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Algorithmic Cooling R.Laflamme Algorithmic cooling R.Laflamme In collaboration with: Guanru Feng Dawei Lu Nayeli Rodriguez-Briones Jonathan Baugh Jun Li Xinhua Peng Tal Mor Yossi Weinstein Improve and Benchmarking initial states Nuclear Overhauser Effect We have two spins, 13C H We irradiate one, the other one cools down!! Nuclear Overhauser Effect We have two spins, We irradiate one, the other one cools down!! When first proposed as a contributed paper at an APS meeting in April 1953, the proposal was met with much skepticism by a formidable array of physics talent. Included among these were notables such as: Felix Bloch (recipient of 1952 Physics Nobel Prize), Edward M. Purcell (recipient of Nobel Prize 1952 with Bloch and session chair), Isidor I. Rabi (recipient of Physics Nobel Prize, 1944) and Norman F. Ramsey (recipient of Physics Nobel Prize, 1989). Eventually everyone was won over. July 27, 1953 Dear Dr. Overhauser: You may recall that at the Washington Meeting of the Physical Society, when you presented your paper on nuclear alignment, Bloch, Rabi, Pearsall, and myself all said that we found it difficult to believe your conclusions and suspected that some fundamental fallacy would turn up in your argument. Subsequent to my coming to Brookhaven from Harvard for the summer, I have had occasion to see the manuscript of your paper. After considerable effort in trying to find the fallacy in your argument, I finally concluded that there was no fundamental fallacy to be found. Indeed, my feeling is that this provides a most intriguing and interesting technique for aligning nuclei. After considerable argument, I also succeeded in convincing Rabi and Bob Pound of the validity of your proposal and I have recently been told by Pound that he subsequently converted Pearsall shortly before Pound left for Europe. I hope that you will have complete success in overcoming the rather formidable experimental problems that still remain. I shall be very interested to hear of what success you have with the method. Sincerely, Norman F. Ramsey Threshold theorem A quantum computation can be as long as required with any desired accuracy using a reasonable amount of resources as long as the noise level is below a threshold value -6,-5,-4,-3,-2,... P < 10 Knill et al.; Science, 279, 342, 1998 Kitaev, Russ. Math Survey 1997 Aharonov & Ben Or, ACM press Preskill, PRSL, 454, 257, 1998 Significance: ... -imperfections and imprecisions are not fundamental objections to quantum computation -it gives criteria for scalability -its requirements are a guide for experimentalists -it is a benchmark to compare different technologies Ingredients for FTQEC ⌅ Parallel operations ⌅ Good quantum control ⌅ Ability to extract entropy ⌅ Knowledge of the noise No lost of qubits • Independent or quasi independent errors • Depolarising model • Memory and gate errors • ... • The need for nearly pure states For quantum error correction The initial state in NMR Benchmarking 1 βH 1 ⇢ = e− ( 1l βH + ...) Z ⇠ Z − Making a pseudo pure state (Cory et al 1996, Gershenfeld et al. 1997) C C C C C C R State State Prep. Meas. 1 1 β!n ( 1l βH) ( 1l ) Z − ! Z − 2n | ih | φ φ Encoding Gradient: Decoding Gradient:-2 Y:90 Fidelity Y:90 - X:90 -Y:90 - _ _ - i _ e i e _ _ 4 e _ - 4 - _ 4 _ i - _ i I Y 2 Y Y _ _ i 2 X _ 4 - Z _ _ 2 φ + _ _ e _ φ X X Z _ _ I I φ - 4 ( ( _ i ( = I ( X 2 X - _ X _ Z φ - X _ + 4 _ i Z ) ( + i I + Y I + i i + X ) X - ( ( Y I I Y I - - - - - ) Y Y Y Z ( ) ) Y Y I + + + + System Error Rate Reference i i Y Y I Ion Trap 0.000056 arxiv:1512.04600v1 (2015) - X Z ) ) ) Silicon (n) 0.0001 J. Phys.: Cond. Mat. 27 154205 (2015) Silicon (e−) 0.0005 J. Phys.: Cond. Mat. 27 154205 (2015) Superconducting 0.0008 Nature 508, 501 (2014) Ion Trap (single) 0.00002 PRA 83 030303(R) (2011) Neutral atoms 0.00014 NJP 12, 113007 (2010) Liquid-State NMR 0.00013 NJP 11 013034 (2009) Superconducting 0.011 PRL 102 090502 (2009) Ion Trap (crystal) 0.0008 QIC 9 920 (2009) Ion Trap (single) 0.00482 PRA 77 012307 (2008) ESR 0.007 Laflamme/Morton Algorithmic cooling polarization = ✏ = Tr[⇢Z] Sorenson, JMR, 1990 Schulman, Vazirani ACM 1999 Algorithmic cooling (PPA or partner pairing ) We have seen that we can cool a subset of spins by swapping states. For excample, with 3 spins, implementing a gate that swaps 011 100 will increase the order of the first spin at the expense of the last| i$ two.| Wei could concatenate this process to reach polarization of order 1. βH 1 ⇢ e− ( 1l β!(Z1 + Z2 + Z3)+...) ⇠ ⇠ 2n − 300 0 0 0 0 0 3000 0 0 0 0 010 0 0 0 0 0 0100 0 0 0 0 0001 0 0 0 0 01 00010 0 0 0 01 β! B000 10 0 0 0C β! B0001 0 0 0 0C ⇢d B − C ⇢d B C thermal ⇡ 8 B000 0 1 0 0 0C () pol ⇡ 8 B0000 10 0 0C B C B − C B000 0 0 10 0C B0000 0 10 0C B − C B − C |011> B000 0 0 0 10C B0000 0 0 10C B − C B − C |100> B000 0 0 0 0 3C B0000 0 0 0 3C B − C B − C @ A @ A d d 3 10 ⇢¯ = Tr 2,3⇢ β! pol pol ⇡ 4 0 1 ✓ − ◆ if want P~1 need ~ 1/ε^2 spins Algorithmic cooling (PPA or partner pairing ) We have seen that we can cool a subset of spins by swapping states. For excample, with 3 spins, implementing a gate that swaps 011 100 will | i$| i increase the order of the first spin at the expense of the last two. We could concatenate this process to reach polarization of order 1. βH 1 ⇢ e− ( 1l β!(Z1 + Z2 + Z3)+...) ⇠ ⇠ 2n − 300 0 0 0 0 0 3000 0 0 0 0 010 0 0 0 0 0 0100 0 0 0 0 0001 0 0 0 0 01 00010 0 0 0 01 β! B000 10 0 0 0C β! B0001 0 0 0 0C ⇢d B − C ⇢d B C thermal ⇡ 8 B000 0 1 0 0 0C () pol ⇡ 8 B0000 10 0 0C B C B − C B000 0 0 10 0C B0000 0 10 0C B − C B − C B000 0 0 0 10C B0000 0 0 10C B − C B − C B000 0 0 0 0 3C B0000 0 0 0 3C B − C B − C @ A @ A d d 3 10 ⇢¯ = Tr 2,3⇢ β! pol pol ⇡ 4 0 1 ✓ − ◆ if want P~1 need ~ 1/ε^2 spins Heat Bath Algorithmic Cooling 0 10 max 1 ∼ -1 ✏ 10 -2 10 Bath 3 qubits 4 qubits maximum polarization 5 qubits 6 qubits -3 10 -2 -1 0 1 2 Schulman, Mor, Weinstein 10 10 10 10 10 PRL 94, 120501 (2005). Scaled Heat-bath polarization ∼ n ✏b 2 n n Schulman/Mor/Weinstein ε<<1/2 => εmax~2 ε Moussa, Master thesis 2005 0 10 max 1 ∼ -1 HBAC ✏ 10 -2 10 3 qubits 4 qubits maximum polarization 5 qubits 6 qubits -3 10 -2 -1 0 1 2 10 10 10 10 10 Scaled Heat-bath polarization ∼ n ✏b 2 Moussa,Master thesis 2005 Schulman/Mor/Weinstein md md (1 + ✏b) (1 ✏b) ✏1l1 = − − . (1 + ✏ )md + (1 ✏ )md b − b if ✏ << m2n0; ✏ md✏ (= m2n0✏ ) N. Rodriguez Briones +RL, PRL 116, 170501, 2016 b 1l1 ⇡ b b 2 n0 if ✏b >> m2 ; ∆max =1 ✏1; ∆max 1l 1+✏b − ⇡ md ln 1 ✏ e − b +1 ⇣ ⌘ 0 10 max 1 ∼ -1 ✏ 10 -2 HBAC 10 3 qubits 4 qubits maximum polarization 5 qubits 6 qubits -3 10 -2 -1 0 1 2 10 10 10 10 10 Scaled Heat-bath polarization ∼ n ✏b 2 Moussa, Master thesis 2005 Schulman/Mor/Weinstein md md (1 + ✏b) (1 ✏b) ✏1l1 = − − . (1 + ✏ )md + (1 ✏ )md b − b For 2qubits: PPA, i.e. SORT + refresh qubits ✏ ✏ 1l1 ! b The deviation matrix |011> 300 0 0 0 0 0 010 0 0 0 0 0 |100> 0001 0 0 0 0 01 3 1 3 d β! B000 10 0 0 0C ⇢thermal 81l = ⇢thermal 8 B − C − ⇡ B000 0 1 0 0 0C B000 0 0 10 0C B C B000 0 0− 0 10C B − C B000 0 0 0 0 3C B − C @ A 200 0 000 0 ⇢2 d β! thermal ⇡ 4 0000 01 000 2 B C @ − A HBAC for 2 qubits ⇢˙ = i[H, ⇢]+ ⇢ − R ⇢ = γ↵ 2L↵⇢L† L† L↵⇢ ⇢L† L↵ R ↵ − ↵ − ↵ ↵ X ⇣ ⌘ Can get polarization enhancement for 2 qubits! Jun Li et al, PRA94, 012312, 2016 Approximation of Reachable Set for Coherently HBAC for 2 qubits Controlled Open Quantum Systems: Application to Quantum State Engineering, J. Li, D. Lu, Z. Luo, RL, X. Peng, and J. Du.l, PRA94, 012312, 2016 n 4n 1 1l − ⇢ = + rkBk 2Nn . Xk=0 P <0 Linblad => ~r˙ = ~r (~r r~ ) H − R − eq . P =0 2 Purity: P = Tr[⇢ ] P˙ = 2n+1~rT (~r r~ ) − R − eq 13C Experimental check H r ,r ,r ,r ,r ,r 0.0532, 0.0918, 0.0798, 0.0212, 0.0000, 0.0022 • { 1 2 3 4 5 6} ≈ { } r t=0=ZZ+4IZ r t=0=ZI+4Z 1/4 0 000 1/4 4 4 d ⎛ ZI ⎞ ⎡ (4r1 + 16r4)ε r1 r4 r5 ⎤ ⎛ ZI ⎞ 2 2 ⟨ ⟩ = − ⟨ ⟩ , dt ⎜ IZ ⎟ ⎢ (4r + 16r )ε r r r ⎥ ⎜ IZ ⎟ 0 0 ⎜ ⟨ ⟩ ⎟ ⎢ − 4 2 4 2 6 ⎥ ⎜ ⟨ ⟩ ⎟ ZI ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ -2 IZ -2 ⎜ ZZ ⎟ ⎢ (4r5 + 16r6)ε r5 r6 r3 ⎥ ⎜ ZZ ⎟ ZZ ⎜ ⟨ ⟩ ⎟ ⎢ − ⎥ ⎜ ⟨ ⟩ ⎟ -4 -4 ⎝ (⎠a) ⎣ ⎦ ⎝ (b⎠ X \ A1 :(ϵ, 4ϵ, 0) 50 0 10 20 30 X 40\ 50 0 10 20 30 A2 : (0, 4.25ϵ, 0) s X \ s A3 : (1.87ϵ, 1.87ϵ, 1.87ϵ) A4 : (1.67ϵ, 1.67ϵ, 1.67ϵ) ê ê A5 : (1.53ϵ, 1.53ϵ, 1.53ϵ) Trajectory: NOE Trajectory: PPS 4ϵ Ry(90◦) Rx(90◦) S ( E A3 3 A4 x 2ϵ A5 P 0 A1 4ϵ A2 2ϵ 4ϵ 2ϵ 0 0 x1 x2 Nuclear Overhauser Effect (NOE) Solomon equations Cross-relaxation d Z1 h i = r ( Z1 Z1 ) r ( Z2 Z2 ), dt − 1 h ih i0 − 12 h ih i0 |11> Z1 Z1 + r12 Z2 0 r 0 Γ Γ2 Γ h i!h i 1 h i 1 1 |01> r (Γ Γ ) |10> 12 2 − 0 Γ = 0 Γ r1 (Γ0 +2Γ1 + Γ2) 1 Γ1 Solomon PR99, 559,1955 |00> ( ( Follow the deviation matrix ( Correlated ( 2 0 2 Flip qubit 2 Relaxation 0 2 2 0 -2 -2 ( -2 ( 0 ( -2 Γ1 Γ 2 |11> Γ1 X Γ2 ε 2 ε |00> This is a quantum information science version of the Nuclear Overhauser effect (NOE) Can do a little better… Γ1 Γ Γ 2 2 Γ1 X Γ1 X ( ( ( ( Qubit 2 Cross( 3 1 2 Flip Relaxation 1 3 3 … -1 -3 -3 ( -3 ( -1 ( -2 Polarization ε 3 ε Solid state NMR with Malonic acid Ensemble qubit Hamiltonian (13C) determined spectroscopically ó depends on crystal orientation C1(Hz) C2(Hz) Cm(Hz) T2*(ms) T1(s) C1 5693 237 828 2.4 162 C2 -- 1748 1020 2.6 326 Cm -- -- -3358 3.1 314 C1(kHz) C2(kHz) Cm(kHz) Hm1(kHz) Hm2(kHz) H1(kHz) H2(kHz) Hm1 1.9 1.7 3.5 -- -23 1.4 0.9 Hm2 -0.7 0.6 -46 -- -- 0.9 1.6 H1 2.3 0.2 0.4 -- -- -- 0.3 H2 0.2 1.4 0.5 -- -- -- -- Multiple Rounds of Algorithmic Cooling C2 C1 Cm Hm1/2 Hbath Refresh Thermal Contact Register Operation • By using heat-bath able to surpass 9 x 10 10 Shannon/Soresnsen bound of 1.5X C 1 C2 9 Cm heat-bath polarization 8 7 Polarization Boost 1.69X Boost 6 w.r.t.
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