physics of quantum information lecture 5: entanglement ii
TRANSCRIPT
Physics of Quantum Information
Lecture 5: Entanglement II
Piège de Paul (RF + statique)
Piège de Paul linéaire
Répulsion de CoulombÞ Cristal ionique
+V0 +V0
~ VRF
~ VRF
W = 1 - 10 MHz
VRF
V0
Innsbrück
Innsbrück
Trapping individual ions
40Ca+ Innsbrück (R. Blatt)
729 nm
1 secLaser cooling,
detection
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Optical qubit and its detection
|1ñ
|0ñ
|1ñ
|0ñ
Chaque point = 200 répétitions
Oscillations de Rabi d’un atome unique
Spectre de la transition entre deux niveaux hyperfins de l’état fondamental du 87Rb (Institut d’optique, Palaiseau)
Oscillations de Rabi d’un atome unique
Oscillation de Rabi entre deux niveaux hyperfins de l’état fondamental du 87Rb (Institut d’optique, Palaiseau)
|1ñ
|0ñ
6.8 GHz
Exemple d’oscillation de Rabi sur un atome (Institut Optique)
Entanglement of two ions: tomography
with an intensified CCD camera separately for each ion.Observation of fluorescence indicates that the ion wasprojected into the S1=2 ! j1i state; no fluorescence revealsthe D5=2 ! j0i state. By repeating the experimental cycle200 times, the average populations of all product basisstates j00i, j01i, j10i, and j11i are determined.
A Bell state is created by applying laser pulses to ion 1and 2 on the blue sideband and the carrier. Using the Paulispin matrices !x,!y,!z [13] and the operators b and by
that annihilate and create a phonon in the breathingmode, we denote single qubit carrier rotations of qubit" by
R""#;$# $ exp!
i#2"!""#
x cos$% !""#y sin$#
"
(1)
and rotations on the blue sideband of the vibrationalbreathing mode by
R&" "#;$# $ exp
!
i#2"!""#
x by cos$% !""#y b sin$#
"
: (2)
The Bell state !' $ "j10i' j01i#=###
2p
is produced bythe pulse sequence U!' $ R&
2 "%;'%=2#R2"%;%=2# (R&1 "%=2;%%=2# applied to the j11i state. The pulse
R&1 "%=2;%%=2# entangles the motional and the internal
degrees of freedom; the next two pulses R&2 "%;'%=2# (
R2"%;%=2# map the motional degree of freedom ontothe internal state of ion 2. Appending another % pulse,U"' $ R2"%; 0#U!', produces the state "' up to aglobal phase. The pulse sequence takes less than 200 &s.
To account for experimental imperfections, the quan-tum state is described by a density matrix '. For itsexperimental determination we expand ' into a super-position ' $ P
i(iOi of mutually orthogonal Hermitianoperators Oi, which form a basis and obey the equationtr"OiOj# $ 4)ij [14]. Then the coefficients (i are relatedto the expectation values of Oi by (i $ tr"'Oi#=4. For atwo-qubit system, a convenient set of operators is given bythe 16 operators !"1#
i ) !"2#j , "i; j $ 0; 1; 2; 3#, where !""#
iruns through the set of Pauli matrices 1;!x;!y;!z, ofqubit ".
The reconstruction of the density matrix ' is accom-plished by measuring the expectation values h!"1#
i )!"2#
j i'. A fluorescence measurement projects the quantumstate into one of the states jx1x2i, xi 2 f0; 1g. By repeat-edly preparing and measuring the quantum state, theaverage population in states jx1x2i is obtained from whichwe calculate the expectation values of !"1#
z , !"2#z , and
!"1#z ) !"2#
z . To measure operators involving !y, we applya transformation U that maps the eigenvectors of !y ontothe eigenvectors of !z, i.e., U!yU%1 $ !z, where U $R"%=2;%#. Similarly, the operator !x is transformed into!z by choosing U $ R"%=2; 3%=2#. Therefore, all expec-tation values can be determined by measuring !"1#
z , !"2#z ,
or !"1#z ) !"2#
z . To obtain all 16 expectation values, ninedifferent settings have to be used. For each setting, theexperiment is repeated 200 times at a repetition rate of
50 Hz. The whole reconstruction process is thereforecompleted in less than 40 s. Since a finite number ofexperiments allows only for an estimation of the expec-tation values h!"1#
i ) !"2#j i', the reconstructed matrix 'R
is not guaranteed to be positive semidefinite [15].We avoid this problem by employing a maximum like-lihood estimation of the density matrix [15,16], fol-lowing the procedure as suggested and implemented inRefs. [16–18].
For the pulse sequence that is designed to produce thestate !& $ "j10i& j01i#=
###
2p
, we obtain the density ma-trix '!& shown in Fig. 1(a). The fidelity F of the recon-structed state is F $ h!&j'!& j!&i $ 0:91. To produce
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Re ρ Im ρ
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Re ρ Im ρ
(b)
(a)
(d)
(c)
FIG. 1. (a) Real and imaginary parts of the density matrix'!& that approximates !& $ "j10i& j01i#=
###
2p
. The measuredfidelity is F!& $ h!&j'!& j!&i $ 0:91. (b) Real and imagi-nary parts of the density matrix '!% that approximates !% $"j10i% j01i#=
###
2p
. The measured fidelity is F!% $ 0:90.(c),(d) Density matrix elements of (c) '"& and (d) '"% .Here, F"& $ 0:91 and F"% $ 0:88.
P H Y S I C A L R E V I E W L E T T E R S week ending4 JUNE 2004VOLUME 92, NUMBER 22
220402-2 220402-2
with an intensified CCD camera separately for each ion.Observation of fluorescence indicates that the ion wasprojected into the S1=2 ! j1i state; no fluorescence revealsthe D5=2 ! j0i state. By repeating the experimental cycle200 times, the average populations of all product basisstates j00i, j01i, j10i, and j11i are determined.
A Bell state is created by applying laser pulses to ion 1and 2 on the blue sideband and the carrier. Using the Paulispin matrices !x,!y,!z [13] and the operators b and by
that annihilate and create a phonon in the breathingmode, we denote single qubit carrier rotations of qubit" by
R""#;$# $ exp!
i#2"!""#
x cos$% !""#y sin$#
"
(1)
and rotations on the blue sideband of the vibrationalbreathing mode by
R&" "#;$# $ exp
!
i#2"!""#
x by cos$% !""#y b sin$#
"
: (2)
The Bell state !' $ "j10i' j01i#=###
2p
is produced bythe pulse sequence U!' $ R&
2 "%;'%=2#R2"%;%=2# (R&1 "%=2;%%=2# applied to the j11i state. The pulse
R&1 "%=2;%%=2# entangles the motional and the internal
degrees of freedom; the next two pulses R&2 "%;'%=2# (
R2"%;%=2# map the motional degree of freedom ontothe internal state of ion 2. Appending another % pulse,U"' $ R2"%; 0#U!', produces the state "' up to aglobal phase. The pulse sequence takes less than 200 &s.
To account for experimental imperfections, the quan-tum state is described by a density matrix '. For itsexperimental determination we expand ' into a super-position ' $ P
i(iOi of mutually orthogonal Hermitianoperators Oi, which form a basis and obey the equationtr"OiOj# $ 4)ij [14]. Then the coefficients (i are relatedto the expectation values of Oi by (i $ tr"'Oi#=4. For atwo-qubit system, a convenient set of operators is given bythe 16 operators !"1#
i ) !"2#j , "i; j $ 0; 1; 2; 3#, where !""#
iruns through the set of Pauli matrices 1;!x;!y;!z, ofqubit ".
The reconstruction of the density matrix ' is accom-plished by measuring the expectation values h!"1#
i )!"2#
j i'. A fluorescence measurement projects the quantumstate into one of the states jx1x2i, xi 2 f0; 1g. By repeat-edly preparing and measuring the quantum state, theaverage population in states jx1x2i is obtained from whichwe calculate the expectation values of !"1#
z , !"2#z , and
!"1#z ) !"2#
z . To measure operators involving !y, we applya transformation U that maps the eigenvectors of !y ontothe eigenvectors of !z, i.e., U!yU%1 $ !z, where U $R"%=2;%#. Similarly, the operator !x is transformed into!z by choosing U $ R"%=2; 3%=2#. Therefore, all expec-tation values can be determined by measuring !"1#
z , !"2#z ,
or !"1#z ) !"2#
z . To obtain all 16 expectation values, ninedifferent settings have to be used. For each setting, theexperiment is repeated 200 times at a repetition rate of
50 Hz. The whole reconstruction process is thereforecompleted in less than 40 s. Since a finite number ofexperiments allows only for an estimation of the expec-tation values h!"1#
i ) !"2#j i', the reconstructed matrix 'R
is not guaranteed to be positive semidefinite [15].We avoid this problem by employing a maximum like-lihood estimation of the density matrix [15,16], fol-lowing the procedure as suggested and implemented inRefs. [16–18].
For the pulse sequence that is designed to produce thestate !& $ "j10i& j01i#=
###
2p
, we obtain the density ma-trix '!& shown in Fig. 1(a). The fidelity F of the recon-structed state is F $ h!&j'!& j!&i $ 0:91. To produce
|00>|01>
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0.5
0
-0.5
0.5
0
-0.5
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|10>|11>
|00> |01> |10>|11>
Re ρ Im ρ
0.5
0
-0.5
0.5
0
-0.5
Re ρ Im ρ
0.5
0
-0.5
0.5
0
-0.5
Re ρ Im ρ
0.5
0
-0.5
0.5
0
-0.5
Re ρ Im ρ
(b)
(a)
(d)
(c)
FIG. 1. (a) Real and imaginary parts of the density matrix'!& that approximates !& $ "j10i& j01i#=
###
2p
. The measuredfidelity is F!& $ h!&j'!& j!&i $ 0:91. (b) Real and imagi-nary parts of the density matrix '!% that approximates !% $"j10i% j01i#=
###
2p
. The measured fidelity is F!% $ 0:90.(c),(d) Density matrix elements of (c) '"& and (d) '"% .Here, F"& $ 0:91 and F"% $ 0:88.
P H Y S I C A L R E V I E W L E T T E R S week ending4 JUNE 2004VOLUME 92, NUMBER 22
220402-2 220402-2
Roos, PRL (2004)
Entanglement of 8 ions: tomography
Häffner, Nature 438, 643 (2005)
8 ions W states, 65531 coefficients10 hours of measurements!
Measuring the Rényi entropy of 4 atoms
Islam et al. Nature 528, 77 (2015)
2
systems of photonic qubits [30] and trapped ion spins [31], butthere is no known scheme to perform tomography for systemsinvolving itinerant delocalized particles. With multiple copiesof a system, however, one can use quantum many-body in-terference to quantify entanglement even in itinerant systems[21, 32, 33].
In this work, we take advantage of the precise control andreadout afforded by our quantum gas microscope [34] to pre-pare and interfere two identical copies of a four-site Bose-Hubbard system. This many-body quantum interference en-ables us to measure quantities that are not directly accessiblein a single system, e.g. quadratic functions of the density ma-trix [21, 32, 33, 35–38]. Such non-linear functions can re-veal entanglement [1]. In our system, we directly measure thequantum purity, Renyi entanglement entropy, and mutual in-formation to probe the entanglement in site occupation num-bers.
BIPARTITE ENTANGLEMENT
To detect entanglement in our system, we use a fundamen-tal property of entanglement between two subsystems (bipar-tite entanglement): ignoring information about one subsys-tem results in the other becoming a classical mixture of purequantum states. This classical mixture in a density matrix⇢ can be quantified by measuring the quantum purity, de-fined as Tr(⇢2). For a pure quantum state the density ma-trix is a projector and Tr(⇢2) = 1, whereas for a mixedstate Tr(⇢2) < 1. In case of a product state, the subsys-tems A and B of a many-body system AB described by awavefunction | ABi (Fig. 1) are individually pure as well,i.e. Tr(⇢2A) = Tr(⇢2B) = Tr(⇢2AB) = 1. Here the re-duced density matrix of A, ⇢A = TrB(⇢AB), where ⇢AB =| ABih AB | is the density matrix of the full system. TrBindicates tracing over or ignoring all information about thesubsystem B. For an entangled state, the subsystems becomeless pure compared to the full system as the correlations be-tween A and B are ignored in the reduced density matrix,Tr(⇢2A) = Tr(⇢2B) < Tr(⇢2AB) = 1. Even if the many-bodystate is mixed (Tr(⇢2AB) < 1), it is still possible to measureentanglement between the subsystems [1]. It is sufficient [39]to prove this entanglement by showing that the subsystems areless pure than the full system, i.e.
Tr(⇢2A) < Tr(⇢2AB),
Tr(⇢2B) < Tr(⇢2AB). (1)
These inequalities provide a powerful tool for detecting entan-glement in the presence of experimental imperfections. Fur-thermore, quantitative bounds on the entanglement present ina mixed many-body state can be obtained from these state pu-rities [40].
Eq.(1) can be framed in terms of entropic quantities [1, 39].A particularly useful and well studied quantity is the n-th or-der Renyi entropy,
Sn(A) =1
1� nlog Tr(⇢nA). (2)
a
beven particle # in Output 2
avarageparity
quantum stateoverlap
purity
even particle # in Output 1 odd particle #
odd particle #
2 identical N-particlebosonic states
Figure 2. Measurement of quantum purity with many-bodybosonic interference of quantum twins. a. When two N parti-cle bosonic systems that are in identical pure quantum states are in-terfered on a 50%-50% beam splitter, they always produce outputstates with even number of particles in each copy. This is due to thedestructive interference of odd outcomes and represents a general-ized Hong-Ou-Mandel interference, in which two identical photonsalways appear in pairs after interfering on a beam splitter. b. If theinput states ⇢1 and ⇢2 are not perfectly identical or not perfectly pure,the interference contrast is reduced. In this case the expectation valueof the parity of particle number hPii in output i = 1, 2 measures thequantum state overlap between the two input states. For two identi-cal input states ⇢1 = ⇢2, the average parity hPii therefore directlymeasures the quantum purity of the states. We only assume that theinput states have no relative macroscopic phase relationship.
From Eq. (2), we see that the second-order (n = 2) Renyientropy and purity are related by S2(A) = � log Tr(⇢2A).S2(A) provides a lower bound for the von Neumann entangle-ment entropy SV N (A) = S1(A) = �Tr(⇢A log ⇢A) exten-sively studied theoretically. The Renyi entropies are rapidlygaining importance in theoretical condensed matter physics,as they can be used to extract information about the “entan-glement spectrum” [41] providing more complete informa-tion about the quantum state than just the von Neuman en-tropy. In terms of the second-order Renyi entropy, the suffi-cient conditions to demonstrate entanglement [1, 39] becomeS2(A) > S2(AB), and S2(B) > S2(AB), i.e. the subsys-tems have more entropy than the full system. These entropicinequalities are more powerful in detecting certain entangledstates than other inequalities like the Clauser-Horne-Shimony-Holt (CHSH) inequality [36, 39].
MEASUREMENT OF QUANTUM PURITY
The quantum purity and hence the second-order Renyi en-tropy can be directly measured by interfering two identicaland independent copies of the quantum state on a 50%-50%beam splitter [21, 32, 33, 36]. For two identical copies of abosonic Fock state, the output ports always have even parti-
Measuring the Rényi entropy of 4 atoms
Islam et al. Nature 528, 77 (2015)
Measuring the Rényi entropy of 4 atoms
Islam et al. Nature 528, 77 (2015)
Measuring the Rényi entropy of 4 atoms
Islam et al. Nature 528, 77 (2015)
Measuring the Rényi entropy in an ion chain
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0 1 2 3 410�2
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101
0 1 2 3 410�2
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100
101
T = 0ms
T = 10ms
a) b)
�1.0 �0.5 0.0 0.5 1.0h�ziu
0.0
0.5
1.0
Figure 1: Measuring second-order Renyi entropies via randomized measure-ments. a) Single qubit Bloch sphere. The purity is directly related to the width of thedistribution of measurement outcomes after applying random rotations ui. Initial purestate (blue) and mixed state (red) cases are shown. See text. b) Generalization to mul-tiple qubits: Measuring up to 10-qubit partitions of a 20-qubit string, as shown (top).Repeated measurements are made to obtain statistics, see text. Experimental data (bot-tom): Histograms of the weighted sum X of cross correlations (as defined in Eq. (2)),with mean values corresponding to the purities (dashed lines). Results are shown for twodi↵erent times during evolution under HXY, starting from a highly pure, separable stateand evolving into a high entropy state.
the system was initially prepared in the Neel ordered product state ⇢0 = | ih | with| i = | #"# .. "i. This state was subsequently time-evolved under HXY (or H) into thestate ⇢(t). The coherent interactions arising from this time evolution generated varyingtypes of entanglement in the system. Subsequently, randomized measurements on ⇢(t)were performed through individual rotations of each qubit by a random unitary (ui),sampled from the CUE (25), followed by a state measurement in the z-basis. Each ui canbe decomposed into three rotations Rz(✓3)Ry(✓2)Rz(✓1), and two random unitaries wereconcatenated to ensure that drawing of the ui was stable against small drifts of physicalparameters controlling the rotation angles ✓i (26). Finally, spatially resolved fluorescencemeasurements realised a projective measurement in the logical z-basis. To measure theentropy of a quantum state, NU sets of single-qubit random unitaries, U = u1 ⌦ · · · ⌦ uN ,were applied. For each set of applied unitaries, U , the measurement was repeated NM
times.In the first experiment, the 10-qubit state ⇢0 was prepared and subsequently time-
evolved under HXY (Eq. (3)), without disorder, for ⌧ = 0, . . . , 5 ms. Fig. 2 shows themeasured purities (a) and entropies (b) of all connected partitions that include qubit1 during this quench. The overall purity (and thus entropy) remained at a constantvalue of Tr [⇢2] = 0.74 ± 0.07, within error, throughout the time evolution, implyingthat the time evolution was approximately unitary. The initial state’s reconstructed
4
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10�2
10�1
0 1 2 3 410�2
10�1
100
101
0 1 2 3 410�2
10�1
100
101
T = 0ms
T = 10ms
a) b)
�1.0 �0.5 0.0 0.5 1.0h�ziu
0.0
0.5
1.0
Figure 1: Measuring second-order Renyi entropies via randomized measure-ments. a) Single qubit Bloch sphere. The purity is directly related to the width of thedistribution of measurement outcomes after applying random rotations ui. Initial purestate (blue) and mixed state (red) cases are shown. See text. b) Generalization to mul-tiple qubits: Measuring up to 10-qubit partitions of a 20-qubit string, as shown (top).Repeated measurements are made to obtain statistics, see text. Experimental data (bot-tom): Histograms of the weighted sum X of cross correlations (as defined in Eq. (2)),with mean values corresponding to the purities (dashed lines). Results are shown for twodi↵erent times during evolution under HXY, starting from a highly pure, separable stateand evolving into a high entropy state.
the system was initially prepared in the Neel ordered product state ⇢0 = | ih | with| i = | #"# .. "i. This state was subsequently time-evolved under HXY (or H) into thestate ⇢(t). The coherent interactions arising from this time evolution generated varyingtypes of entanglement in the system. Subsequently, randomized measurements on ⇢(t)were performed through individual rotations of each qubit by a random unitary (ui),sampled from the CUE (25), followed by a state measurement in the z-basis. Each ui canbe decomposed into three rotations Rz(✓3)Ry(✓2)Rz(✓1), and two random unitaries wereconcatenated to ensure that drawing of the ui was stable against small drifts of physicalparameters controlling the rotation angles ✓i (26). Finally, spatially resolved fluorescencemeasurements realised a projective measurement in the logical z-basis. To measure theentropy of a quantum state, NU sets of single-qubit random unitaries, U = u1 ⌦ · · · ⌦ uN ,were applied. For each set of applied unitaries, U , the measurement was repeated NM
times.In the first experiment, the 10-qubit state ⇢0 was prepared and subsequently time-
evolved under HXY (Eq. (3)), without disorder, for ⌧ = 0, . . . , 5 ms. Fig. 2 shows themeasured purities (a) and entropies (b) of all connected partitions that include qubit1 during this quench. The overall purity (and thus entropy) remained at a constantvalue of Tr [⇢2] = 0.74 ± 0.07, within error, throughout the time evolution, implyingthat the time evolution was approximately unitary. The initial state’s reconstructed
4
1 qubit Ion chain
Brydges et al. Science 364, 260 (2019)
Large scale entanglement GHZ statesI. POGORELOV et al. PRX QUANTUM 2, 020343 (2021)
two 19-inch racks. We have presented mechanical, optical,and electrical systems along with characterizing experi-ments. This experimental platform improves upon conven-tional hardware implementations in terms of modularity,integration, and remote control. In our characterizationmeasurements we find the system performance to be on parwith conventional, laboratory-based hardware implemen-tations in terms of experimentally relevant performancecriteria. We find that mechanical stability, optical read-out and addressing performance, heating rates, coherencetimes, and Mølmer-Sørensen entangling fidelities in thecurrent implementation do not suffer relative to tradi-tional optical table setups. Using the compound system,we are able to produce maximally entangled Greenberger-Horne-Zeilinger states with up to 24 qubits with a fidelityof 54.4% ± 0.7%. To our knowledge, this is the largestmaximally entangled state yet generated in any systemwithout the use of error mitigation or postselection, anddemonstrates the capabilities of our system.
In addition, we presented site-selective qubit operationsto implement a complete gate set for universal quantumcomputation using two distinct approaches to addressing: amicro-optics approach with fixed, rigid waveguides, and anacousto-optic deflector approach. Both of these approachesoffer advantages over the other in particular settings.
The micro-optics approach readily offers itself for simul-taneous multisite addressing without producing off-axisspots that can lead to resonant and off-resonant crosstalk.The power scaling in such a scenario is linear in thenumber of channels, assuming a suitable power distribu-tion system is at hand. Parallel radial sideband coolingis one direct beneficiary of such capabilities, as is directgeneration of interaction between more than two qubits.Individual control over amplitude, phase, and frequency ofeach channel is a fundamental advantage of this approachbut requires one FAOM per qubit. The positional stabilityis not affected by the oscillator stability of rf sources, andextension to larger registers are not limited by the speed ofsound or similar quantities such as in AOD-based devices.
The AOD approach on the other hand is technologi-cally simpler, thus offering superior performance at thisstage. Optical quality of macroscopic components is oftenalso superior to micro-optics, in particular in prototyp-ing scenarios such as here, which reduces abberations.The addressing is inherently flexible, such that it can beadjusted for qubit registers from 2 ions to 40 ions in ourconfiguration. Adjustment of power and optical phase ofindividual channels is possible without an optical modula-tor per ion, which significantly reduces the technologicaloverhead compared to the micro-optics approach. Thisunit is fed by a single fiber, and no prior power dis-tribution capabilities are required. The switching speedin AODs is limited ultimately by the speed of soundin the deflecting crystal, which therefore also limits thespeed at which operations can be performed. The quadratic
(a)
(b)
FIG. 19. Ion images of (a) 24 and (b) 50 ions in the demon-strator. The 24-ion chain is used to demonstrate 24-partite entan-glement within the setup without the use of postselection or errormitigation. Fifty-ion chains are the midterm control target andcan already be trapped and cooled. Nonuniform brightness stemsfrom the finite size of the detection beam.
power loss for multisite addressing, and off-axis spots limitthis technology to the simultaneous addressing of a smallnumber of ions.
With both of these approaches we demonstrate single-qubit and pairwise-entangling gates on registers up to10 ions. From randomized benchmarking we obtain afidelity of Fgate = 99.86% ± 0.01% per addressed single-ion gate. Measurements of resonant crosstalk are shownto be below 1% across the entire ten-ion register with theAOD approach, while nonresonant crosstalk is measured tobe less than 1.25 × 10−4 in the same string. Together withthe pairwise-entangling operations with fidelities between97% and 99% we show all the basic operations for a fullyprogrammable quantum system. Benchmarking of largerregisters, and with more complete suites of benchmarkingtools [96] will be undertaken as part of the next round ofhardware integration and software improvements.
These near- and midterm upgrades to the hardware andsoftware stacks will further improve upon the demonstra-tor’s capabilities. Use of an external master oscillator togenerate the narrow-linewidth qubit laser will no longerbe required after installation of the compact diode-lasersource that is currently under construction as part of theAQTION collaboration. Similarly, single-site addressingcapabilities will be improved in mode quality and the num-ber of channels. This will allow the setup to implementmore complex algorithms by moving from axial gates toradial quantum gates enhanced by established quantumcontrol techniques [97,98]. Upgrades to M-ACTION, aswell as complimentary developments to the control andremote access software stack will enable easier accessto the demonstrator capabilities in a hardware-agnosticfashion.
Already, the device presented is capable of operatingwith qubit numbers on par with state-of-the-art conven-tional laboratory setups. An image of such a qubit registeris shown in Fig. 19(a). The midterm upgrades shouldenable us to increase this number to the AQTION controltarget of 50 qubits and beyond. We have already demon-strated basic capabilities of control in larger qubit chainsas shown in Fig. 19(b), with a 50-ion chain already crys-tallized in our trap. In the long term, we hope that thedemonstrator’s features and engineering solutions mean
020343-20
24 ions
PRXQuantum 2, 020343 (2021)
GHZ, F=0.54
20 atoms