and C);
(B and C); or (A, B, and C).”

[0134] It should be understood that the foregoing descrip-
tion is only illustrative of the present disclosure. Various
alternatives and modifications can be devised by those
skilled in the art without departing from the disclosure.
Accordingly, the present disclosure is intended to embrace
all such alternatives, modifications, and variances. The
embodiments described with reference to the attached draw-
ing figures are presented only to demonstrate certain
examples of the disclosure. Other elements, steps, methods,
and techniques that are insubstantially different from those
described above and/or in the appended claims are also
intended to be within the scope of the disclosure.

What is claimed is:

1. A system for quantum state transfer and entanglement
generation, the system comprising:

a quantum system including a plurality of qubits;

a processor; and

a memory, including instructions stored thereon, which,

when executed by the processor, cause the quantum

system to:

access a signal of the quantum system;

encode unknown coefficients in one qubit of the plu-
rality of qubits:

initialize each of the remaining qubits of the plurality of
qubits in state 10);

group the plurality of qubits into a plurality of subsys-
tems;

in each of the plurality of subsystems: encode quantum
information into Greenberger-Horne-Zeilinger-like
(GH7Z-like) states using nearest-neighbor interac-
tions; and

apply a generalized controlled-phase gate between the
plurality of subsystems to merge the GHZ-like states
into an entangled state between of the plurality of
subsystems.

2. The system of claim 1, wherein the instructions, when
executed by the processor, further cause the quantum system
to rotate the entangled state into a desired GHZ-like state by
concentrating an entanglement in each subsystem of the
plurality of subsystems onto one qubit ¢, of each of the
plurality of subsystems.

3. The system of claim 2, wherein the instructions, when
executed by the processor, further cause the quantum system
to apply a single-qubit rotation to c,

4, The system of claim 3, wherein the instructions, when.
executed by the processor, further cause the quantum system
to redistribute the entanglement to the plurality of subsys-
tems as a resulting GHZ-like state.

5. The system of claim 4, wherein the instructions, when
executed by the processor, further cause the quantum system
to repeatedly feed the resulting GHZ-like state back into
applying a generalized controlled-phase gate between the
plurality of subsystems to merge the GHZ-like states into an
entangled state between of the plurality of subsystems to
yield larger GHZ-like states.

6. The system of claim 3, wherein the system further
includes a Hadamard gate, and wherein the rotation of c, is
applied using the Hadamard gate.

7. The system of claim 1, wherein the subsystems include
hypercubes.

8. The system of claim 1, wherein the unknown coeffi-
cients include a, b

Apr. 27, 2023

9. The system of claim 8, wherein the unknown coefli-
cients of the one qubit of the plurality of qubits is encoded

as al0) +bi1)).

10. The system of claim 1, wherein the quantum system
includes a power-law interacting Hamiltonian.

11. A method for quantum state transfer and entanglement
generation, the method comprising:

accessing a signal of a quantum system, the quantum

system includes a plurality of qubits;

encoding unknown coefficients in one qubit of the plu-

rality of qubits;

initializing each of the remaining qubits of the plurality of

qubits in state 10);

grouping the plurality of qubits into a plurality of sub-

systems;

in each of the plurality of subsystems: encoding quantum

information into Greenberger-Home-Zeilinger-like
(GHZ-like) states using nearest-neighbor interactions;
and

applying a generalized controlled-phase gate between the

plurality of subsystems to merge the GHZ-like states
into an entangled state between of the plurality of
subsystems.

12. The method of claim 11, further comprising rotating
the entangled state into a desired GHZ-like state by con-
centrating an entanglement in each subsystem of the plural-
ity of subsystems onto one qubit c, of each of the plurality
of subsystems.

13. The method of claim 12, further comprising applying
a single-qubit rotation to c,

14. The method of claim 13, further comprising redistrib-
uting the entanglement to the plurality of subsystems as a
resulting GHZ-like state.

15. The method of claim 14, further comprising repeat-
edly feeding the resulting GHZ-like state back into applying
a generalized controlled-phase gate between the plurality of
subsystems to merge the GH7Z-like states into an entangled
state between of the plurality of subsystems to yield larger
GHZ-like states.

16. The method of claim 11, wherein the subsystems
include hypercubes.

17. The method of claim 11, wherein the unknown coef-
ficients include a, b.

18. The method of claim 17, wherein the unknown
coeflicients of the one qubit of the plurality of qubits is
encoded as al) +bi1) .

19. The method of claim 11, wherein the rotation of ¢, is
applied using a Hadamard gate.

20. A method for quantum state transfer and entanglement
generation, the method comprising:

accessing a signal of a quantum system, the quantum

system includes a plurality of qubits;

encoding al0) +bl1) in one qubit of the plurality of qubits;
initializing each of the remaining qubits of the plurality of

qubits in state 10);

grouping the plurality of qubits into a plurality of hyper-
cubes;

in each of the hypercubes of the plurality of hypercubes
encoding quantum information into Greenberger-
Horne-Zeilinger-like (GHZ-like) states;

merging the GH7Z-like states into an entangled state
between of the plurality of hypercubes;

rotating the entangled state into a desired GHZ-like state;


US 2023/0129562 Al

ent site in C, a quantum state may be transferred from ceC
to any other site ce C in time 2t.

[0058] The disclosed method recursively builds the GHZ-
like state in a large hypercube from the GHZ-like states of
smaller hypercubes. For the base case, hypercubes of finite
lengths, i.e., rStq for some fixed rp, can always be generated
in times that satisfy Eqn. 3 for some suitably large (but
constant) prefactor K,,. Assuming that information can be
encoded into a GHZ-like state in hypercubes of length r, in
time t, satisfying Eqn. 3, the following subroutine encodes
information into a GHZ-like state in an arbitrary hypercube
C of length r=mr, containing ¢, the site initially holding the
phase information a, b. Here m is an a-dependent number to
be chosen later.

[0059] At step 120, the hypercube C is divided into m4
smaller hypercubes C,, . . . C,,é, each of length r,. Without
loss of generality, C, is the hypercube that contains c. Let
V=r,“ be the number of sites in each C;, a, b are simulta-

neously encoded into |GHZ(a, b)) ¢, and prepare IGHZ) ¢; for
all j , m4, which, takes time

log’@ny if d<a<2d,
hs Kyx4 oP if a= 2d, and
if Wd <asddt1

Eqn. 4)

[0060] By the end of this step, the hypercube C is in the
state

wf )., +e Ean. 5)
(ald) +6T))o, @ +
£

re

[0061] Next, at step 130 the following Hamiltonian is
applied to the hypercube C:

Eqn. 6)

1 wit
= oaay out ee Dace) Mn eI

[0062] This Hamiltonian effectively generates the
so-called controlled-PHASE gate between the hypercubes,
with C, being the control hypercube and C,,.... C,, being
the target hypercubes. The interactions between qubits in
Eqn. 6 are selected to be identical for simplicity. If the
interactions were to vary between qubits, the interaction
between C, and C, may be tured off once the total phase
accumulated by C, reaches x. The prefactor 1/(mr,¥d)*
ensures that this Hamiltonian satisfies the condition of a
power-law interacting Hamiltonian. It is straightforward to
verify that, under this evolution, the state of the hypercube
C rotates to:

+, (Eqn. 7)

2

De .;-D.,

+4IDe, 8

nl
ac, @

2

[0063] after time t,=nd”(mr,)"/V2.
[0064] To obtain the desired state |GHZ(a, b)) ¢, it remains
to apply a Hadamard gate on the effective qubit {10) oT

Apr. 27, 2023

Je} for j , m?, The Hadamard gate is applied in the
following three steps by first concentrating the information
stored in hypercube C, onto a single site c¢ C, (Step 3), then
applying a Hadamard gate on c, (Step 4), and then unfolding
the information back onto the full hypercube C, (Step 5).
(0065] Next, at step 140, for each hypercube G (=2

, mé) and given a designated site eC, there exists a
(time-dependent) Hamiltonian H; that generates aunitary U;
such that

uy Eqn. 8)
(ol) +UrlD).,De,e; 4 YolDe, + HIDe,

[0066] for all complex coefficients Wo and yy, in time t,
satisfying Eqn. 4. By linearity, this property applies even if
C; is entangled with other hypercubes. Consequently, back-
ward time evolution under H, generates U,', which “undoes”
the GHZ-like state of the j th hypercube:

(Eqn. 9)

Vado, +UalDo, “> J el AI Dejey

(0067] for any Wo, y. In this step, U,Y is simultaneously
applied to C; for all ..,m¢, These unitaries rotate the
state of C to:

nf nfl Eqn. 10)
ADC BHD, +N o, BI Dey

where 1+) = (10) # IL)y/2.
[0068] Next, at step 150 a Hadamard gate is applied,

1

2

(Eqn. 11) to the site c, of each hypercubes C,, j=2,...,m%.
These Hadamard gates can be implemented arbitrarily fast
since any constraints are not assumed on the single-site
terms of the Hamiltonian. The state of C by the end of this
step is:

wf (Eqn. 12)

a1Dc,
f

DW. Dege, + ADe, B le/lDe,

ney
7

[0069] Next at step 160, U, is applied again to each
hypercube C, m4) to obtain the desired GHZ-like
state:

nf ~ (Eqn. 13)
ac, @ Dc, + HIT
mo

Ye, = IGHZ(a, b)re-

[0070] At the end of this routine, the unitary satisfying
Eqn. 2 in time t=3t,+t,=3t,+0d’?m“r,?24 (Eqn. 14) has
been implemented.


US 2023/0129562 Al

Apr. 27, 2023 Sheet 4 of 9

Patent Application Publication

vSld

ul Cb CLL UL
CH Col dol <4l <b an l Col <ol «tl <b ae ol al “ol Ja (bl lg .
(ol <ol <ol <ol «ol <ole a +} (ol (ol «ol Col ole tl a tl a (ol <ol «ole
201 sa}eIS ZHO
RU) sm Ot? SP sins
e808 8 e 9 @ =F
uol}e}0:
POL OL gz] ano ‘albus og1-7% gz) PO! ajbuewuoun © oz gz, OL n
ajbuejus-oy re, op mH, uz}
ORC KORO RCROC a ORC ROMA TLCRO RCL Sul JOWW
(ol <ol <ol <ol <ol Cole (ol <ol ol ‘Nol dol «ole
OLL~% gz, 70! zA a7 Mh

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US 2023/0129562 Al
11

applying a single-qubit rotation to one bit of one of the
plurality of hypercubes; and

redistributing the entanglement to the plurality of hyper-
cubes.

Apr. 27, 2023


US 2023/0129562 Al

(0 I1, M [A there are no interactions between Imo[a and
Im, 2 This results in the dipole-dipole interactions:

1-3

5

29 (Eqn. 31)

Hy = Bey [lod + ah)Zids + (oi — a2 + Z)),

[0108] providing positive Ising interactions. Note that
negative interactions cannot be achieved in this way, so
another scheme must be used in combination with this
approach.

neoding Im, Alo, of +
Bland Im,, 2] 10,,of2. where a is defined by Q, A.
For any electric field strength E, the value of a can always
be tuned so that there are only negative Ising interactions.
Unlike the second scheme, only negative Ising interactions
can be achieved, so by combining the two schemes, one may
avoid the complications associated with using high N states
in the first scheme.

[0110] Thus, by transferring the Im, [2] population to the
Im,, Q]state and changing which of the above schemes is
used, it is possible to implement the state transfer method via
polar molecules.

[0111] In aspects, the state transfer method may be imple-
mented using Rydberg atoms, which will share several
similarities with the disclosed implementation in polar mol-
ecules. In order to do so, Imof2] lm, (2 is encoded in linear
combinations of four different states 1g] |sQ pof] and |p,
-orresponding to a ground state, a Rydberg s state, and
two Rydberg p states. In the case of the latter two states, 0/-+
correspond to the polarization of the drive needed to couple
the s state to the corresponding p state. Additionally, the p
states may have similar values of the principal quantum
number n compared to the s state, leading to strong dipole-
dipole interactions between s states and p states. These states
are illustrated in FIGS. 6A and 6B. The disclosed systems
and methods enable much stronger interactions, providing
faster state transfer with fewer errors due to dissipation.
[0112] In order to realize H,,, the Is[qjstatemay be
strongly dressed with the Ip state with a strong resonant
microwave field. In the absence of any external fields, these
two states interact according to the dipole-dipole interaction

\cos"8 (Eqn. 32)

|s:pj 07.0812

Hag = Hi
i)

eB

[0113] where r,, is the distance between atoms i and j, ®,
is the angle the displacement vector makes with the quan-
tization axis, and uo=[] poldo|s is the corresponding tran-
sition dipole moment, where d,=é,,d is a component of the
dipole operator d and 6=2, 6,=+(8+i9)/V2. Using the strong
resonant microwave field, Imo =lg[2] and Im, [] =(/s2]+1P5

may be encoded. In the limit of a strong microwave
i frequency Q,,,,>>H,, the dipole-dipole interactions in
the Imp[2] Im, [] basis may be expressed as:

(Eqn. 33)

Ba zy 2)
“(= 20 = Z)

Apr. 27, 2023

[0114] In order to realize -H,,,, the same approach as for
, may be used. However, rather than dressing the Is
state with IpoQ the Is] state is resonantly dressed with
In this case, the corresponding dipole-dipole interac-

(Eqn. 34)

2 1 =3cos@
Fr RPS

a

Hig = Bay —

Note that, aside from a con-
stant factor, the dipole-dipole interaction here only differs by
a sign from H,,. As such, if Im,, 2] =(IsP]+!p,B))/V2 is
encoded and again the limit of a strong microwave Rabi
frequency Q,,,,,>>H 4, is taken, the resulting interactions are:

; 12 1~3e0s% Eqn. 35)
Hm = Daas B a ZN 23),

9

[0116] which differs from H,,,, by a negative constant.
Thus, by transferring the Im,fJ population to the Im,
state and changing the polarization of the microwave
field, it is possible to implement the state transfer methods
via Rydberg atoms. Note that when the polarization of the
drive is changed, one must ensure that the qubit state is
temporarily not encoded in IQ] |PoQ] !P.21

[0117] Using similar principles to the above, it is also
possible to realize the desired control-target interactions
without the spin-echo approach. This is achieved by turning
on both drives Qy,, with detunings Ao, and using different
principal quantum numbers n for |po,,2 With a proper
choice of parameters, two of the resulting dressed states Ic
interact with each other but not themselves. Thus, by
forming the control qubits with Ic] and the target qubits
with It(2] the desired pure control-target interactions can be
realized, although a spin-echo sequence must be applied to
all qubits to eliminate an effective longitudinal field. The
details of engineering this form of interactions can be
generalized to polar molecules and magnetic atoms as well.
[0118] FIGS. 6A and 68, illustrate a dressing scheme for
realizing (FIG. 6A) H,,, and (FIG. 6B) H,,,’ via microwave
dressing, with Im SEI+!poB) V2 and Im,
+lp,.21/V2. In each case, Qy+>>H,, may be used
the microwave-dressed basis may be used.

[0119] In aspects, 1/° vdW interactions may be used for
state transfer in 3D. While this will not be as fast as using
Ir? dipole-dipole interactions, it provides the benefit of
being less complicated since the vdW interactions will have
no significant angular dependence and there will be no
microwave drive necessary. Moreover, it provides a natural
means of experimentally demonstrating that vdW interac-
tions are long-range in three dimensions.

[0120] One option to realize the desired sign change in the
vdW interactions is by inducing a Forster resonance for a s
state via an external magnetic field. Provided one keeps the
atoms sufliciently far apart outside of the dipole-dipole
interaction regime, where the interaction causes pairs of s
states hybridize with pairs of p states, the sign of C, will be
opposite on either side of the Forster resonance.

[0121] Alternatively, there is another approach which does
not require the use of external fields. Here, Imp =z



Patent Application Publication

800~

Apr. 27, 2023 Sheet 9 of 9

US 2023/0129562 Al

100

FIG. 8

Oo
foe)

oO}



Patent Application Publication Apr. 27, 2023 Sheet 6 of 9 US 2023/0129562 Al



Patent Application Publication Apr. 27,2023 Sheet 5 of 9 US 2023/0129562 Al

(a)
N=3
N=2
N=1 ----- 0 -----
N=0 F=0
| TM )
L y
FIG. 5A
r ~
(b) J,>0
E#0
N=2 00000 7 [im )
Net -----
N=0
| TM )
L y

FIG. 5B


US 2023/0129562 Al

[1 and Img, are encoded, where |s{]] Is’ are
two different Rydberg s states. The Rydberg interactions will
generally have the form:

(Eqn. 36)
tha =~), 5 IGS REBD + °

Cor [315254] + Cals) s ABs} 1 + Hc.)],

[0122] where Cg Cg, Ce°: denote the vdW interaction
coefficients for the s interaction, the s’ interactions, and the
cross interaction between s and s’.

[0123] The resulting interactions for ImpQ] Im, Pi takes the
form:

237
1-Z)-Z), Eqn. 37)

ly 6
Hye =-Z> 0, FF

[0124] naturally realizing the desired Ising interactions. In
contrast, the resulting interactions for Imo,

(Eqn. 38)
Hy = ide, Benen ar+

&
(1 = Z yl +Z))+ BU +Z 0 -Z)).
4

Ce
E+ Zyl + Z) +
* a

z

[0125] In this case, the resulting Ising interactions have
strength (Cp/4+Cg/4-C,/2)°. Thus, in order to ensure
that H’,,, differs by a negative sign from H,,,, then CotCo.
-2C,°<0, which is achievable for reasonable choices of
Rydberg states, One minor complication arises via the fact
that the coefficients of the Z terms relative to the Ising
interactions are not the same for each Hamiltonian, How-
ever, this is easily remedied by applying an X gate to all
atoms at once as in the usual spin-echo procedure. Since it
is applied to all atoms, it will not affect the Ising interactions
themselves, but it will reverse the evolution due to the Z.
terms, allowing one to ensure that their contribution is 0 at
the end of the evolution.

[0126] FIG. 7 illustrates the effective van der Walls (vW)
interactions between controls and targets for negative Ising
interactions, normalized to the case of positive interactions.
The Rydberg states used are n and n'=n-1.

[0127] In aspects, the state transfer method may be imple-
mented in a two-dimensional lattice of ultracold dysprosium.
atoms, in particular, bosonic isotopes of dysprosium (such as
162Dy, !©*Dy). The nuclear spin may be ignored. The com-
putation states mp, m, in the Zeeman manifold of the
electronic ground state of the dysprosium may be encoded.
The interactions between a pair of dysprosium atoms i, j are
given by magnetic dipole-dipole interactions (in units of
Hol Sia) /400):

(Eqn. 38)
Hy = 40-3008 o)fsrgj — Hor +r a] Gan. 38)
5

Apr. 27, 2023

-continued

Fsintole*'yj +H

[0128] Here, J=8 is the electronic angular momentum of
the ground state, and 8, @ are the polar and azimuthal angles
with respect to the quantization axis (which is set by a
magnetic field which also breaks the degeneracy of the
Zeeman levels M, —8SMS8).

[0129] The simplest way of realizing the diagonal inter-
actions that are used is to make use of the J,J,* term in Eqn.
38. For instance, the choice m,=—8, m,=8 yields the inter-
action

Hing = 216 AZ jy

up to potential longitudinal terms (Z,, Z;) depending on the
energies of the two states mg, m,.

[0130] In order to realize the reverse Hamiltonian, i.e., the
same Hamiltonian but with opposite sign, the flip-flop
interactions in Eqn. 38 may be used. Using microwave and
optical dressing. for example, the following dressed states
may be selected: mg=(4+—4)/V2, and m,=a8+b7+c-8, where
the a, b, c amplitudes are tunable via the dressing param-
eters. This choice yields mpJ.m=0 and by properly tuning
a, b,c that m,J.m,=0. In this case, only the flip-flip inter-
actions from Eqn. 38 contribute, and in fact, the only
nonzero matrix element is m,m,H,,m,m,, thus realizing the
same Z,Z, interactions, as before, but with the opposite sign.
In general, this would also give rise to longitudinal terms
that depend on both the interactions strength and the energy
splitting of the states. These can be dealt with the spin echo
procedure as explained previously.

(0131] Certain embodiments of the present disclosure may
include some, all, or none of the above advantages and/or
‘one or more other advantages readily apparent to those
skilled in the art from the drawings, descriptions, and claims
included herein. Moreover, while specific advantages have
been enumerated above, the various embodiments of the
present disclosure may include all, some, or none of the
enumerated advantages and/or other advantages not specifi-
cally enumerated above.

[0132] The embodiments disclosed herein are examples of
the disclosure and may be embodied in various forms. For
instance, although certain embodiments herein are described
as separate embodiments, each of the embodiments herein
may be combined with one or more of the other embodi-
ments herein. Specific structural and functional details dis-
closed herein are not to be interpreted as limiting, but as a
basis for the claims and as a representative basis for teaching
one skilled in the art to variously employ the present
disclosure in virtually any appropriately detailed structure.
Like reference numerals may refer to similar or identical
elements throughout the description of the figures.

(0133] The phrases “in an embodiment,” “in embodi-
ments,” “in various embodiments,” “in some embodiments,”
or “in other embodiments” may each refer to one or more of
the same or different example embodiments provided in the
present disclosure. A phrase in the form “A or B” means
*(A), (B), or (A and B).” A phrase in the form “at least one


Patent Application Publication Apr. 27, 2023 Sheet 8 of 9 US 2023/0129562 Al

7 80
n
FIG.7

T
60

T
50



as) United States
a2 Patent Application Publication 1) Pub. No.: US 2023/0129562 Al

TRAN et al.

US 202301295624.

1

(43) Pub. Date: Apr. 27, 2023

(54)

(29)

(72)

(21)
(22)

(60)

SYSTEMS AND METHODS FOR OPTIMAL
STATE TRANSFER AND ENTANGLEMENT
GENERATION IN POWER-LAW
INTERACTING SYSTEMS.

Applicants: University of Maryland, College Park,

Inventors:

College Park, MD (US); The Regents
of the University of Colorado, a body
corporate, Denver, CO (US);
Government of the United States of
America as Represented by the
Secretary of Commerce, Gaithersburg,
MD (US)

Minh C. TRAN, College Park, MD
(US); Abhinav DESHPANDE, College
Park, MD (US); Andrew Y. GUO,
College Park, MD (US); Andrew
LUCAS, Boulder, CO (US); Alexey
GORSHKOV, Potomac, MD (US)

Appl. No.: 17/959,901

Filed:

Oct. 4, 2022

Related U.S. Application Data

Provisional application No. 63/262,085, filed on Oct.
4, 2021, provisional application No. 63/378,210, filed

on Oct.

Publication Classification

(51) Int. Cl.
GO6N 10/20 (2006.01)
(52) US. CL
CPC vascsssnenneneintes GOON 10/20 (2022.01)
(57) ABSTRACT

A system for quantum state transfer and entanglement
generation includes a quantum system including a plurality
of qubits, a processor, and a memory. The memory includes
instructions stored thereon, which, when executed by the
processor, cause the quantum system to: access a signal of
the quantum system; encode unknown coefficients in one
qubit of the plurality of qubits; initialize each of the remain-
ing qubits of the plurality of qubits in state 10) ; group the
plurality of qubits into a plurality of subsystems; in each of
the plurality of subsystems: encode quantum information
into Greenberger-Horne-Zeilinger-like (GHZ-like) states
using nearest-neighbor interactions; and apply a generalized
controlled-phase gate between the plurality of subsystems to
merge the GHZ-like states into an entangled state between

3, 2022. of the plurality of subsystems.
300
TM~ (Start)
+40,
[Encode unknown coefficients in one qubit _]
1205,

Encode quantum information into GHZ-like
states using nearest-neighbor interactions

130,

Apply a generalized controlled-phase gate
between the plurality of hypercubes to merge
the GHZ-like states into an entangled state
between of the plurality of subsystems

1405,

Rotate the entangled state into a desired
GHZ-like state by concentrating an
entanglement in each hypercube of the
plurality of hypercubes onto one qubit
cj of each of the plurality of hypercubes

150,

[ Apply a single-qubit rotation to cj ]

160~,

Redistribute the entanglement to the plurality
of hypercubes as a resulting GHZ-like state



US 2023/0129562 Al

[0022] In yet another aspect of the present disclosure, the
unknown coefficients may include a, b

[0023] In a further aspect of the present disclosure, the
unknown coefficients of the one qubit of the plurality of
qubits may be encoded as al0) +bI1)

[0024] In a further aspect of the present disclosure, the
rotation of c, may be applied using a Hadamard gate.
[0025] An aspect of the present disclosure provides a
method for quantum state transfer and entanglement gen-
eration. The method includes: accessing a signal of a quan-
tum system, the quantum system including a plurality of
qubits; encoding al0) +bl1) in one qubit of the plurality of
qubits; initializing each of the remaining qubits of the
plurality of qubits in state 10); grouping the plurality of
qubits into a plurality of hypercubes; in each of the hyper-
cubes of the plurality of hypercubes encoding quantum
information into Greenberger-Home-Zeilinger-like (GHZ-
like) states; merging the GHZ-like states into an entangled
state between of the plurality of hypercubes; rotating the
entangled state into a desired GH7Z-like state; applying a
single-qubit rotation to one bit of one of the plurality of
hypercubes; and redistributing the entanglement to the plu-
rality of hypercubes.

[0026] Further details and aspects of exemplary embodi-
ments of the present disclosure are described in more detail
below with reference to the appended figures.

BRIEF DESCRIPTION OF THE DRAWINGS

[0027] A better understanding of the features and advan-
tages of the present disclosure will be obtained by reference
to the following detailed description that sets forth illustra-
tive embodiments, in which the principles of the present
disclosure are utilized, and the accompanying drawings of
which:

[0028] FIG. 1 is a diagram of an exemplary quantum
system diagram, in accordance with examples of the present
disclosure;

[0029] FIG. 2 is a diagram of a method for quantum state
transfer and entanglement generation for the quantum sys-
tem of FIG. 1, in accordance with examples of the present
disclosure;

[0030] FIG. 3 illustrates a flow diagram of the method for
encoding an arbitrary qubit into a multiqubit and subse-
quently transferring the quantum state for the quantum
system of FIG. 1, in accordance with examples of the present
disclosure;

[0031] FIG. 4 is a diagram illustrating quantum state
transfer of FIG. 1, in accordance with examples of the
present disclosure;

[0032] FIGS. 5A-C is a diagram illustrating a scheme for
realizing dipole-dipole Ising interactions using polar mol-
ecules, in accordance with examples of the present disclo-
sure;

[0033] FIGS. 6A and 6B, illustrate a dressing scheme for
realizing H,,, and H,,, via microwave dressing, with Im,
=(Isfl +po [21 )/V2. and |Im,.) =(Is 7] +!p,,[21)/V2. in accor-
dance with examples of the present disclosure;

[0034] FIG. 7 is a diagram illustrating the effective van der
Walls (vdW) interactions between controls and targets for
negative Ising interactions, in accordance with examples of
the present disclosure; and

Apr. 27, 2023

[0035] FIG. 8 is a schematic diagram of an exemplary
processing system diagram for use with the system of FIG.
1, in accordance with examples of the present disclosure.

DETAILED DESCRIPTION

[0036] The present disclosure relates generally to the field
of quantum operations. More specifically, the present dis-
closure provides a method for encoding an arbitrary qubit
into a multiqubit and subsequently transferring the quantum
state.

[0037] Embodiments of the present disclosure are
described in detail with reference to the drawings wherein
like reference numerals identify similar or identical ele-
ments.

[0038] Although the present disclosure will be described
in terms of specific examples, it will be readily apparent to
those skilled in this art that various modifications, rearrange-
ments, and substitutions may be made without departing
from the spirit of the present disclosure. The scope of the
present disclosure is defined by the claims appended hereto.

[0039] For purposes of promoting an understanding of the
principles of the present disclosure, reference will now be
made to exemplary embodiments illustrated in the drawings,
and specific language will be used to describe the same. It
will nevertheless be understood that no limitation of the
scope of the present disclosure is thereby intended. Any
alterations and further modifications of the novel features
illustrated herein, and any additional applications of the
principles of the present disclosure as illustrated herein,
which would occur to one skilled in the relevant art and
having possession of this disclosure, are to be considered
within the scope of the present disclosure.

[0040] Referring to FIG. 1, a diagram of an example
quantum system 100 (e.g., hypercube C), is shown. The
quantum system 100 may include two or more qubits 102
Qubits 102 may be arranged in a one-or more dimensional
system 100. For example, the qubits 102 may be grouped
into hypercubic latices. Although qubit systems are used as
an example, it is contemplated that the disclosed technology
may be used in qudit systems (arbitrary finite-level systems)
as well,

[0041] Harnessing entanglement between many particles
is key to a quantum advantage in applications including
sensing and timekeeping, secure communication, and quan-
tum computing. For example, encoding quantum informa-
tion into a multiqubit Greenberger-Hore-Zeilinger-like
(GHZ-like) state is particularly desirable as a subroutine in
many quantum applications, including metrology, quantum
computing, anonymous quantum communication, and quan-
tum secret sharing.

[0042] The speed at which one can unitarily encode an
unknown qubit state al0) +bl1) into a GHZ-like state al00

O)+bI11 . . . 1) of a large system is constrained by
Lieb-Robinson bounds and depends on the nature of the
interactions in the system. In systems with finite-range
interactions and power-law interactions decaying with dis-
tance ras 1/r* for all a=2d+1, where d is the dimension of the
system, the Lieb-Robinson bounds imply a linear light cone
for the propagation of quantum information. Consequently,
in such systems, the linear size of a GHZ-like state that can
be prepared from unentangled particles cannot grow faster
than linearly with time.


US 2023/0129562 Al

SYSTEMS AND METHODS FOR OPTIMAL
STATE TRANSFER AND ENTANGLEMENT
GENERATION IN POWER-LAW
INTERACTING SYSTEMS

CROSS-REFERENCE TO RELATED
APPLICATIONS/CLAIM OF PRIORITY

[0001] This application claims the benefit of, and priority
to, U.S. Provisional Patent Application No. 63/262,085, filed
on Oct. 4, 2021, and U.S. Provisional Patent Application No.
63/378.210, filed on Oct. 3, 2022, the entire contents of each
are hereby incorporated herein by reference.

GOVERNMENT SUPPORT

[0002] This invention was made with government support
under W911NF1610349 awarded by the Department of the
Army; DGE1840340 awarded by the National Science
Foundation; and DMR1420541 awarded by the National
Science Foundation (NSF). The government has certain
rights in the invention.

TECHNICAL FIELD

[0003] The present disclosure relates generally to the field
of quantum operations. More specifically, the present dis-
closure provides at least a system and method for encoding
an arbitrary qubit into a multiqubit and subsequently trans-
ferring the quantum state.

BACKGROUND

[0004] Harnessing entanglement between many particles
is key to a quantum advantage in applications including
sensing and timekeeping, secure communication, and quan-
tum computing. Encoding quantum information into a mul-
tiqubit Greenberger-Horne-Zeilinger-like state is particu-
larly desirable as a subroutine in many quantum applications
including metrology, quantum computing, anonymous quan-
tum communication, and quantum secret sharing.

[0005] Accordingly, there is interest in encoding quantum
information into a multiqubit Greenberger-Horner-Zeil-
inger-like state to improve harnessing entanglement.

SUMMARY

[0006] An aspect of the present disclosure provides a
system for quantum state transfer and entanglement genera-
tion. The system includes a quantum system including a
plurality of qubits, a processor, and a memory. The memory
includes instructions stored thereon, which, when executed
by the processor, cause the quantum system to: access a
signal of the quantum system; encode unknown coefficients
in one qubit of the plurality of qubits; initialize each of the
remaining qubits of the plurality of qubits in state |0) ; group
the plurality of qubits into a plurality of subsystems; in each
of the plurality of subsystems: encode quantum information
into Greenberger-Home-Zeilinger-like (GHZ-like) states
using nearest-neighbor interactions; and apply a generalized
controlled-phase gate between the plurality of subsystems to
merge the GHZ-like states into an entangled state between
of the plurality of subsystems.

[0007] In accordance with aspects of the disclosure, the
instructions, when executed by the processor, may further
cause the quantum system to rotate the entangled state into
a desired GHZ-like state by concentrating an entanglement

Apr. 27, 2023

in each subsystem of the plurality of subsystems onto one
qubit c, of each of the plurality of subsystems.

[0008] In an aspect of the present disclosure, the instruc-
tions, when executed by the processor, may further cause the
quantum system to apply a single-qubit rotation to c,.
[0009] In another aspect of the present disclosure, the
instructions, when executed by the processor, may further
cause the quantum system to redistribute the entanglement
to the plurality of subsystems as a resulting GHZ-like state.
[0010] In yet another aspect of the present disclosure, the
instructions, when executed by the processor, may further
cause the quantum system to repeatedly feed the resulting
GH7Z-like state back into applying a generalized controlled-
phase gate between the plurality of subsystems to merge the
GHZ-like states into an entangled state between of the
plurality of subsystems to yield larger GHZ-like states.
[0011] In accordance with further aspects of the present
disclosure, the system may further include a Hadamard gate.
The rotation of c, is applied using the Hadamard gate.
[0012] In an aspect of the present disclosure, the subsys-
tems may include hypercubes.

[0013] In another aspect of the present disclosure, the
unknown coefficients may include a, b.

[0014] In yet another aspect of the present disclosure, the
unknown coefficients of the one qubit of the plurality of

qubits may be encoded as al0) +bI1)

[0015] In yet another aspect of the present disclosure, the
quantum system may include a power-law interacting Ham-
iltonian.

[0016] An aspect of the present disclosure provides a
method for quantum state transfer and entanglement gen-
eration. The method includes: accessing a signal of a quan-
tum system, the quantum system including a plurality of
qubits; encoding unknown coefficients in one qubit of the
plurality of qubits; initializing each of the remaining qubits
of the plurality of qubits in state 10); grouping the plurality
of qubits into a plurality of subsystems; in each of the
plurality of subsystems encoding quantum information into
Greenberger-Home-Zeilinger-like (GHZ-like) states using
nearest-neighbor interactions; and applying a generalized
controlled-phase gate between the plurality of subsystems to
merge the GHZ-like states into an entangled state between
of the plurality of subsystems.

[0017] In another aspect of the present disclosure, the
method may further include rotating the entangled state into
a desired GHZ-like state by concentrating an entanglement
in each subsystem of the plurality of subsystems onto one
qubit c, of each of the plurality of subsystems.

[0018] In yet another aspect of the present disclosure, the
method may further include applying a single-qubit rotation
toc).

[0019] In yet another aspect of the present disclosure, the
method may further include redistributing the entanglement
to the plurality of subsystems as a resulting GHZ-like state.
[0020] In accordance with further aspects of the present
disclosure, the method may further include repeatedly feed-
ing the resulting GHZ-like state back into applying’a gen-
eralized controlled-phase gate between the plurality of sub-
systems to merge the GHZ-like states into an entangled state
between of the plurality of subsystems to yield larger
GHZ-like states.

[0021] In another aspect of the present disclosure, the
subsystems may include hypercubes


Patent Application Publication Apr. 27, 2023 Sheet 7 of 9 US 2023/0129562 Al

(~ TM
|Po
np ----- TT
i
i
Oy
i
ns _v
|s)
NX S
| m )
FIG. 6A
(~ TM
np -----
ns
|s)
XQ S

| ms)

FIG. 6B


Patent Application Publication Apr. 27, 2023 Sheet 3 of 9 US 2023/0129562 Al

300
> (Start)
10~ v
Encode unknown coefficients in one qubit
120~ Y

Encode quantum information into GHZ-like
states using nearest-neighbor interactions

130, t

Apply a generalized controlled-phase gate
between the plurality of hypercubes to merge
the GHZ-like states into an entangled state
between of the plurality of subsystems

140% |

Rotate the entangled state into a desired
GHZ-like state by concentrating an
entanglement in each hypercube of the
plurality of hypercubes onto one qubit
cj of each of the plurality of hypercubes

150~

v
Apply a single-qubit rotation to cj

160~, {

Redistribute the entanglement to the plurality
of hypercubes as a resulting GHZ-like state

FIG. 3


US 2023/0129562 Al

[0071] Three cases corresponding to different ranges of a
are considered and it is shown that if t,(r)) satisfies Eqn. 3,
then t(r) also satisfies Eqn. 3.

[0072] For ae (2d, 2d+1], t, <K,,r,-°4. Selecting m>1 to be
a constant integer provides:

3K -_ 2 2 Eqn. 15
(SS nih 2 Kae, (Fan 15)
mi
nang (Eqn. 16)
where m > 34 and Ky = aie
1-5
net
nd”
is selected.

me

[0073] For ae(d, 2d), m is chosen to scale with r, such that
r,*"'<ms2r,* where A=2d/a. The length of the larger cube
C is then r=mr,>r,* and, therefore, the total time is:

15 3Kylog’ ry + x(2d Jr Der (Eqn. 17)
4Ky ‘Eqn. 18;
= Foe" (rt) s Katou", (Fan 18)

[0074] where K,=log 4/log(2d/a) is selected and K,
log®r,27(2Vd)" is assumed to simplify the expression. The
factor log 4 in the definition of x, may be made arbitrarily
close to log 3 by increasing K,,.

[0075] Finally, for a=2d, m is chosen such that

exp( Flow )sms 2exo(% ioar }

where ¥=3Vd. Substituting t,<Kexp(yVTogr,) into Eqn. 14:
153K 42°nd*)e% TORT. (Eqn. 19)

[0076] Assuming r,2exp(8/d), it is straightforward to
prove that ~\/Togr,<ylog(mr,)-2. Applying this condition
on the above inequality:

te LK, 12nd) 5 KF, (Ean. 20)
3

[0077] _ where r=mr, is the length of the resulting GHZ-like
state and K,22°nd“/(e?-3) is selected. Eqn. 15, Eqn. 18,
and Eqn. 20 prove that t satisfies Eqn. 3. Repeatedly
applying this routine yields larger and larger GHZ-like
states.

[0078] In the disclosed method, quantum information is
simultaneously encoded into the GHZ-like state over C, and
the symmetric GHZ states are created over other multiqubit
subsystems C3, ...¢,,é As a result, the implementation of
the Controlled operations in step 130 (FIG. 1) is enhanced
quadratically by the volume of each subsystems.

[0079] In aspects, the size of the resulting GHZ-like state
may be maximized at the end of each iteration by allowing
m to depend on a and on the size of the existing GHZ-like
states. When the disclosed method is used for state transfer,
this strategy results in most of the qubits between the source

Apr. 27, 2023

and the target sites participating in the transfer, significantly
speeding up the disclosed method.

[0080] First. the disclosed method enables encoding an
unknown qubit into a multiqubit GHZ-like state and, sub-
sequently, performing state transfer at unprecedented
speeds. For d<a<2d, which applies, for example, to dipole-
dipole interactions (a=3) in two dimensions and to the
effective interactions between trapped ions (ae [0,3]) in one
and two dimensions, the disclosed method encodes infor-
mation into GHZ-like states and transfers information in
polylogarithmic time, exponentially faster than methods
available in the literature. Even for the seemingly weakly
long-range interactions with a=2d, such as van der Waals
interactions between Rydberg atoms (a=6) in three dimen-
sions, the disclosed method still takes only sub-polynomial
time to entangle an entire system and to transfer a quantum
state. When applied to the preparation of GHZ states, these
speedups enable potential improvements to quantum sensors
built from nitrogen-vacancy centers, Rydberg atoms, and
polar molecules, as well as to atomic clocks based on
trapped ions.

[0081] The optimality of the disclosed systems and meth-
‘ods for aé(1,3) in one dimension and ae(d, 2d] in d>1
dimensions also lays the foundation for optimal quantum
information processing in power-law interacting systems.
Using quantum state transfer between auxiliary qubits and
encoding qubits into large GHZ-like states as subroutines,
the disclosed systems and methods leads to optimal imple-
mentations of quantum gates between distant qubits in large
quantum computers. In particular, the faster encoding of
information into a GHZ-like state of ancillary qubits speeds
up the implementation of the quantum fanout—a powerful
multiqubit quantum gate. At the same time, the faster state
transfer speeds up the constructions of multiscale entangle-
ment renormalization ansatz (MERA) states, commonly
used to represent highly entangled states, such as topologi-
cally ordered states. Specifically, a fanout gate may be
implemented on qubits in a hypercube of volume n and
prepare a MERA state on these qubits in time t~polylog(n)
for ae (d. 2d),

for a=2d, which are both exponential speedups compared to
the previous best, and t~n2” for ae (2d, 2d+1). The
optimality of these operations is again guaranteed (up to
sub-polynomial corrections) by the matching lower limits
imposed by the Lieb-Robinson bounds.

[0082] In practice, using single-site Hamiltonians to
implement the echoing technique the controlled-PHASE
gate in step 130 of the disclosed systems and methods can
be realized starting from time-independent power-law inter-
actions between all sites of the system. The method therefore
does not require explicit time-dependent control of indi-
vidual two-qubit Hamiltonians, making it appealing for
implementation on available experimental platforms. How-
ever, because the diameter of the GHZ-like state increases
by more than twofold in every iteration of the method, the
scaling in Eqn. 3 may only be observed in large systems.
[0083] _Information-propagation speed limits.—Conceptu-
ally, since the disclosed systems and methods saturate (up to
sub-polynomial corrections) the Lieb-Robinson bounds for


US 2023/0129562 Al

een if d<as2d, (Eqn. 25)
a=

Of!) if Wd <as2d+1

[0095] The gap between this state-of-the-art upper bound
and the lower bound Q(n) of the disclosed systems and
methods hints at the possibility of a more efficient algorithm
for simulating power-law systems.

[0096] Regarding FIGS. SA-C, a diagram illustrating a
scheme for realizing dipole-dipole Ising interactions using
polar molecules is shown. In the state transfer method, there
are three types of qubits: control qubits, target qubits, and
uninvolved qubits. The uninvolved qubits, which are con-
fined to a single state, can generally be prepared easily by
storing them in a state which does not interact with the other
two types of qubit. For the control and target qubits, a
Hamiltonian with long-range Ising interactions between
control and target qubits is used. However, the control qubits
must not interact with themselves, and similarly the target
qubits must not interact with themselves. However, this
behavior does not generically occur and must be engineered.
[0097] The spin-echo-like approach is used. This approach
relies on the ability to realize a Hamiltonian of the form

(Eqn. 26)

[0098] as well as its negative —cH,,,, where Z, denotes the
Pauli-Z matrix for the ith qubit. By evolving under H,,, for
a time t, applying a Pauli-X gate to either the control or the
target qubits, and evolving under —cH,,, for a time Uc, the
evolution from the control-control and target-target interac-
tions are undone while the control-target interactions
remain, as desired. In light of this, most of the following
discussion will be centered around how to realize both signs
of the Ising interactions. Local control may be achieved in
tweezer arrays of Rydberg atoms and polar molecules or by
using quantum gas microscopes for polar molecules and
dysprosium.

[0099] Referring to FIG. 5A, by either dressing rotational
states with Qo (Im,f]) or Q,(Im,,2]), both positive and
negative Ising interactions may be realized. With reference
to FIG. 5B, by turning on an electric field, the molecular
states gain permanent dipole moments, resulting in positive
interactions. Referring to FIG. SC, by combining the appli-
cation of an electric field and dressing, negative Ising
interactions can be realized. In FIGS. 5B and 5C, N denotes
which rotational states the molecular states are adiabatically
connected to (in E).

[0100] The state-transfer procedures may be implemented
using dipole-dipole interactions in a system of polar alkali-
metal dimers. A negligible tunneling of molecules and a
single molecule per lattice site may be assumed. Three
different example schemes for realizing the desired interac-
tions are presented. In all three, the electronic and vibra-
tional ground states, can be described in terms of the angular
momentum operator N. In the first, both positive and nega-
tive Ising interactions can be realized at the cost of involving
large rotational states (N=3). In the second, only positive
Ising interactions can be realized, but only N<2 rotational
states are needed. In the third, only negative Ising interac-

Apr. 27, 2023

tions can be realized, but only NS1 rotational states are
needed. These are shown in FIGS. 5A, 5B, and 5C, respec-
tively.

{0101} In the first scheme, the IN=2/] and IN=3[] states
are dressed with no electric field. In order to realize positive
interactions, only Qo is turned on, which resonantly drives
the it transitions. Although any pair of states dressed by Qo,
IN=2, M: is used as an example. The
corresponding dipole-dipole interaction is:

[0102] where ris the distance between molecules i and j,
6; is the angle the displacement vector makes with the
quantization axis, and j=] 3,2Id9I2,2[Q] is the correspond-
ing transition dipole moment, where d,=é,-d is a component
f the dipole operator d and é,=2, é,=H(&+i9V/V2. If Imp
(0,0) and im, 2,20)+13,2 V2 is encoded, the
dipole-dipole interactions in the Im, asis becomes:

7 1 1=3c0s?6 Zl-z (Eqn. 28)
ap ery ier
[0103] thus realizing the desired positive Ising interac-

tions. Note that in addition to the Ising interactions, the
above Hamiltonian also has terms proportional to Z. How-
ever, this is not an issue, as upon applying the spin-echo
procedure using —cH,,,,, these terms will disappear. Addi-
tionally, because N=2,3 is used rather than N=1,2, there are
no interactions associated with Imo[] due to selection rules.
Finally, note that for the above interactions to hold, a strong
drive Q,>>H,, must be used.

[0104] To realize the negative interactions, only Q, is
turned on, driving the o, transitions. Considering only the
states |2,2(q 13,3] the corresponding dipole-dipole interac
tions take the form:

(Eqn. 29)
2.2%, 3.37 23,3iF2 21.

[0105] _ where uo=P2] 3.31d, 12,28 is the corresponding tran
sition dipole moment. In this case, Im). =(I2.221 13.31)
V2 is encoded, leading to the resulting interaction Hamilto-
nian in the Imp] Im, basis:

HE 1 = 3c0s?6 (Eqn. 30)
Finn = = 1-4) - Z)),
i 8
[0106] thus realizing the desired negative Ising interac-

tions.
[0107] In the second scheme, an electric field may be
applied. Defining 1y/f2).as the states which are adiabati-
cally connected (as E is turned on) to IN, M I |mof2] =!o,0
and im are encoded. Due to the electric field E,
these two states gain permanent dipole moments 1, and Hy,
Additionally, because the electric field does not couple 12,2



US 2023/0129562 Al

Patent Application Publication Apr. 27, 2023 Sheet 2 of 9

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Patent Application Publication Apr. 27,2023 Sheet 1 of 9 US 2023/0129562 Al

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US 2023/0129562 Al

[0043] The disclosed method enables the encoding of a
qubit into a GHZ-like state in a one-dimensional quantum
system 100. Although a one dimensional system is used as
an example, systems of larger dimensionality are contem-
plated.

[0044] Referring to FIGS. 1, 2, 3, and 8, a method 300 and
illustrative schematic 800 for quantum state transfer and
entanglement generation for the quantum system of FIG. 1
is shown. The system 800 for quantum state transfer and
entanglement generation may inchide a processor system
810 (FIG. 8) and a memory 811, including instructions
stored thereon, which when executed by the processor 810,
cause the quantum system 100 to perform the steps of
method 300.

[0045] The processor 810 may be connected to a com-
puter-readable storage medium or a memory 811. The com-
puter-readable storage medium or memory 811 may be a
volatile type of memory, e.g., RAM, or a non-volatile type
of memory, e.g., flash media, disk media, etc. In various
aspects of the disclosure, the processor 810 may be any type
of processor such as a digital signal processor, a micropro-
cessor, an ASIC, a graphics processing unit (GPU), a field-
programmable gate array (FPGA), or a central processing
unit (CPU).

[0046] In aspects of the disclosure, the memory 811 can be
random access memory, read-only memory, magnetic disk
memory, solid-state memory, optical disc memory, and/or
another type of memory. In some aspects of the disclosure,
the memory 811 can be separate from the processor and can
communicate with the processor through communication
buses of a circuit board and/or through communication
cables such as serial ATA cables or other types of cables. The
memory 811 includes computer-readable instructions that
are executable by the processor 810 to operate the processor.
In other aspects of the disclosure, the system 800 may
include a network interface to communicate with other
computers or to a server. A storage device may be used for
storing data.

[0047] Initially at step 110, the unknown coefficients a, b
are encoded in one qubit 104 while the other qubits 102 are
each initialized in state |0) . The next step 120 of the method
300 assumes the ability to encode information into GHZ-like
states in subsystems C,,... , Cy 106 using, for example,
nearest-neighbor interactions. Next at step 130, a general-
ized controlled-phase gate (Eqn. 6) is applied between the
subsystems 106 to “merge” the GHZ-like states into an
entangled state between all sites (FIGS. 2 and 4).

[0048] At step 140, the entangled state is rotated into the
desired GHZ-like state by concentrating the entanglement in
each subsystem 126 onto one qubit 152 (FIGS. 2 and 4).
[0049] At step 150, single-qubit rotations are applied. For
example, qubit 152 c; of each subsystem 126 may be rotated.
[0050] At step 160 the entanglement is redistributed to the
rest of the quantum system 100. Repeatedly feeding the
resulting GHZ-like state back into step 130 of the method
300 yields larger and larger GHZ-like states.

[0051] The Lieb-Robinson bounds become less stringent
for longer-range interactions, ie. those with a<2d+1. The
bounds theoretically allow quantum information to travel a
distance r in time t that scales sublinearly with r. However,
no method in the present literature can saturate these bounds.
In particular, existing methods for ae (d, 2d] are exponen-
tially slower than what is allowed by the corresponding
bounds. Up until now, the existence of this gap between the

Apr. 27, 2023

Lieb-Robinson bounds and the achievable methods has
meant that at least one of the two is not yet optimal, hinting
at either a tighter Lieb-Robinson bound or the possibility of
speeding up many quantum information processing tasks.
[0052] The disclosed technology closes the gap for all
ae(d, 2d+1] in d1 by designing method 300 for encoding
an arbitrary qubit into a multiqubit GHZ-like state and,
subsequently, transferring information at the limits imposed
by the Lieb-Robinson bounds. There are at least three
implications of the method. First, within these regimes of a,
the disclosed technology establishes the tightness of the
Lieb-Robinson bounds, up to sub-polynomial corrections,
and effectively puts an end to the fifteen-year search for a
tighter bound. The disclosed technology may provide opti-
mal designs for power-law interacting systems, including
trapped ions (a€ [0,3]) in one and two dimensions, ultracold
atoms in photonic crystals, van-der-Waals interacting Ryd-
berg atoms (a=6) in three dimensions, as well as the very
common case of dipolar interactions in nitrogen-vacancy
centers, polar molecules, and dipole-dipole interacting Ryd-
berg atoms (a=3) in two dimensions. Finally, the disclosed
technology implies a lower bound on the gate count in
simulating power-law interacting systems on a quantum
computer, providing a benchmark for the performance of
quantum simulation algorithms.

[0053] For simplicity, a d-dimensional hypercubic lattice
A and a two-level system located at every site of the lattice
is considered. It is contemplated that the disclosed technol-
ogy generalizes straightforwardly to all regular lattices.
Without loss of generality, it is assumed that the lattice
spacing is one. A power-law interacting Hamiltonian H(t)
=2, je. hy(t), where h,(t) is a Hamiltonian supported on sites
i, j such that, at all times t and for all ij, provides
[h,|iL/distG, j)*, where dist(i, j) is the distance between i, j,
|s\'is the operator norm, and a20 is a constant. |GHZ(a, b)
)s is used to denote the GHZ-like state over sites in SCA:

IGHZ(a,b)) =a10) +6T) 5. (Eqn. 1)

[0054] where Ix) .) =®,- 51x); (x=0.1) are product states
over all sites in S and a, b are complex numbers such that
lal?+IbI?=1. In particular, GHZ) is used to denote the sym-
metric state a=b=1/Y2.
[0055] Given a d-dimensional hypercube CCA of length
121, encode a possibly unknown state al0)+bI1) of a site c
EC into the GHZ-like state |GHZ(a, b))_ over C, assuming
that all sites in C, except for c, are initially in the state 10) .
Specifically, a time-dependent, power-law interacting Ham-
iltonian H(t) that generates U()=J" exp (-i, J, dsH(s)) is
constructed that satisfies:

UH(Al0}#b11)) LD)..=A1D) 4bIT) (Eqn. 2)

[0056] at time:

log'er if d<a<2d,
17) 3 Kz) ONE if a = 2d, and
PH if dd <as2d+1

(Eqn. 3)

[0057] Here, y=3V4d, x, and x, are constants independent
of tand r. Additionally, by reversing the unitary in Eqn. 2 to

“concentrate” the information in IGHZ(a, b)) onto a differ


US 2023/0129562 Al

d<as2d+1 for all d and, additionally, 2<a<3 for d=1, the
tightness of these fundamental bounds in these regimes is
demonstrated. In particular, the sub-polynomial entangle-
ment time for aS2d disproves the conjecture where a gap in
the provable heating times of periodically driven, power-law
interacting systems had suggested the existence of a tighter
Lieb-Robinson bound with an algebraic light cone in this
regime of a. Additionally, for 2d<a<2d+1, the disclosed
systems and methods suggests that t2r°-*4 is the tightest
possible light cone, providing strong evidence for the con-
jectured generalization of the Lieb-Robinson bound to d>1.
[0084] Since the best known generalizations of these
bounds to k-body, power-law interacting Hamiltonians—
those described by H=Z, hy, where the sum is over all
subsets XCA of at most k sites and Ex; j|hy|[S1/dist(i, j)*
for all i#j, have the same scaling as the best known 2-body
bounds when d<a<2d, the scaling of the disclosed 2-body
method is also optimal even if one allows for k-body
interactions. In other words, in this regime of a, allowing for
k-body interactions cannot enable a qualitative speedup
relative to 2-body interactions.

[0085] The disclosed systems and methods also general-
izes straightforwardly from two-level to arbitrary finite-level
systems. Given a q-level system at each site of the lattice, an
arbitrary state hy) =5%} a,l€) of site c E C may be uni-
tarily encoded, where a, are complex coefficients and C is a
hypercube of linear size r, into a multi-qudit state:

Wy) 10) EES aad?) Eqn. 21)

[0086] in time t(r) satisfying Eqn. 3. This can be done by
replacing the Hamiltonian in Eqn. 6 with:

(Eqn. 22)

6 Bac) Efpra ll MMe BM

1
invay

[0087] and replacing the single-qubit Hadamard gate in
step 4 by a q-by-q discrete Fourier transform matrix. Since
the Lieb-Robinson bounds have the same light cones for any
finite-level systems, the disclosed systems and methods also
saturates these bounds a E (d, 2d+1] in d21 dimensions.

[0088] In aspects, al0)+bl1) may be assumed to be a
possibly unknown state. Encoding such a state into the
GHZ-like state is at least as hard as generating a GHZ-like
state with known coefficients a, b. In fact, the latter task is
not known to be sufficient for state transfer and, therefore, is
not directly constrained by the Lieb-Robinson bounds.
Instead, one often indirectly obtains a speed limit for this
task by applying the Lieb-Robinson bounds on the growth of
two-point connected correlators. Consequently, the task of
generating a known GHZ-like state could potentially be
constrained by a weaker light cone than that of encoding an
unknown qubit state into a GHZ-like state. Nevertheless, the
disclosed systems and methods for encoding into a GHZ-
like state saturates (up to sub-polynomial corrections) the
bound tlog r on the growth of connected correlators when
d<a<2d, implying that knowing the coefficients a, b does not
speed up the preparation of the GHZ-like state in this
regime. The same statement may hold for ae (2d, 2d+1).

[0089] The disclosed violates the so-called Frobenius light
cone, for a>1 in one dimension as part of a hierarchy of
speed limits for different types of information propagation in
long-range interacting systems and later extended to regimes

Apr. 27, 2023

of smaller a, The Frobenius bound, which considers infor-
mation propagation from the operator-spreading perspec-
tive, constrains information-propagation tasks that are more
demanding than the tasks that saturate the Lieb-Robinson
bound, and therefore has a more stringent light cone. For
example, quantum state transfer given intermediate qubits in
arbitrary initial states (ie., universal state transfer) is con
strained by the Frobenius light cone, whereas state transfer
assuming initialized intermediate qubits is constrained by
the Lieb-Robinson bound and can actually violate the Frobe-
nius light cone. The presently disclosed methods prove for
the first time that the task of encoding information into
GHZ-like state, which is at least as hard as state transfer with
initialization—is not constrained by the Frobenius light
cone, but is instead tightly constrained (up to sub-polyno-
mial corrections) by the Lieb-Robinson bound. In particular,
when d<a<2d, the disclosed systems and methods prove that
state transfer with initialization can be implemented expo-
nentially faster than state transfer without initialization,
which is constrained by polynomial light cones in this
regime. Furthermore, since the disclosed systems and meth-
ods for encoding into a GHZ-like state can also be used to
prepare a known GHZ-like state, the disclosed systems and
methods also prove for the first time that preparing a known
GHZ-like state is not constrained by the Frobenius light
cone.

[0090] In aspects, the disclosed method can be generalized
to the regime O<a<d, where there are still substantial gaps
between the Lieb-Robinson bounds and achievable methods.
The bounds suggest that, in addition to the distance, the
information-propagation time also depends on the total
number of sites on the lattice.

[0091] The disclosed systems and methods also give the
first known example of a lower bound on the gate count in
simulating power-law systems on a quantum computer: it
takes Q(n) elementary quantum gates to simulate an n-qubit
power-law system evolving for time t2t., where:

Ollogn) if d<a<2d,
1 =f fe") if a = 2a, and

One!)

(Eqn. 23)

if Wd<asdd+1

[0092]

[0093] Lower bounds on the simulation gate count are
valuable benchmarks for the performance of quantum algo-
rithms. To date, despite progressively more efficient quan-
tum simulation algorithms in recent literature, no saturable
lower bounds are known for power-law systems. For
example, the analysis of the Suzuki-Trotter product formulas
results in upper bounds,

to constant error.

if d<a@<2d, (Eqn. 24)

_- (re

Olay!) if a> 2d

[0094] for simulating an n-qubit power-law system for
time t. At t=t, given in Eqn. 23, the corresponding upper
bounds reduce to