2. Simple quantum algorithms with an electron in a Penning Trap David Vitali, Giacomo Ciaramicoli, Irene Marzoli, and Paolo Tombesi Dip. di Matematica e Fisica and Unità INFM, Università di Camerino, Italy

3. Single electron confined with STATIC FIELDS (less noisy than rf potentials): - a uniform magnetic field (parallel to the z axis) - a quadrupole electric field magnetic field  radial confinement (cyclotron motion) quadrupole field  axial confinement  slow circular motion in the radial plane superimposed to the cyclotron motion (magnetron motion)  c /2   100 GHz  z /2   50 MHz  m /2   10 kHz

4. Three independent harmonic oscillators (and a 1/2 spin) H 0 = h  c (a c + a c +1/2) + h  z (a z + a z +1/2) - h  m (a m + a m +1/2) + (h/2)  s  z Energy levels: E 0 (n,k,l,s) = nh  c + kh  z - lh  m + s (h/2)  s n,k,l = 0,1,2,3…. s=  1 magnetron motion decay time of order of years it can be cooled to  l   1000 only axial motion decay time of order of days It can be cooled to its ground state cyclotron motion decay time  13s (inhibited by cavity effects) it can be cooled to its ground state spin motion decay time of order of years very low decoherence

5. A single trapped electron is not obviously scalable HOWEVER High control of quantum dynamics (it is used for high precision measurements) Very low decoherence it can be used for the implementation of simple quantum algorithms with few qubits, using different electronic degrees of freedom spin: “natural” qubit cyclotron and axial motion: multi-level systems with equally spaced energy states: anharmonicity is needed to encode quantum information

6. Anharmonic corrections to transition frequencies  c (n,s) = [E 0 (n+1,k,l,s) – E 0 (n,k,l,s)]/h   c -  (n+1) -  s/2  s (n) = [E 0 (n,k,l,+1) – E 0 (n,k,l,-1)]/h   s -  (n+1/2)  z (k) = [E 0 (n,k+1,l,s) – E 0 (n,k,l,s)]/h   z +  1 (k+1)  = h  c 2 /mc 2 Relativistic corrections, yielding anharmonicity on the cyclotron motion (even if transition degeneracy is not completely removed)  1 corrections to the ideal quadrupole potential, yielding anharmonicity on the axial oscillator After introducing anharmonicities, quantum information can be encoded in the following three qubits: {  0  c,  1  c } {  0  z,  1  z } { ,  }

7. Cyclotron state control: p c - pulses Apply a microwave field with frequency close to  c ’ H ic  h  /2 (a c e -i  + a c + e i  ) Spin state control: p s -pulses Apply a rotating magnetic field on the x-y plane with frequency close to  s H is  h  /2 (  + e -i  +  - e i  ) Axial state control: p z -pulses Apply an oscillating potential on the trap electrodes with frequency close to  z H ic  h  /2 (a z e -i  + a z + e i  )

8.  c - 5  /2  c - 3  /2  c -  /2  c - 5  /2  c - 3  /2 n = 0 n = 1 n = 2 n = 3 n = 0 n = 1 n = 2 s = -1s = 1 Spin-cyclotron levels with degenerate transitions

9. CONDITIONAL DYNAMICS BETWEEN QUBITS : apply an inhomogeneous magnetic field (magnetic bottle)  B = B 2 [(z 2 – (x 2 + y 2 )/2)k – z(x i + y j)] State-dependent transition frequencies  c (n,s,k)   c -  (n + 1) -  s/2 +  2 (k + 1/2)  s (n,k)   s -  (n + 1/2 ) +  2 (k + 1)  z (k,n,s)   z +  1 (k + 1) +  2 (n + 1/2 + s/2) (  2  B 2 ) p c, p s, p z pulses modify the cyclotron, spin, axial states respectively, conditioned on the state of the other degrees of freedom However, we have still transition degeneracy:  k,n,    k+1,n,  and  k,n+1,    k+1,n+1,  have the same frequency

10. CYCLOTRON QUBIT: multifrequency p c pulses have to be supplemented by appropriate p s pulses in order to avoid undesired transitions due to degeneracy AXIAL QUBIT: multifrequency p z pulse SPIN QUBIT: multifrequency p s pulse One-qubit rotations Example: 1) multifrequency p c pulse  c (n=0,s=-1,k=0,1) 2) multifrequency p s pulse flipping the spin states 3) p c pulse identical to the first one 4) multifrequency p s pulse flipping the spin states

11. n = 0 n = 1 n = 2 n = 3 n = 0 n = 1 n = 2 s = -1s = 1  c - 3  /2  c -  /2 Spin one-qubit rotations

12. spin conditioned to cyclotron: a multifrequency p s pulse  s (n=n 0,k=0,1) spin conditioned to axial: a multifrequency p s pulse  s (n=0,1,k=k 0 ) cyclotron conditioned to spin: 1) p s pulse flipping the spin states  2) multifrequency p c pulse  c (n=0,s=-1,k=0,1) 3) p s pulse flipping the spin states  1) multifrequency p c pulse  c (n=0,s=-1,k=0,1) Two-qubit conditional dynamics cyclotron conditioned to axial: 1) p c pulse  c (n=0,s=-1,k=k 0 ) 2) p s pulse flipping the spin states 3) p c pulse identical to the first one 4) p s pulse flipping the spin states

13. axial conditioned to spin: 1) p s pulse flipping the spin only if the cyclotron state is  1  c 2) p z pulse  z (k=0,n=0,s=1)(degenerate transition, acting on  the transition  0  z  1  z only if cyclotron-spin state is  0  c  and  1  c  ) 3) p s pulse flipping the spin only if the cyclotron state is  1  c axial conditioned to cyclotron: 1) multifrequency  p c pulse  c (n=0,s=-1,k=0,1)+  c (n=1,s=1,k=0,1) 2) multifrequency p z pulse  z (k=0,n=0,s=-1)+  c (k=0,n=2,k=1)  1  c (it acts on the transition  0  z  1  z only if cyclotron-spin state is  2  c  and  0  c  ) 3) multifrequency  p c pulse  c (n=0,s=-1,k=0,1)+  c (n=1,s=1,k=0,1) the various C-NOT gates require 3.3 pulses on average

14. Three-qubits conditional dynamics three-qubits gates can be very easily implemented, e.g. Conditional phase shift  11   -  11  (used in Grover algorithm) in one pulse only: a single 2  p c pulse  c (n=1,s=1,k=1) Toffoli gate with spin as target  11    11 ,  11    11  in two pulses: 1) a  p s pulse  s (n=1,k=1) 2) a 2  p c pulse  c (n=1,s=1,k=1)

15. State detection The axial state can be measured by transferring it into the cyclotron and spin states and then measuring them In fact, with 7 pulses one realizes the following mapping of the logical states |n c,n z,s> onto |n’ c,0 z,s’> 00  00  01  01  10  10  11  11  00  10  00  10  20  30  20  30  Cyclotron and spin levels can be directly measured as state-dependent axial frequency shifts  z =  z (gs/4 + n + 1/2)

16. Implementation of quantum algorithms Quantum Fourier Transform 20 pulses Three-qubits quantum error correction 26 pulses Deutsch-Josza algorithm 17 pulses Grover algorithm 44 pulses (+ 7 pulses for the detection)