Parallel Entangling Operations on a Universal Ion Trap Quantum Computer
C. Figgatt, A. Ostrander, N. M. Linke, K. A. Landsman, D. Zhu, D. Maslov, C. Monroe
PParallel Entangling Operations on a Universal Ion Trap Quantum Computer
C. Figgatt, A. Ostrander, N. M. Linke, K. A. Landsman, D. Zhu, D. Maslov,
3, 1 and C. Monroe
1, 4 Joint Quantum Institute, Department of Physics,and Joint Center for Quantum Information and Computer Science,University of Maryland, College Park, MD 20742, USA Department of Physics and Joint Center for Quantum Information and Computer Science,University of Maryland, College Park, MD 20742, USA National Science Foundation, Alexandria, VA 22314, USA IonQ Inc., College Park, MD 20742, USA (Dated: October 30, 2018)
The circuit model of a quantum computer consistsof sequences of gate operations between quantum bits(qubits), drawn from a universal family of discrete op-erations [1]. The ability to execute parallel entanglingquantum gates offers clear efficiency gains in numerousquantum circuits [2–4] as well as for entire algorithmssuch as Shor’s factoring algorithm [5] and quantum sim-ulations [6, 7]. In cases such as full adders and multiple-control Toffoli gates, parallelism can provide an exponen-tial improvement in overall execution time. More impor-tantly, quantum gate parallelism is essential for the prac-tical fault-tolerant error correction of qubits that sufferfrom idle errors [8, 9]. The implementation of parallelquantum gates is complicated by potential crosstalk, es-pecially between qubits fully connected by a common-mode bus, such as in Coulomb-coupled trapped atomicions [10, 11] or cavity-coupled superconducting trans-mons [12]. Here, we present the first experimental resultsfor parallel 2-qubit entangling gates in an array of fully-connected trapped ion qubits. We demonstrate an appli-cation of this capability by performing a 1-bit full addi-tion operation on a quantum computer using a depth-4quantum circuit [4, 13, 14]. These results exploit thepower of highly connected qubit systems through clas-sical control techniques, and provide an advance towardspeeding up quantum circuits and achieving fault toler-ance with trapped ion quantum computers.Trapped atomic ions are among the most advancedqubit platforms [10, 11], with atomic clock precision andthe ability to perform gates in a fully-connected and re-configurable qubit network [15]. The high connectivitybetween trapped ion qubits [16] is mediated by opti-cal forces on their collective motion [17–20], and can bescaled in a modular fashion using a variety of methods[10, 11]. Within a single large chain of ions, gates canbe performed by appropriately shaping the laser pulsesthat drive select ions within the chain. Here, the targetqubits become entangled through their Coulomb-coupledmotion, and the laser pulse is modulated such that themotional modes are disentangled from the qubits at theend of the operation [21–23]. The execution of multipleparallel gates in this way requires more complex pulseshapes, not only to disentangle the motion but also to entangle exclusively the intended qubit pairs. We achievethis type of parallel operation by designing appropriateoptical pulses using nonlinear optimization techniques.We perform parallel gates on a chain of five atomic Yb + ions, with resonant laser radiation used to laser-cool, initialize, and measure the qubits. Coherent quan-tum gate operations are achieved by applying counter-propagating Raman beams from a single mode-lockedlaser, which form beat notes near the qubit differencefrequency. Single-qubit gates are generated by tuningthe Raman beatnote to the qubit frequency splitting ω and driving resonant Rabi rotations ( r gates) of definedphase and duration. Two-qubit ( xx ) gates are realizedby illuminating two ions with beams that have beat-notefrequencies near the motional sidebands, creating an ef-fective Ising interaction between the ions via transiententanglement through the modes of motion [18–20]. Weuse an amplitude-modulated pulse-shaping scheme thatprovides high-fidelity entangling gates on any ion pair[15, 22, 23]; frequency [24] or phase [25] modulation ofthe laser pulses would also suffice. (See Methods for ad-ditional experimental setup details.)In order to perform parallel entangling operations in-volving M independent pairs of qubits in a chain of N ≥ M ions with N motional modes at frequencies ω k , a shaped qubit-state-dependent force is applied tothe involved ions using bichromatic beat notes at ω ± µ ,resulting in the evolution operator [21, 22] U || ( τ ) = exp M (cid:88) i =0 ˆ φ i ( τ ) σ xi + i M (cid:88) i 962 MHz, exhibitthe greatest displacement, and contribute the most to thefinal spin-spin entanglement by enclosing a larger area ofphase space. Negative-amplitude pulses are implementedby applying a π phase shift to the control signal, allow-ing the entangling pairs to continue accumulating entan-glement while cancelling out accumulated entanglementwith cross-talk pairs. Consequently, all pulse solutionsfeature similar patterns with symmetric phase flips onone pair to cancel out crosstalk entanglement. Pulseshapes and phase space trajectories for the additionalsolutions are given in [26].We characterize the experimental gate fidelities bymeasuring the selected output qubits in different basesand extracting the parity as a witness operator [27–29],as described in the Methods. Fitted parity curves areshown in Figure 2. Entangling gate fidelities are typi-cally between 96-99%, with crosstalk of a few percent.Crosstalk errors are estimated by fitting the crosstalkpair populations and parity in the same way as above.Crosstalk fidelities for all pairs were close to 25%, whichindicate a complete statistical mixture. The given er-rors are statistical. All data has been corrected for statepreparation and measurement (SPAM) errors of 3-5%, asdescribed in [15, 26, 30].As an example application of a parallel operation use-ful for error correction codes [3], we perform a pair of cnot gates in parallel on two pairs of ions. The cnot gate sequence (compiled version with r and xx gatesshown in [15]) is performed simultaneously on the pair(1 , , cnot gates are performed for each of the 16possible bitwise inputs, and population data for the 16possible bitwise outputs is shown in Figure 3 with anaverage process fidelity of 94.5(2)%.Another application that benefits from the use of par-allel entangling operations is the quantum full adder. Inmodern classical computing, a full adder is a basic circuitthat can be cascaded to add many-bit numbers, and canbe found in processors as components of arithmetic logicunits (ALU’s) and performing low-level operations suchas computing register addresses. In quantum computing,adders can be used in a similar fashion to perform arith-metic operations over quantum registers (e.g. [6]); somealgorithms are dominated by the adders, notably Shor’sinteger factoring algorithm. The quantum full adder re-quires 4 qubits, 3 for the primary inputs x , y , and thecarry bit C in , and the fourth initialized to | (cid:105) . The fouroutputs consist of the first input, x , simply continuing (a) ions i = (1 , (b) ions i = (2 , α i,k for each mode k correlatedwith ion i (right five panels) for parallel xx gates on (a) ions (1,4) and (b) ions (2,5). The pulse shape solutions are expressedin terms of the time-dependent Rabi frequency Ω i ( t ) experienced by both ions in each pair, and is broken into S = 60 segmentswith a total gate time of 250 µ s. Negative Rabi frequencies correspond to an inverted phase of the beatnote. The 5 modes ofmotion have frequencies ω k / π = { . , . , . , . , . } MHz, and with the constant laser beatnote detuning set to µ = 2 . 962 MHz, the nearby modes 4 and 5 feel the largest displacements. The phase space trajectories begin at the blue circlesand follow the continuous path to the green star, with the color shading of the trajectory corresponding to the pulse shape intime at left. The sum of the normalized area enclosed by all 5 modes is set to π/ (a) (b) Rotation Axis -1-0.8-0.6-0.4-0.200.20.40.60.81 P a r i t y (1,4)(2,5)(1,2)(1,5)(2,4)(4,5) 0 /4 /2 3 /4 5 /4 3 /2 7 /4 2 Rotation Axis -1-0.8-0.6-0.4-0.200.20.40.60.81 P a r i t y (1,4)(2,3)(1,2)(1,3)(2,4)(3,4) FIG. 2. Parity curves and fidelities for parallel xx gates on two example sets of ions. Circles indicate data, with matching-colorlines indicating calculated fit. Additional data is shown in the Methods. (a) Ions (1,4) and (2,5), yielding fidelities of 96.5(4)%and 97.8(3)% on the respective entangled pairs, with an average crosstalk error of 3.6(3)%. (b) Ions (1,4) and (2,3), yieldingfidelities of 98.8(3)% and 99.0(3)% on the respective entangled pairs, with an average crosstalk error of 1.4(3)%. The quotederrors are statistical. P r o b a b ili t y Detected Input FIG. 3. Data for sumiltaneous cnot gates on ions (1,4) and(2,3), with an average process fidelity of 94.5(2)%. through; y (cid:48) , which carries x ⊕ y (an additional cnot canbe added to extract y if desired); and the sum S and out-put carry C out , which together comprise the 2-bit resultof summing x , y , and C in , where C out is the most sig-nificant bit and hence becomes the carry bit to the nextadder in a cascade, and S is the least significant bit. Wecan also write the sum as S = x ⊕ y ⊕ C in and the outputcarry as C out = ( x · y ) ⊕ ( C in · ( x ⊕ y )). Feynman firstdesigned such a circuit using cnot and Toffoli gates [13],shown in Figure 4(a), which would require 12 xx gatesto implement on an ion trap quantum computer. A moreefficient circuit requires at most 6 2-qubit interactions[4], and features a gate depth of only 4 if simultaneous2-qubit operations are available, as shown by the dashedoutlines in Figure 4(b).The full adder is implemented using 2 different parallel xx gate configurations, as well as the single-qubit rota-tions and additional xx gates shown in Figure 8 in theMethods. The parallel gates require different amountsof entanglement, equivalent to implementing a fully-entangling xx (cid:0) χ ij = π (cid:1) gate and a partially-entangling xx (cid:0) π (cid:1) gate in parallel. This is experimentally imple-mented by independently adjusting the optical powersupplied to each gate; discussions of the calibration in-dependence of these parallel gates and fidelity data forsuch an operation are given in the Methods. The in-puts x , y , C in , and 0 are mapped to the qubits (1 , , , xx gates in a chain of N ions requires 4 N + 6 ∼ O ( N ) constraints, so the problemsize growth is linear in N . Entangling more pairs in par-allel enlarges the problem size quadratically: entangling M pairs involves the interactions of 2 M ions, yielding (cid:0) M (cid:1) = 2 M − M ∼ O ( M ) spin-spin interactions tocontrol and 2 M N spin-motional entanglements to close.Scaling both the number of entangled pairs M and thenumber of ions N in the chain therefore gives a total num-ber of constraints of 2 M N + 2 M − M ∼ O ( M + M N ).On very long chains, not all ion-ion connections will bedirectly available [31], reducing the number of quadraticconstraints on crosstalk pairs that must be considered,indicating that this is an upper bound on the scaling.Several lines of future inquiry may help increase thetheoretical solution fidelity. Easing constraints on thepower needed may allow for higher-fidelity solutions tobe calculated, although increasing power on the experi-ment can exacerbate errors due to Raman beam noise.Investigating whether the constraint matricies in Equa-tion 14 (see Methods) can be modified to become positiveor negative semidefinite may provide improvements, asconvex QCQP’s are readily solved using semidefinite pro-gramming techniques and could allow for higher-fidelitysolutions. However, these problems are all ones of over-head. Once a high-quality gate solution is implementedon the experiment, no further calculations are needed;only a single calibration is required to compensate forRabi frequency drifts. (a) (c) x • • xy • • • y (cid:48) C in • S C out (b) x • • xy • • y (cid:48) C in • • S V V V V † C out FIG. 4. (a) Feynman’s original quantum full adder [13]. (b) Optimized full adder with 2-qubit gate depth 4 [4]. The two parallel2-qubit operations are outlined in dashed boxes. 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Rev. A , 012316(2014).[39] K. R. Brown, A. W. Harrow, and I. L. Chuang, Phys.Rev. A , 052318 (2004). Acknowledgements We thank S. Wang, Z. Gong,S. Debnath, P. H. Leung, Y. Wu, and L. Duan forhelpful discussions. Circuits were drawn using the qcir-cuit.tex package. This work was supported by the AROwith funds from the IARPA LogiQ program, the AFOSRMURI program, and the NSF Physics Frontier Center atJQI.This material was partially based on work supportedby the National Science Foundation during D.M.’s as-signment at the Foundation. Any opinion, finding, andconclusions or recommendations expressed in this mate-rial are those of the authors and do not necessarily reflectthe views of the National Science Foundation. Author contributions C.F., A.O., N.M.L., K.A.L.,D.M., and C.M. designed the research; C.F., N.M.L.,K.A.L., D.Z., and C.M. collected and analyzed data; C.F.performed the theory derivations; A.O. performed thepulse sequence optimizations; and C.F., A.O., N.M.L.,K.A.L., D.Z., D.M., and C.M. all contributed to themanuscript. Corresponding author Correspondence should beaddressed to C.F. (email: cfi[email protected]). Declaration of competing financial interests C.M. is co-founder and Chief Scientist at IonQ, Inc. METHODS CONSTRAINT PROBLEM FOR OPTIMALPARALLEL OPERATIONS Here, we discuss in detail the constraint problem con-structed when performing operations in parallel betweentwo ion pairs ( M = 2), as is the case for the data shownin this paper. In order to perform parallel entangling op-erations involving 2 pairs of qubits ( i, j ) and ( m, n ) in achain of N ions with N motional modes ω k , the evolutionoperator [21–23] here can be written as: U || ( τ ) = exp (cid:88) i =0 φ i ( τ ) σ xi + i (cid:88) i 00 0 (cid:19) (cid:18) Ω ij Ω mn (cid:19) = χ ideal ij (cid:0) Ω Tij Ω Tmn (cid:1) (cid:18) D mn (cid:19) (cid:18) Ω ij Ω mn (cid:19) = χ ideal mn (cid:0) Ω Tij Ω Tmn (cid:1) (cid:18) D cross (cid:19) (cid:18) Ω ij Ω mn (cid:19) = 0 , (14)where (cid:8) C i , C j , C m , C n (cid:9) are the S × N spin-motioninteraction matricies for each segment on each ion, (cid:8) D ij , D mn (cid:9) are the two S × S spin-spin interaction ma-tricies for each segment on the entangling ion pairs,and D cross = (cid:8) D im , D in , D jm , D jn (cid:9) are the four S × S spin-spin interaction matricies for each segment on thecrosstalk ion pairs. Here we see that while the C -constraints are linear, the 6 D -constraints on the spin-spin interaction terms are not. OPTIMIZATION METHODS The constraint problem on α and χ was converted toan unconstrained optimization problem using the penaltymethod, specifically by minimizing an objective functionthat is quadratic in the deviations of α and χ from theirideal values. This objective function also penalized highpower pulse sequences. The objective function was min-imized using the built-in MATLAB function fminunc.The initial guess used for the pulse shapes in the op-timization protocol was that one pair would have an all-positive-amplitude shape, while the other pair would seepositive amplitudes for the first half, and negative am-plitudes for the second half of the gate. FIDELITY OF PARALLEL xx OPERATIONS Here, we calculate the fidelity of simultaneous xx gateoperations as a function of the above control parameters.The fidelity is given by F || = (cid:104) ψ init | U † ideal ρ r U ideal | ψ init (cid:105) , (15)where ρ r is the density matrix for the experimental op-eration traced over the motion, ρ r = T r m (cid:104) U expt | ψ init (cid:105) (cid:104) ψ init | U † expt (cid:105) . (16)Plugging in all values and solving, we derive the fidelity, F || (cid:0) α { i,j,m,n } ,k , χ ij , χ ideal ij , χ mn , χ ideal mn , χ im , χ in , χ jm , χ jn (cid:1) =1128 (8 + Γ + −−− + Γ + −− + + Γ + − + − + Γ + − ++ + Γ ++ −− + Γ ++ − + + Γ +++ − + Γ ++++ + 2 (Γ −− + Γ +000 ) cos [2 (∆ χ ij − χ im − χ in )] + 2 (Γ − + Γ +000 ) cos [2 (∆ χ ij + χ im − χ in )]+ 2 (Γ − + + Γ +000 ) cos [2 (∆ χ ij − χ im + χ in )] + 2 (Γ + Γ +000 ) cos [2 (∆ χ ij + χ im + χ in )]+ 2 (Γ + Γ +0 −− ) cos [2 (∆ χ ij − χ jm − χ jn )] + 2 (Γ + Γ +0+ − ) cos [2 (∆ χ ij + χ jm − χ jn )]+ 2 (Γ + Γ +0 − + ) cos [2 (∆ χ ij − χ jm + χ jn )] + 2 (Γ + Γ +0++ ) cos [2 (∆ χ ij + χ jm + χ jn )]+ 2 (Γ + Γ + − ) cos [2 ( χ im − χ jm + ∆ χ mn )] + 2 (Γ + Γ ++0+ ) cos [2 ( χ im + χ jm + ∆ χ mn )]+ 2 (Γ + Γ + − − ) cos [2 ( χ im − χ jm − ∆ χ mn )] + 2 (Γ + Γ ++0 − ) cos [2 ( χ im + χ jm − ∆ χ mn )]+ 2 (Γ + Γ + − +0 ) cos [2 ( χ in − χ jn + ∆ χ mn )] + 2 (Γ + Γ +++0 ) cos [2 ( χ in + χ jn + ∆ χ mn )]+ 2 (Γ + Γ + −− ) cos [2 ( χ in − χ jn − ∆ χ mn )] + 2 (Γ + Γ ++ − ) cos [2 ( χ in + χ jn − ∆ χ mn )]+ 2 (Γ + Γ + − ) cos [2 ( χ im + χ in − χ jm − χ jn )]+ 2 (Γ − + Γ ++00 ) cos [2 ( χ im − χ in + χ jm − χ jn )]+ 2 (Γ − + Γ + − ) cos [2 ( χ im − χ in − χ jm + χ jn )]+ 2 (Γ + Γ ++00 ) cos [2 ( χ im + χ in + χ jm + χ jn )]+ 2 (Γ − + Γ +0 − ) cos [2 (∆ χ ij − χ in − χ jm + ∆ χ mn )]+ 2 (Γ + Γ +0+0 ) cos [2 (∆ χ ij + χ in + χ jm + ∆ χ mn )]+ 2 (Γ + Γ +0 − ) cos [2 (∆ χ ij + χ in − χ jm − ∆ χ mn )]+ 2 (Γ − + Γ +0+0 ) cos [2 (∆ χ ij − χ in + χ jm − ∆ χ mn )]+ 2 (Γ − + Γ +00 − ) cos [2 (∆ χ ij − χ im − χ jn + ∆ χ mn )]+ 2 (Γ + Γ +00+ ) cos [2 (∆ χ ij + χ im + χ jn + ∆ χ mn )]+ 2 (Γ + Γ +00 − ) cos [2 (∆ χ ij + χ im − χ jn − ∆ χ mn )]+ 2 (Γ − + Γ +00+ ) cos [2 (∆ χ ij − χ im + χ jn − ∆ χ mn )]) . (17)Here, Γ A i A j A m A n = exp (cid:32) − (cid:88) k β k | A i α i,k + A j α j,k + A m α m,k + A n α n,k ) | (cid:33) , (18)where the parameters { A i , A j , A m , A n } can be { , ± } ,and we indicate them as { → , +1 → + , − → −} inEquations 17 and 23 for display purposes. The inversemode temperature β k is β k = coth (cid:18) (cid:126) ω k k B T (cid:19) = coth (cid:20) 12 ln (cid:18) n k (cid:19)(cid:21) , (19)where ¯ n k is the average phonon number in the k th mode.We additionally use∆ χ ij = χ ij − χ ideal ij ∆ χ mn = χ mn − χ ideal mn . (20) Plugging in the ideal-case parameters, where α { i,j,m,n } ,k ( τ ) = 0, χ im = χ in = χ jm = χ jn = 0, χ ij = χ ideal ij , and χ mn = χ ideal mn , we indeed get F || = 1.See [26] for a more detailed derivation. TOWARD A SINGLE-OPERATION GHZ STATE This control scheme for parallel 2-qubit entanglinggates in ions also suggests a method for performing multi-qubit entanglement in a single operation. Of particularinterest is the creation of GHZ states [32], which are aclass of non-biseparable maximally-entangled multi-qubitstates. Setting all 6 spin-spin interaction terms to χ = π in Equation 1 yields the unitary U idealGHZ = U (cid:16) α { i,j,m,n } ,k = 0 , χ ij = χ im = χ in = χ jm = χ jn = χ mn = π (cid:17) = 1 √ (cid:16) I − i ( σ x ) ⊗ (cid:17) = 1 √ − i − i 00 0 1 0 0 0 0 0 0 0 0 0 0 − i − i − i − i − i − i − i − i − i − i − i − i − i − i , (21)and adding one Z rotation R z ( θ ) = (cid:32) e − i θ e i θ (cid:33) with θ = π produces a 4-qubit GHZ state: R z (cid:16) π (cid:17) · U idealGHZ | (cid:105) = 1 √ | (cid:105) + | (cid:105) ) . (22)Following a similar derivation as with parallel gates, we therefore calculate the objective fidelity function to be F GHZ (cid:0) α { i,j,m,n } ,k , χ ideal ij , χ ideal im , χ ideal in , χ ideal jm , χ ideal jn , χ ideal mn , χ ij , χ im , χ in , χ jm , χ jn , χ mn (cid:1) =1128 (8 + Γ + −−− + Γ + −− + + Γ + − + − + Γ + − ++ + Γ ++ −− + Γ ++ − + + Γ +++ − + Γ ++++ + 2 (Γ + Γ +000 ) cos [2 (∆ χ ij + ∆ χ im + ∆ χ in )] + 2 (Γ − + + Γ +000 ) cos [2 (∆ χ ij − ∆ χ im + ∆ χ in )]+ 2 (Γ − + Γ +000 ) cos [2 (∆ χ ij + ∆ χ im − ∆ χ in )] + 2 (Γ −− + Γ +000 ) cos [2 (∆ χ ij − ∆ χ im − ∆ χ in )]+ 2 (Γ + Γ +0++ ) cos [2 (∆ χ ij + ∆ χ jm + ∆ χ jn )] + 2 (Γ + Γ +0 − + ) cos [2 (∆ χ ij − ∆ χ jm + ∆ χ jn )]+ 2 (Γ + Γ +0+ − ) cos [2 (∆ χ ij + ∆ χ jm − ∆ χ jn )] + 2 (Γ + Γ +0 −− ) cos [2 (∆ χ ij − ∆ χ jm − ∆ χ jn )]+ 2 (Γ + Γ ++0+ ) cos [2 (∆ χ im + ∆ χ jm + ∆ χ mn )] + 2 (Γ + Γ + − ) cos [2 (∆ χ im − ∆ χ jm + ∆ χ mn )]+ 2 (Γ + Γ ++0 − ) cos [2 (∆ χ im + ∆ χ jm − ∆ χ mn )] + 2 (Γ + Γ + − − ) cos [2 (∆ χ im − ∆ χ jm − ∆ χ mn )]+ 2 (Γ + Γ +++0 ) cos [2 (∆ χ in + ∆ χ jn + ∆ χ mn )] + 2 (Γ + Γ + − +0 ) cos [2 (∆ χ in − ∆ χ jn + ∆ χ mn )]+ 2 (Γ + Γ ++ − ) cos [2 (∆ χ in + ∆ χ jn − ∆ χ mn )] + 2 (Γ + 2Γ + −− ) cos [2 (∆ χ in − ∆ χ jn − ∆ χ mn )]+ 2 (Γ + Γ ++00 ) cos [2 (∆ χ im + ∆ χ in + ∆ χ jm + ∆ χ jn )]+ 2 (Γ − + Γ + − ) cos [2 (∆ χ im − ∆ χ in − ∆ χ jm + ∆ χ jn )]+ 2 (Γ − + Γ ++00 ) cos [2 (∆ χ im − ∆ χ in + ∆ χ jm − ∆ χ jn )]+ 2 (Γ + Γ + − ) cos [2 (∆ χ im + ∆ χ in − ∆ χ jm − ∆ χ jn )]+ 2 (Γ + Γ +0+0 ) cos [2 (∆ χ ij + ∆ χ in + ∆ χ jm + ∆ χ mn )]+ 2 (Γ − + Γ +0 − ) cos [2 (∆ χ ij − ∆ χ in − ∆ χ jm + ∆ χ mn )]+ 2 (Γ − + Γ +0+0 ) cos [2 (∆ χ ij − ∆ χ in + ∆ χ jm − ∆ χ mn )]+ 2 (Γ + Γ +0 − ) cos [2 (∆ χ ij + ∆ χ in − ∆ χ jm − ∆ χ mn )]+ 2 (Γ + Γ +00+ ) cos [2 (∆ χ ij + ∆ χ im + ∆ χ jn + ∆ χ mn )]+ 2 (Γ − + Γ +00 − ) cos [2 (∆ χ ij − ∆ χ im − ∆ χ jn + ∆ χ mn )]0+ 2 (Γ − + Γ +00+ ) cos [2 (∆ χ ij − ∆ χ im + ∆ χ jn − ∆ χ mn )]+ 2 (Γ + Γ +00 − ) cos [2 (∆ χ ij + ∆ χ im − ∆ χ jn − ∆ χ mn )]) , (23)where ∆ χ ij = χ ij − χ ideal ij ∆ χ mn = χ mn − χ ideal mn ∆ χ im = χ im − χ ideal im ∆ χ in = χ in − χ ideal in ∆ χ jm = χ jm − χ ideal jm ∆ χ jn = χ jn − χ ideal jn . (24)This indicates we may be able to use the same op-timization approach to produce pulse shapes that willcreate GHZ states when applied to the ions. Unlike withparallel gates, however, it may be necessary to allow in-dependent pulse shapes on all 4 ions, rather than solvingfor pairwise solutions; this will provide more free param-eters. Additional challenges will include finding effectivecalibration techniques when implementing such gates onthe experiment, since there will be 6 interactions that willall need to be at the same strength, but only 4 controlsignals. Our current approach of calibrating a 2-qubitgate by adjusting the overall power for the pulse shapeapplied by the control signal may no longer work; newtechniques with more degrees of freedom may be needed,such as independently adjusting the power for a few dif-ferent sections of the pulse shape on each ion.The benefits of implementing GHZ states with fewergates would be significant, as it would substantially re-duce the circuit depth of several important algorithms.While the use of axial modes for multi-qubit GHZ stateshas already been shown [33], this scheme represents anew method for use with radial mode interactions. Withonly 2-qubit gates available, building a GHZ state ofsize N requires O ( N ) 2-qubit gates. With parallel 2-qubit gates available, the gate depth required to builda GHZ state is reduced to O (log( N )); this is accom-plished with a binary tree algorithm by dividing all qubitsinto pairs and entangling those pairs in parallel, thenentangling pairs of these pairs, and so on until all areentangled. A single-operation GHZ state would dropthis circuit depth to unity. Single-operation GHZ stateconstruction will greatly enhance the efficiency of sev-eral algorithms; for example, arbitrary stabilizer circuitsrequire O ( N log( N ) ) cnot gates [34], but could be imple-mented in O ( N ) gates with single-operation GHZ statecircuitry [35]. Single-operation GHZ state creation willalso benefit applications such as quantum secret sharing[36], Toffoli- N gates, the quantum Fourier transform, andquantum Fourier adder circuits [35]. EXPERIMENTAL SETUP The experiments are performed on a linear chain of fivetrapped Yb + ions that are laser cooled to near theirground state. We designate the qubit as the | (cid:105) ≡ | F =0; m F = 0 (cid:105) and | (cid:105) ≡ | F = 1; m F = 0 (cid:105) hyperfine-splitelectronic states of the ion’s S / manifold [37], whichare first-order magnetic-field-insensitive clock states witha splitting of 12.642821 GHz. Coherent operations areperformed by counterpropagating Raman beams from asingle 355 nm mode-locked laser. The first Raman beamis a global beam applied to the entire chain, while thesecond is split into individual addressing beams to targeteach ion qubit [15]. Additionally, a multi-channel arbi-trary waveform generator (AWG) provides separate RFcontrol signals to each ion’s individual addressing beam,providing the individual phase, frequency, and amplitudecontrols necessary to execute independent 2-qubit oper-ations in parallel. Qubits are initialized to the | (cid:105) stateusing optical pumping, and read out by separate channelsof a multi-channel photomultiplier tube (PMT) array us-ing state-dependent fluorescence. CALCULATING FIDELITIES OF 2-QUBITENTANGLING GATES The fidelity of a 2-qubit xx ( χ ) entangling gate can bemeasured by scanning the phase φ of a global π rotationapplied after performing the xx gate and calculating theparity at each point of the scan [27–29]. Here, a rotationgate is defined as R ( θ, φ ) = (cid:18) cos θ − ie − iφ sin θ − ie iφ sin θ cos θ (cid:19) .The analysis pulses used experimentally are rotations us-ing the SK1 composite pulse for increased robustnessagainst errors in the rotation angle [38, 39]. For the fourions involved in each operation, the parity analysis wasperformed for all 6 possible pairs within the set, allowingfor analysis of the 2 entangled ion pairs as well as the 4crosstalk pairs.We start with a global rotation on 2 qubits, R G (cid:16) π , φ (cid:17) = R (cid:16) π , φ (cid:17) ⊗ R (cid:16) π , φ (cid:17) = 12 − ie − iφ − ie − iφ − e − iφ − ie iφ − − ie − iφ − ie iφ − − ie − iφ − e iφ − ie iφ − ie iφ , (25)and a general 2-qubit density matrix ρ g that representsthe density matrix produced after experimentally per-1forming an xx gate, ρ g = ρ ρ ρ ρ ρ ∗ ρ ρ ρ ρ ∗ ρ ∗ ρ ρ ρ ∗ ρ ∗ ρ ∗ ρ , (26)where ρ = (cid:104) | xx | (cid:105) , ρ = (cid:104) | xx | (cid:105) , . . . , ρ = (cid:104) | xx | (cid:105) , ρ = (cid:104) | xx | (cid:105) . After performing theanalysis pulse r G (cid:0) π , φ (cid:1) , the new density matrix ρ a = r G (cid:0) π , φ (cid:1) · ρ g · R † G (cid:0) π , φ (cid:1) is used to calculate the parity:Π ( ρ a , φ ) = (cid:0) ρ a + ρ a (cid:1) − (cid:0) ρ a + ρ a (cid:1) = 2 A cos φ − A cos(2 φ − φ )= 2 A cos φ − A Π cos(2 φ − φ ) , (27)where parity is defined as the sum of the even parity pop-ulations minus the sum of the odd parity populations andthe coherences (off-diagonal density matrix elements)from ρ g are re-written in the form ρ xy = A xy e − iφ xy . Letus also define the parity amplitude A Π ≡ A .Now we calculate the fidelity of an xx ( χ ) gate. Usingthe xx ( χ ) gate unitary, xx ( χ ) = cos( χ ) 0 0 − i sin( χ )0 cos( χ ) − i sin( χ ) 00 − i sin( χ ) cos( χ ) 0 − i sin( χ ) 0 0 cos( χ ) , (28)we construct the ideal density matrix after an xx ( χ ) gate, ρ ideal = xx ( χ ) · | (cid:105) (cid:104) | · xx ( χ ) † = cos ( χ ) 0 0 i cos( χ ) sin( χ )0 0 0 00 0 0 0 − i cos( χ ) sin( χ ) 0 0 sin ( χ ) . (29)The fidelity of the general fidelity matrix ρ g with respectto the ideal fidelity matrix ρ ideal is given by F ( χ ) = T r (cid:104) ρ ideal ( χ ) · ρ g · ρ † ideal ( χ ) (cid:105) . (30)Plugging in equations 26 and 29, using A Π ≡ A , andsimplifying yields F ( χ ) = ρ cos ( χ )+ ρ sin ( χ )+ A Π cos( χ ) sin( χ ) (31)as the fidelity of an xx ( χ ) gate. Specifically for maxi-mally entangling gates, we use χ = π and get F (cid:16) χ = π (cid:17) = 12 ( ρ + ρ ) + 12 A Π . (32)While ρ and ρ are simply the populations in | (cid:105) and | (cid:105) respectively after an xx gate, we still need the A Π term. We can extract this from a parity scan usingEquation 27. Given a perfect xx ( χ ) gate where A = 0, φ = − π , and A Π = 2 A = 2 cos( χ ) sin( χ ) (from Equa-tion 29), scanning the analysis phase φ from 0 to 2 π andmeasuring the parity at each point will yield a sine curveof amplitude 2 cos( χ ) sin( χ ) with 2 periods in the rangefrom 0 to 2 π . (For a fully entangling xx ( χ = π ) gate, thesine curve should have amplitude 1.) Consequently, byfitting a sine curve to this measured parity curve, we canestimate the parity amplitude A Π and use it in Equation31 to calculate the gate fidelity. ADDITIONAL PARITY CURVES AND FIDELITYDATA FOR 2-QUBIT ENTANGLING GATES Additional parity curves and corresponding gate fideli-ties are shown in Figure 5, with typical fidelities of 96-99%. An exception is the { (1,2), (4,5) } gate, for whichthe (4,5) gate has a fidelity of 91% (Figure 5(d)); how-ever, its phase space closure diagram in [26] shows thatthis low fidelity is because the pulse solution found is notideal. FIDELITY OF PARALLEL 2-QUBITENTANGLING GATES WITH DIFFERENTDEGREES OF ENTANGLEMENT Since the xx gates in this parallelization scheme haveindependent calibration (see next section of Methods),the χ parameters of the two xx gates are independent.The continuously-variable parameter χ is directly relatedto the amount of entanglement generated between thetwo qubits, given by xx ( χ ) | (cid:105) = 1 √ χ ) | (cid:105) − i sin ( χ ) | (cid:105) ) , (33)and can be adjusted on the experiment by scaling thepower of the overall gate. Consequently, we can simulta-neously implement two xx gates with different degrees ofentanglement, which may prove useful for some applica-tions. For example, the full adder implementation in themain text requires simultaneously performing an xx (cid:0) π (cid:1) gate on one pair of qubits, and an xx (cid:0) π (cid:1) gate on anotherpair of qubits. To demonstrate this capability, Figure 6shows parity scan data for a simultaneous xx (cid:0) π (cid:1) gateon ions (1,5) and an xx (cid:0) π (cid:1) gate on ions (2,4). The datais analyzed as in Figures 2 and 5, but while we use Equa-tion 32 (setting χ = π ) to calculate the fidelity for the(1,5) gate, we use Equation 31 and set χ = π for the (2,4)gate. The respective gate fidelities are therefore 96.4(3)%and 99.4(3)%, with an average crosstalk error of 2.2(3)%.2 (a) (b) Rotation Axis -1-0.8-0.6-0.4-0.200.20.40.60.81 P a r i t y (1,2)(3,4)(1,3)(1,4)(2,3)(2,4) 0 /4 /2 3 /4 5 /4 3 /2 7 /4 2 Rotation Axis -1-0.8-0.6-0.4-0.200.20.40.60.81 P a r i t y (1,5)(2,4)(1,2)(1,4)(2,5)(4,5) (c) (d) Rotation Axis -1-0.8-0.6-0.4-0.200.20.40.60.81 P a r i t y (1,3)(2,5)(1,2)(1,5)(2,3)(3,5) 0 /4 /2 3 /4 5 /4 3 /2 7 /4 2 Rotation Axis -1-0.8-0.6-0.4-0.200.20.40.60.81 P a r i t y (1,2)(4,5)(1,4)(1,5)(2,4)(2,5) FIG. 5. Parity curves and fidelities for parallel xx gates on several sets of ions. Circles indicate data, with matching-colorlines indicating calculated fit. (a) Ions (1,2) and (3,4), yielding fidelities of 98.4(3)% and 97.7(3)% on the respective entangledpairs, with an average crosstalk error of 0.6(3)%. (b) Ions (1,5) and (2,4), yielding fidelities of 96.8(3)% and 98.1(2)% on therespective entangled pairs, with an average crosstalk error of 1.7(3)%. (c) Ions (1,3) and (2,5), yielding fidelities of 98.3(3)%and 97.5(2)% on the respective entangled pairs, with an average crosstalk error of 0.8(4)%. (d) Ions (1,2) and (4,5), yieldingfidelities of 97.2(3)% and 91.9(3)% on the respective entangled pairs, with an average crosstalk error of 0.9(3)%. INDEPENDENCE OF PARALLEL GATECALIBRATION Parallel gates can be calibrated independently fromone another by adjusting a scaling factor that controlsthe overall power on the gate without modifying the pulseshape. Furthermore, adjusting a scaling factor that con-trols the power on a single ion only affects the gate inwhich it participates by modifying the total amount ofentanglement, without any apparent ill effects on the gatequality. This was confirmed experimentally using paral-lel operations on ions (1,2) and (3,4) by scanning over thescaling factors associated with ions 1 and 2. Figures 7(a-b) show two such scans over the scaling factors for ions 1 and 2 while keeping the (3,4) gate “on”, with the scalingfactor for those two ions set near to a fully-entanglinggate. Figure 7(a) shows a scan of just the scaling factorfor ion 1 while holding the scale factor for ion 2 constant,and Figure 7(b) shows a scan over the scaling factor forions 1 and 2 together. Figures 7(c-d) show scans over thescaling factors for ions 1 and 2 while keeping the interac-tion on (3,4) “off”; the scaling factor for the (3,4) gate isset to 0, so the ions see no light and therefore perform nointeraction during the gate. Figure 7(c) scans the scalefactor just on ion 2 while holding the scale on ion 1 con-stant, and Figure 7(d) scans the overall scaling factor forions 1 and 2 together. For all of these scans, as the scal-ing factors are increased, the population in | (cid:105) for ions3 Rotation Axis -1-0.8-0.6-0.4-0.200.20.40.60.81 P a r i t y (1,5)(2,4)(1,2)(1,4)(2,5)(4,5) FIG. 6. Parity curve for parallel xx ( χ ) gates on ions (1,5) and(2,4), where an xx (cid:0) π (cid:1) gate is performed on ions (1,5), andan xx (cid:0) π (cid:1) gate is performed on ions (2,4). Circles indicatedata, with matching-color lines indicating calculated fit. Thisyields fidelities of 96.4(3)% and 99.4(3)% on the respectiveentangled pairs, with an average crosstalk error of 2.2(3)%.Parallel Gate Pairs R || , Pair 1 R || , Pair 2(1,4) and (2,5) 4.3 1.8(1,2) and (3,4) 7.9 5.0(1,5) and (2,4) 2.1 1.6(1,4) and (2,3) 4.3 3.8(1,3) and (2,5) 0.9 1.5(1,2) and (4,5) 2.2 2.2TABLE I. For each pair of parallel xx gates implemented, wecompare the optical power required to perform each compo-nent xx with its corresponding stand-alone 2-qubit xx gateby calculating the power ratio R || . | (cid:105) decreasescorrespondingly), while the | (cid:105) and | (cid:105) populations forthe (3,4) gate remain unchanged. OPTICAL POWER REQUIREMENTS While the gate time τ gate = 250 µ s for running 2 xx gates in parallel is comparable to that of a single xx gate(and consequently, half the time it would take to exe-cute two xx gates in series), the parallel gates schemerequires more optical power. The Rabi frequency Ω isproportional to the square root of the beam intensity I ,Ω ∝ √ I I , where I and I are the beam intensitiesfor the individual and global beams. We can thereforecalculate the ratio R || of the power for a gate executedin parallel to the power required for a single xx gate onthe same ions as R || = P || P xx = I || I xx = (cid:16) Ω || Ω xx (cid:17) . Intensityis power per unit area, and since the beam sizes do notnot vary, the areas cancel out. Measured power ratios for each experimentally implemented gate are shown in Ta-ble I. While some gates required substantially more power(for example, we had some trouble finding a solution for(1,2), (3,4) that was both high-quality and low power),most gates performed in parallel require about two tofour times as much power as their singly-performed coun-terparts. However, a full accounting of power require-ments on this experiment must also take into accountpower wasted by unused beams, and the total time re-quired to perform equivalent operations. Since the indi-vidual addressing system has all individual beams on atall times and are dumped after the AOM when not in use(see [15, 26]), any ion not illuminated corresponds to anindividual beam wasting power. Running 2 xx gates inparallel takes τ gate = 250 µ s and uses beams each withpower P to illuminate 4 ions, but performing those same2 gates in series using stand-alone xx gates requires time2 τ gate and uses 4 beams each with power P/ P/ OPTIMIZED ADDER CIRCUIT The optimized full adder circuit to be implemented onthe experiment, shown in Figure 8, is constructed fromFigure 4(b) by combining the cnot , C ( V ), and C ( V † )gates from Figure 5.12 of [26] and further optimizing therotations per the method described in Section 5.2.1 of[26]. The two parallel 2-qubit operations are outlined indashed boxes.4 (a) (b) Scale Factor for ion 1 P opu l a t i on (1,2) 00(3,4) 00(1,2) 01(3,4) 01(1,2) 10(3,4) 10(1,2) 11(3,4) 11 0 0.2 0.4 0.6 0.8 1 Scale Factor for ions (1,2) P opu l a t i on (1,2) 00(3,4) 00(1,2) 01(3,4) 01(1,2) 10(3,4) 10(1,2) 11(3,4) 11 (c) (d) Scale Factor for ion 2 P opu l a t i on (1,2) 00(3,4) 00(1,2) 01(3,4) 01(1,2) 10(3,4) 10(1,2) 11(3,4) 11 0.7 0.72 0.74 0.76 0.78 0.8 0.82 Scale Factor for ions (1,2) P opu l a t i on (1,2) 00(3,4) 00(1,2) 01(3,4) 01(1,2) 10(3,4) 10(1,2) 11(3,4) 11 FIG. 7. Parallel gates can be calibrated independently. (a) Scanning the scale factor on ion 1, with ions (3,4) performing anentangling gate. (b) Scanning the scale factor on ions (1,2), with ions (3,4) performing an entangling gate. (c) Scanning thescale factor on ion 2, with no light on ions (3,4). (d) Scanning the scale factor on ions (1,2), with no light on ions (3,4). x R z ( − π ) R y ( π ) XX ( π ) y R z ( − π ) R y ( π ) XX ( π ) R y ( − π ) R z ( − π ) C in R z ( π ) R y ( π ) XX ( π ) R z ( − π )0 R x ( − π ) XX ( π ) · · · XX ( π ) R y ( − π ) x · · · XX ( π ) R y ( π ) R z ( π ) y (cid:48) · · · R y ( π ) XX ( − π ) R y ( − π ) R z ( π ) S · · · XX ( π ) C out FIG. 8. Application-optimized full adder implementation using xx ( χ ), r x ( θ ), and r y ( θθ