Parallel Position-Controlled Composite Quantum Logic Gates with Trapped Ions
Michael S. Gutierrez, Guang Hao Low, Richard Rines, Helena Zhang
PParallel Position-Controlled Composite Quantum Logic Gates with Trapped Ions
Michael S. Gutierrez, ∗ Guang Hao Low, Richard Rines, and Helena Zhang
Center for Ultracold Atoms, Department of Physics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: September 18, 2018)We demonstrate parallel composite quantum logic gates with phases implemented locally throughnanoscale movement of ions within a global laser beam of fixed pulse duration. We show that a simplefour-pulse sequence suffices for constructing ideal arbitrary single-qubit rotations in the presenceof large intensity inhomogeneities across the ion trap due to laser beam-pointing or beam-focusing.Using such sequences, we perform parallel arbitrary rotations on ions in two trapping zones separatedby 700 µ m with fidelities comparable to those of our standard laser-controlled gates. Our schemeimproves on current transport or zone-dependent quantum gates to include phase modulation withlocal control of the ion’s confinement potential. This enables a scalable implementation of an arbitrarynumber of parallel operations on densely packed qubits with a single laser modulator and beam path. Quantum processors with hundreds or more qubitspromise to deliver significant computational speedupsover the best classical systems [1]. The limits of physicalcoherence make large-scale parallelization of primitivequantum operations crucial for realizing a fault-tolerantdevice [2]. Trapped ions are one of the leading candidatesfor physical qubits due to their consistency—all ions areidentical—and large ratio of coherence [3] to gate [4]times. Few-qubit ion systems can now demonstrate simplequantum algorithms [5, 6] as well as single- and multi-qubit operations within the fault-tolerant regime [7–10].However, current ion trap systems rely upon bulky free-space optical components and high-power radio-frequencylaser modulators, both of which pose daunting technicaldifficulties [11, 12] to scaling to hundred- or thousand-qubit parallel systems [13]. A major challenge movingforward is managing and optimizing physical resources re-quired to implement high-performing quantum operationsat scale. To this end, many proposed architectures iden-tify key resources that offer clear and ready paths towardscaling up to an arbitrary number of qubits [14–16].One promising resource is fine voltage control of trapelectrodes, which can be harnessed to displace the confin-ing potential of single ions. Local targeted qubit opera-tions have been performed using potential displacementin conjunction with static laser interaction zones [15–18]or magnetic field gradients [19, 20]. These schemes cangreatly reduce complexity of optical addressing systems,and replace the numerous high-power laser modulatorswith low-power voltage generators which can be readilyintegrated on-chip with existing technology [21]. How-ever, quantum control techniques proposed thus far havefocused on using local voltage changes to gate the interac-tion time by transporting ions to or through designatedoperation zones. This approach requires each ion to betransported over large distances ( > µ m), greatly lim-iting the speed and density of parallel operations.In this Letter, we propose an alternative approach usingnanoscale ion movements parallel to the laser beam toimplement local phase changes of the global beam. This scheme offers several significant advantages over previ-ous works. First, the number of parallel ion movementsper beam pass is limited only by the number of inde-pendently controlled electrodes, which is highly scalable.Second, movement operations are local and space efficient:ions remain within a single trapping zone and only un-dergo sub-micron displacements. Third, we demonstrateposition-controlled composite sequences that enable ar-bitrary and ideal single-qubit operations on each ion inparallel, despite the large inhomogeneities that can arisein a global beam.We implement our scheme experimentally using easilyscalable trapping and control technologies. We first ap-ply local, position-controlled phases to a single ion usinga fixed-frequency, fixed-phase laser beam to perform aRamsey experiment. To enable parallel operations de-spite zone-dependent global beam intensity, we constructa simple and efficient four-pulse composite sequence togenerate ideal arbitrary single-qubit rotations. We provethe scalability of such an approach by performing par-allel single-qubit operations on two ions in independenttrapping zones separated by 700 µ m with no additionaloptics, modulators, or timing overhead. We find the fideli-ties of these parallel gates to be comparable to standard,optically modulated gates. We describe the limitationsfor these nanoscale movements in our current setup andfind them comparable in timing and phase precision totypical direct digital synthesizers (DDSs) used for lasermodulation.Our qubit of choice is stored in the electronic states | (cid:105) = 5 S / [ m = − /
2] and | (cid:105) = 4 D / [ m = − /
2] ofthe Strontium ion. We confine single ions 68 µ m abovea Sandia Laboratories High Optical Access trap [22]. A4 Gˆ y static uniform magnetic field is used to spectrallyisolate these Zeeman states, where the ˆ y -axis is normalto the trap surface. Qubit states are manipulated using anarrow linewidth λ = 674 nm laser passing along the trapaxis of symmetry (ˆ z -axis) with a waist of w = 25 µ m.The ˆ z -secular frequency is 2 π × .
25 MHz. The confiningpotential is generated via 188 control surfaces connected a r X i v : . [ qu a n t - ph ] F e b to a 48-channel home-built arbitrary waveform generator(AWG) with ±
10 V output range, 20-bit precision, and 250kHz asynchronous update rate, enabling sub-nanometerposition control. Voltages from the AWG are low-passfiltered at 600 kHz en route to the trap electrodes. Twotrapping zones separated by 700 µ m, Zone 1 (Z1) andZone 2 (Z2), are used throughout this work and depictedin Fig. 1a along with beam geometries.Each experimental sequence is preceded by 300 µ sof zone-addressed Doppler cooling, 100 µ s of globalfrequency-selective optical pumping and 500 µ s of side-band cooling [23], preparing the ion in the | (cid:105) state with ∼ .
5% fidelity. After the sequence, zone-addressed statedetection is performed by recording state-dependent fluo-rescence with a single-photon photomultiplier tube [24],elapsing 200 µ s and with readout fidelity of ∼ . ∼
1% level, yieldingzone-dependent Rabi frequencies of Ω Z = 2 π ×
166 kHzand Ω Z = 2 π ×
159 kHz.Local phase control of qubit operations is achieved bymoving each ion along the ˆ z -axis between identical globallaser pulses of length t p = 1 . µ s, as depicted in Fig. 1b.We demonstrate phase control by performing a Ramseyexperiment on a single ion, varying the displacement ofthe confining potential, with results shown in Fig. 1c. Thedisplacement is generated with a single voltage set update,with an 8 µ s delay between π/ (∆ φ )behavior calibrates voltages to the ion’s displacement ∆ z via the observed phase shift φ = 2 π ∆ z/λ , which is within ∼
1% of boundary element method simulations [22].The Ramsey sequence suffices for accessing the full 4 π rotation space in a single zone. However, fully paralleloperations with O (10 − ) infidelity would require intensitydifferences of no more than ∼
1% across the trap. Whileit is possible to reduce alignment error sufficiently, thiswould still decrease the usable interaction region of aGaussian beam to one-hundredth of a Rayleigh range.For our beam geometry, this would reduce the usableinteraction region to just 30 µ m, less than 1% of the4 mm trapping region of our device.To extend this range, we construct a composite quan-tum gate of fixed pulse durations and variable phasesfollowing the methodology of Low et. al. [25]. We requirethat all possible ideal single-qubit rotations are achiev-able by the composite gate within a continuous rangeof Rabi frequencies such that we are completely insensi-tive to beam intensity inhomogeneities across the trap.The shortest composite gate satisfying this constraintis a sequence of length four. The total duration of thelaser pulses is 2 π , so the sequence does not add exces-sive overhead. Details for extracting phases for each ofthe four pulses and generating a specific target rotation θ Target given a base rotation θ z = (cid:82) t p Ω z dt are outlinedin the Supplemental Material. We define the region-of- D o p . D e t . D o p . D e t . Qubit Op. ˆ x ˆ y ˆ z µm Z1 Z2 ˆ z Time φ t move φ t p t p Dopplercooling
Stateprep.
Detection − π − π/ − π − π/ π/ π π/ π ∆ φ P ( | i ) − nm − nm nm nm a)b)c) FIG. 1: (a) Image of the surface trap indicatinggeometries of Doppler, detection, and qubit operationbeams. The two trapping zones of interest, Z1 and Z2,are labeled with ion fluorescence shown. (b) The Ramseytiming sequence, with pulse duration t p = 1.5 µ s andmovement duration t move = 8 µ s for a single ion (blackcircle) in one zone. Blue arrows represent zone-addressedDoppler cooling and state-detection. Red arrows indicateglobal qubit operations (solid) and global statepreparation (shaded). (c) Experimental Ramseysequence data, showing the | (cid:105) population vs. distancemoved from start position expressed in phase, with 500repeats per data point. The fit gives a 99 . ± . π -rotationspace is achievable for base rotations varying between θ z ∈ { π/ , . π } or equivalently, intensity variationsbetween { I min , . I min } . This extends the usable inter-action region to two Rayleigh ranges, two hundred timeslarger than that of the basic Ramsey sequence. With ourbeam geometry, this now covers the entire length of thetrapping region.To empirically demonstrate the region-of-validity ofthe four-pulse composite gate, we implement scans ofthe target rotation from 0 to 4 π using two different baserotations: one along the minimum boundary, θ z = π/ θ z = 0 . π , ˆ z Time φ t move φ t move φ t move φ t p t p t p t p Dopplercooling
Stateprep
Detection π/ . π π θ z π π π π θ T a r g e t . π π/ π π/ π π/ π π/ π θ Target . . . . . . P ( | i ) ideal θ base = 0 . πθ base = 0 . π a)b)c) FIG. 2: (a) Composite four-pulse sequence andmovement timing, with each experiment pulse lasting t p = 1 . µ s and each movement period lasting t move = 8 µ s for a total gate time of 30 µ s. (b)Composite sequence solutions space: shaded red regionindicates a four-pulse solution exists for the desiredtarget rotation ( θ Target ) and base rotation ( θ z ), andcross-hatched region is where the full 4 π target rotationis achievable. Angle θ z = 0 . π , corresponding toapproximately twice the intensity at θ z = π/
2, isindicated. (c) Experimental data for the | (cid:105) -statepopulation vs target rotation for two base rotations θ z = π/ θ z = 0 . π (diamonds), withcontrast given by 99 . ± .
5% and 98 . ± . . ± .
5% and 98 . ± . µ m. In order tominimize cross-talk, the confinement potential is solvedto satisfy simultaneous constraints for both zones whilespecifying position, ˆ z -secular frequency, tilt in xy -plane,and stray-field compensation using a fast multipole expan-sion for each collection of AWG-controlled electrodes [26],accounting for fields from both neighboring electrodeswhich share the same AWG and electrodes from the othertrapping zone. We then scan the target rotation in onezone while implementing one of three target rotations in ˆ z Time φ Z , t move φ Z , t move φ Z , t move φ Z , φ Z , t move φ Z , t move φ Z , t move φ Z , t p t p t p t p Dopplercooling
Stateprep
Detection π/ π π/ π π/ π π/ θ Z ,Target . . . . . . . P ( | i ) Zone 1 π/ π π/ π π/ π π/ π θ Z ,Target Zone 2 − .
08 0 .
00 0 . Z2 − . . . Z π/ π π/ π π/ π π/ θ Z ,Target . . . . . . . P ( | i ) π/ π π/ π π/ π π/ π θ Z ,Target − .
08 0 .
00 0 . Z1 − . . . Z a)b)c) FIG. 3: (a) Two-zone parallel composite sequence pulseand movement timing. Doppler cooling, state-prep,experimental sequence and state detection are allperformed simultaneously, with t p = 1 . µ s and t move = 8 µ s. (b) Experimental data for Z1 and Z2 | (cid:105) -population for a Z2 target rotation scan and aconstant Z1 target rotation: Identity (diamonds), π/ π (squares) (c) Experimental data for Z1and Z2 | (cid:105) -population for a Z1 target rotation scan and aconstant Z2 target rotation: Identity (diamonds), π/ π (squares). (b-c) insets show Z1 and Z2residuals for the π/ π/ π rotation, with theresults shown in Fig. 3. For each case, the scanned targetrotation shows a contrast of 99 . ± .
4% and 99 . ± . (cid:13)(cid:13)(cid:13)(cid:13) Cov( Z ,Z √ Var( Z Z (cid:13)(cid:13)(cid:13)(cid:13) < .
08. To further characterize thefidelity of each gate and the cross talk would require ran-domized benchmarking [27] or gate set tomography [28],which is beyond the scope of this work.The fundamental limits to the performance of thesegates will be set by the phase resolution achievable forthe specific trap geometry and AWG used. To estimatethe phase resolution achievable in our current setup, wesimulate the electric fields generated by the ion’s nearest-neighbor electrodes to be 0.25 V/mm. Combined withour AWG resolution and ˆ z -secular frequency, this resultsin roughly 12 bits of resolution over one λ of movement orapproximately 0 . π phase shift. Furtherresolution could be achieved through increased ˆ z -secularfrequency, optimized trap designs, or increased AWGresolution. However, we note that the resolution achievedhere is already comparable to most DDSs used for lasermodulation, which have 10 to 16 bits of phase precision.Limitations of a more technical nature will occur dueto the motional excitation induced from the cascade ofion movements. In this work, the relatively slow 8 µ smovement time keeps the motion in the adiabatic regime.However, several groups have already demonstrated iontransport that is both fast and induces minimal motionalexcitation [29–31]. Incorporating these works and increas-ing AWG update rates would reduce the movement timeto sub-microsecond levels, comparable to typical DDSprogramming times.Beyond technical improvements, the composite gateitself can be extended by looking at longer sequencelengths, adding constraint equations for more exotic quan-tum response functions, or extending to non-equiangularsequences. Longer sequences will offer an increased range-of-validity, allowing more tightly focused beams or longertraps. Incorporating additional constraints can allow forerror-resilient broadband rotations and compensate forarchitecture-specific errors, such as detuning and imper-fect movement. Non-equiangular composite sequences cancompensate for intensity gradients over the λ -movementrange, making this work applicable to tightly-focusedradial beams and thus adding phase control to priorwork such as de Clercq et. al. [17]. Combined with al-ready demonstrated work incorporating CMOS compati-bility [21] and integrated optics [32], this scheme wouldenable a completely integrated trapped ion system, avoid-ing the need for complex, active optics.In summary, we have demonstrated parallel position-controlled composite quantum logic gates, where phasesare implemented by nanoscale movements of each ionwithin a global laser beam. We described how to over-come imperfect rotations caused by zone-dependent lightintensities through the construction of a four-pulse com-posite gate, allowing the intensity to vary by more thana factor of two between interaction zones. Our schemeprovides a pathway toward dense parallelized quantumoperations on ions with minimal optics and external mod-ulators.We would like to thank Peter Maunz and Daniel Stickfor useful discussions and providing documentation forthe High Optical Access trap. This work was funded inpart by the IARPA MQCO program and by the NSF Center for Ultracold Atoms. Guang Hao Low would liketo acknowledge funding by the ARO Quantum AlgorithmsProgram and NSF RQCC Project No.1111337. ∗ email:m [email protected][1] R. Van Meter and C. Horsman, Commun. ACM , 84(2013).[2] D. Gottesman, Quantum Inf. Comput. , 1338 (2014).[3] T. Ruster, C. T. Schmiegelow, H. Kaufmann,C. Warschburger, F. Schmidt-Kaler, and U. G.Poschinger, Appl. Phys. B Lasers Opt. , 254 (2016).[4] J. 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A principled and systematic approach to the design of composite quantum gates is surprisingly challenging. Given asequence of L primitive rotations ˆ R φ [ θ ], the essential problem is finding an intuitive characterization of the functionalform of all possible composite quantum gates ˆ U [ θ ] of the formˆ U [ θ ] = ˆ R φ L − [ θ ] · · · ˆ R φ [ θ ] ˆ R φ [ θ ] , ˆ R φ [ θ ] = e − i θ (cos ( φ )ˆ σ x +sin ( φ )ˆ σ y ) . (1)Note that ˆ U [ θ ] is an SU(2) operator, thus it can always be decomposed into the Pauli basis { ˆ1 , ˆ σ x , ˆ σ y , ˆ σ z } :ˆ U [ θ ] = A [ θ ]ˆ1 + i ( B [ θ ]ˆ σ z + C [ θ ]ˆ σ x + D [ θ ]ˆ σ y ) , (2)where A [ θ ] , B [ θ ] , C [ θ ] , D [ θ ] are real functions of θ . Though one approach to this problem is to find the best-fit (cid:126)φ ∈ R L to some objective for the A, B, C, D by gradient-descent, this approach is not particularly insightful, and has anexponentially increasing computational complexity with respect to L .An alternative approach arises from noting that the fidelity F of ˆ U [ θ ] with respect to a target single-qubit rotationˆ R [ θ T ] depends only on the functions A [ θ ] and C [ θ ]. F = 12 (cid:12)(cid:12)(cid:12) tr (cid:110) ˆ R [ θ T ] ˆ U † [ θ ] (cid:111)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) θ T (cid:19) A [ θ ] − sin (cid:18) θ T (cid:19) C [ θ ] (cid:12)(cid:12)(cid:12)(cid:12) . (3)Thus designing ˆ U [ θ ] to implement ˆ R [ θ T ] at a specific value θ = θ or in its neighborhood requires an understandingof what functions A [ θ ] , C [ θ ] can be implemented by some choice of (cid:126)φ ∈ R L . Note that the ˆ σ z rotations required toimplement ˆ R φ [ θ T ] are trivially obtained by a global shift of all φ k → φ k + φ . This understanding was previouslyprovided by one of the authors [25]: Theorem 1.
A choice of A [ θ ] , C [ θ ] is achievable by some (cid:126)φ ∈ R L if and only if all the following are true(1) A [0] = 1 ,(2) A [ θ ] + C [ θ ] ≤ ,(3; L odd) A [ θ ] = (cid:80) Lk odd a k cos k ( θ/ , ∀ k, a k ∈ R ,(4; L odd) C [ θ ] = (cid:80) Lk odd c k sin k ( θ/ , ∀ k, c k ∈ R .(3; L even) A [ θ ] = (cid:80) Lk even a k cos k ( θ/ , ∀ k, a k ∈ R ,(4; L even) C [ θ ] = cos ( θ/ (cid:80) Lk odd c k sin k ( θ/ , ∀ k, c k ∈ R .Moreover, (cid:126)φ ∈ R L can be computed from A [ θ ] , C [ θ ] in poly ( L ) time. These constraints are particularly intuitive. (1) arises from considering θ = 0. There, ˆ U [0] is identity, thus A [0] = 1.(2) is simply a statement that probabilities are bounded by 1. (3,4) arise from a direct expansion of Eq. 1 and restrictthe form of A [ θ ] , C [ θ ] to simple functions.In other words, A [ θ ] , C [ θ ] are trigonometric polynomials in cos ( θ/
2) and sin ( θ/
2) with a bounded norm. Using theChebyshev polynomials of the first and second kind T k [cos ( θ )] = cos ( kθ ) and U k [cos ( θ )] = sin (( k +1) θ )sin ( θ ) , the case ofeven L can be simplified to A [ θ ] = L (cid:88) k even a k cos k ( θ/
2) = L/ (cid:88) k =0 a (cid:48) k cos ( kθ ) , C [ θ ] = cos ( θ/ L (cid:88) k odd c k sin k ( θ/
2) = L/ (cid:88) k =1 c (cid:48) k sin ( kθ ) . (4)Thus A [ θ ] , C [ θ ] are any Fourier series with a bounded norm. In the following, all even L sequences are represented bythis Fourier series, and so we drop the primes on the coefficients.As the implementing phases (cid:126)φ can be efficiently computed from A [ θ ] , C [ θ ], it suffices to specify the compositequantum gate only through some choice of A [ θ ] and C [ θ ]. This representation has the significant advantage of verydirectly describing the fidelity response function of the composite quantum gate, in contrast to (cid:126)φ , which provide nodirect information about what the implemented composite gate does. It is also often the case that the expression forthe trigonometric polynomials or Fourier series A [ θ ] , C [ θ ], as a function of say ( θ , θ T ), is much simpler than that of (cid:126)φ .Note however that multiple different, but valid, solutions of (cid:126)φ can be obtained for each choice of A [ θ ] , C [ θ ]. We nowapply this characterization to the design of composite quantum gates.Our movement-controlled composite gates should ideally satisfy the following properties:1. For some fixed value of θ = θ , there exist (cid:126)φ such that ˆ U [ θ ] = ˆ R [ θ T ] for all θ T ∈ [0 , π ). This ensures that wecan implement all possible single qubit rotations in a single sequence.2. Property (1) holds for a continuous range of θ ∈ [ θ min , θ max ], 0 < θ min < θ max . This ensures that for allvariations in θ induced by say, an inhomogeneous laser beam, an ideal arbitrary single-qubit gate can still beimplemented in a single sequence, so long as θ is known.Combined with Thm. 1, the design of our desired composite quantum gates reduces to finding Fourier series A [ θ ] , C [ θ ]that satisfies these properties, which can be contrasted to the more direct, but less efficient and less insightful numericalsearch for (cid:126)φ for every pair ( θ , θ T ). In particular, Thm. 1 and these properties furnish a set of linear constraints on thecoefficients { a k , c k } : A [0] = 1 , Thm. 1.1 , (5) A [ θ ] = cos ( θ T / , Eq. 3: F = 1 ,C [ θ ] = − sin ( θ T / , Eq. 3: F = 1 , (cid:20) cos ( θ T / dA [ θ ] dθ − sin ( θ T / dC [ θ ] dθ (cid:21) θ = θ = 0 , Eq. 3: d F dθ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ = 0 . Thus we have 4 linear equations for L + 1 coefficients of terms in A [ θ ] , C [ θ ]. All that remains is the choose L − c = 0 that leads to the satisfaction of condition Thm. 1(2). As any fully-determinedsystem of linear equations can be easily solved, we consider a closed-form specification of such a system of linearequations to be equivalent to finding the A [ θ ] , C [ θ ] in closed-form, which is then equivalent, through Thm. 1, tofinding (cid:126)φ in closed-form. Note that many such choices of these remaining linear equation are possible, and could beconstructed to impose additional desirable properties such as flatness of the fidelity response function F with respectto variations in θ . Length Composite Gates
Given the 4 linear constraints of Eq. 5, the shortest composite gate that could possibly satisfy them all simultaneouslymust have 4 total coefficients in the A [ θ ] , C [ θ ] terms. This corresponds to L = 3. When expanded fully, these constraintsare 1 = a + a , (6)0 = a cos ( θ /
2) + a cos ( θ / − cos ( θ T / , c sin ( θ /
2) + c sin ( θ /
2) + sin ( θ T / , (cid:18) θ T (cid:19) sin ( θ / (cid:0) a + 3 a cos ( θ / (cid:1) + sin (cid:18) θ T (cid:19) cos ( θ / (cid:0) c + 3 c sin ( θ / (cid:1) . As these are a system of linear equations, they can solved easily for the coefficients ( a , a , c , c ) which then furnish A [ θ ] , C [ θ ]. Provided that A [ θ ] , C [ θ ] satisfy the conditions of Thm. 1, we are guaranteed that the phases (cid:126)φ implementingthis gate can be efficiently computed.It then remains to determine the parameter space of ( θ , θ T ) such that A [ θ ] , C [ θ ] is achievable. By construction,conditions (1,3,4) of Thm. 1 are satisfied by the A [ θ ] , C [ θ ] obtained from Eq. 6. Thus all that remains to guaranteethat (cid:126)φ exists is to check condition (2) that A [ θ ] + C [ θ ] ≤
1. We now derive necessary and sufficient conditions for( θ , θ T ) that satisfy this condition. Let us expand A [ θ ] + C [ θ ] about θ = 0 , θ : A [ θ ] + C [ θ ] = 1 + θ θ / − cos ( θ T / (cos (3 θ / − cos ( θ T / ( θ ) cos ( θ /
2) sin ( θ T /
2) + O ( θ ) ≤ A [ θ ] + C [ θ ] = 1 + ( θ − θ ) (cos ( θ / − cos ( θ T / (cos (3 θ / − cos ( θ T / ( θ ) sin ( θ T /
2) + O (( θ − θ ) ) ≤ . By construction through Eq. 6, A [ θ ] + C [ θ ] = 1 at θ = 0 , θ , π − θ – the point θ = π − θ arises from the symmetry A [ θ ] + C [ θ ] = A [ − θ ] + C [ − θ ] – and these are also stationary points. By choosing g ( θ , θ T ) = (cos ( θ / − cos ( θ T / θ / − cos ( θ T / < , (8)so that the second derivative of A [ θ ] + C [ θ ] about θ = 0 is negative, the intermediate value theorem tells us that atleast 3 additional stationary points also develop at A [ θ ] + C [ θ ] < θ . Thus we have identifiedat least 6 stationary points of A [ θ ] + C [ θ ] that all have value ≤
1. However, as A [ θ ] + C [ θ ] is a Fourier series ofdegree of degree 3, it has at most 6 stationary points in θ ∈ [0 , π ). Thus we conclude that ∀ θ ∈ R , A [ θ ] + C [ θ ] ≤ { ( θ , θ T ) | g ( θ , θ T ) < } . This can be strengthened to if and only if { ( θ , θ T ) | g ( θ , θ T ) ≤ } , plotted in Fig. 5, bynoting that A [ θ ] + C [ θ ] = 1 when g ( θ , θ T ) = 0.Observe from Fig. 4 that the full range of θ T ∈ [ − π, π ] is not achievable for any fixed θ . Thus we must explorelonger sequences of (cid:126)φ . - - / π θ T / π FIG. 4: Shaded region indicates ( θ , θ T ) such that the trigonometric polynomial in Eq. 6 is achievable by some choiceof (cid:126)φ ∈ R . Length Composite Gates
When L = 4, A [ θ ] , C [ θ ] are Fourier series constrained by1 = a + a + a , (9)0 = a + a cos ( θ ) + a cos (2 θ ) − cos ( θ T / , c sin ( θ ) + c sin (2 θ ) + sin ( θ T / , (cid:18) θ T (cid:19) ( a sin ( θ ) + 2 a sin (2 θ )) + sin (cid:18) θ T (cid:19) ( c cos ( θ ) + 2 c cos (2 θ )) . Thus we have 4 linear equations for 5 free parameters. All that remains is to choose one more linear equation, say c = 0, that leads to the satisfaction of condition Thm. 1.2. Note that many such choices of this last linear equationare possible. We explore two possibilities in the following. - - / π θ T / π - - / π θ T / π FIG. 5: Shaded region indicates ( θ , θ T ) such that the Fourier series in Eq. 11 (left) or Eq. 16 (right) is achievable bysome choice of (cid:126)φ ∈ R . Symmetric Length 4 Composite Gate
One particularly simple choice for the last linear equation is A [ π ] = 1 ⇒ a − a + a = 1 . (10)Together with Eq. 9, solving for { a , , , c , } produces the Fourier series A [ θ ] = 1 + sin (cid:0) θ T (cid:1) sin ( θ ) (cos (2 θ ) −
1) (11) C [ θ ] = tan (cid:0) θ T (cid:1) sin ( θ ) (cid:34) − θ ) + sin (cid:0) θ T (cid:1) sin ( θ ) [2 sin ( θ ) − cos ( θ ) sin (2 θ )] (cid:35) . The phases (cid:126)φ implementing this gate can be efficiently computed in principle from Eq. 11 using Thm. 1, and theparameter space of achievable ( θ , θ T ) can be obtained numerically by checking that the maximum of A [ θ ] + C [ θ ] is 1.Alternatively, a more elegant analytic approach is enabled by our use of the necessary and sufficient conditionsof Thm. 1. By construction, conditions (1,3,4) of Thm. 1 are satisfied by our choice of A, C . Thus all that remainsto guarantee that (cid:126)φ exists is to check condition (2) that A [ θ ] + C [ θ ] ≤
1. We now derive necessary and sufficientconditions for ( θ , θ T ) that satisfy this condition. Let us expand A [ θ ] + C [ θ ] about θ = 0 , πA [ θ ] + C [ θ ] = 1 + θ (cid:20) (cos (2 θ ) − cos ( θ T / ( θ T /
4) tan ( θ T / ( θ /
2) cos ( θ / (cid:21) + O ( θ ) ≤ A [ θ ] + C [ θ ] = 1 + ( θ − π ) (cid:20) (cos (2 θ ) − cos ( θ T / ( θ T /
4) tan ( θ T / ( θ /
2) cos ( θ / (cid:21) + O (( θ − π ) ) ≤ . By construction through the system of linear constraints, A [ θ ] + C [ θ ] = 1 at θ = 0 , θ , , π − θ and these are alsostationary points. By choosing g ( θ , θ T ) = cos (2 θ ) − cos ( θ T / < , (13)the intermediate value theorem tells us that at least 4 additional stationary points also develop at A [ θ ] + C [ θ ] < θ . Thus we have identified at least 8 stationary points of A [ θ ] + C [ θ ] that all have value ≤
1. However, as A [ θ ] + C [ θ ] is a Fourier series of degree 4, it has at most 8 stationary points in θ ∈ [0 , π ). Thuswe conclude that ∀ θ ∈ R , A [ θ ] + C [ θ ] ≤ { ( θ a , θ T ) | g ( θ , θ T ) < } . This can be strengthened to if and only if { ( θ a , θ T ) | g ( θ , θ T ) ≤ } , plotted in Fig. 5, by noting that A [ θ ] + C [ θ ] = 1 when g ( θ , θ T ) = 0.This example also has a simple closed-form solution valid for { ( θ a , θ T ) | g ( θ , θ T ) ≤ ∧ θ T ≥ } : φ = − π/ γ, φ = π/ γ + χ, φ = φ , φ = φ , (14) χ = − cos − (cid:34) − (cid:0) θ T (cid:1) sin ( θ ) (cid:35) , γ = tan − cos ( χ/ χ/
2) cos ( θ ) sin ( θ )sin (cid:16) θT (cid:17) − . Note that it suffices to consider only this range of θ T ≥ R [ θ ] = ˆ R π [ − θ ] = ˆ R π [4 π − θ ] . (15)Thus the phases (cid:126)φ implementing the composite gates for θ T ∈ [ − π,
0) can be obtained by adding π to the phases ofcomposite gates for θ T ∈ (0 , π ].Observe from Fig. 5 that the full range of θ T ∈ [ − π, π ] can only be achieved by symmetric length 4 compositegates with a base rotation angle of θ = π/
2. In comparison, there exist sequences that implement θ T ∈ [ − π, π ] for any θ ∈ [ π/ , π/ θ T can be covered by some range of θ . Anti-Symmetric Length 4 Composite Gate
Another choice for the last linear equation for 0 < θ T ≤ θ is2 c + 4 c = cot (cid:18) θ (cid:19) tan (cid:18) θ T (cid:19) (cid:32) − sign[sin ( θ T / (cid:115) θ T / ( θ / (cid:18) ( θ / − ( θ T / (cid:19)(cid:33) . (16)As described in Eq. 15, it suffices to consider only composite gates implementing θ T ∈ [0 , π ].Eq. 16 combined with Eq. 9 can in principle be solved for the { a , , , c , } to produce the Fourier series of A [ θ ] , C [ θ ]–though an extremely complicated expression, it is nevertheless obtainable in closed form. Thm. 1 assures us thatfor choices of ( θ , θ T ) that satisfy its conditions, (cid:126)φ implementing the composite gate corresponding to Eq. 16 can becomputed using the techniques described in [25].Though more complicated than the symmetric composite gates, the utility of composite gates corresponding toEq. 16 is evident by plotting in Fig. 5 the region of achievable ( θ , θ T ) where A [ θ ] + C [ θ ] ≤
1. Unlike the symmetriccomposite gates, observe that for a range of base rotation angles θ ∈ [ π/ , . π ], the full range of θ T ∈ [ − π, π ] isaccessible.While (cid:126)φ can in principle be expressed in closed-form as a function of ( θ , θ T ), the result is extremely lengthy andneither insightful nor practical. A more practical option is to compute (cid:126)φ , also via [25], from A [ θ ] , C [ θ ] with ( θ , θ T )substituted with numerical values. In this manner, we plot in Fig. 6 (cid:126)φ as a function of θ T for θ = 0 . π and θ = 0 . π .Note that for these composite gates, φ = − φ , φ = − φ . (17) θ πϕ ϕ ϕ ϕ - - - - θ T / π ϕ / π θ πϕ ϕ ϕ ϕ - - - - θ T / π ϕ / π FIG. 6: Phases (cid:126)φ implementing the composite gate corresponding to Eq. 16 for θ = 0 . π (left) and θ = 0 . ππ