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**Three-qubit quantum error correction with superconducting circuits**

M. Reed et al. Nature 482, 382 (2012) Abraham Asfaw Princeton University

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**Outline Quantum Error Correction Circuit QED Bit-flip QEC**

Three-qubit quantum error correction with superconducting circuits Outline Quantum Error Correction Bit-flip QEC Phase-flip QEC QEC Codes Circuit QED Relevant energy level transitions Implementation of a CCPHASE gate QEC experimental results Alternate scheme(s)

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**Quantum Error Correction**

Three-qubit quantum error correction with superconducting circuits Quantum Error Correction Condition for the existence of a recovery operator In other words, Nielsen & Chuang, Kaye, Laflamme & Mosca

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**Quantum Error Correction**

Three-qubit quantum error correction with superconducting circuits Quantum Error Correction 𝑙, 𝑚∈{0,1} Nielsen & Chuang, Kaye, Laflamme & Mosca

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**Bit-Flip Quantum Error Correction**

Three-qubit quantum error correction with superconducting circuits Bit-Flip Quantum Error Correction Encoding scheme produces entangled GHZ-like states Codewords are +1 eigenstates of the ZiZj operators Nielsen & Chuang, Kaye, Laflamme & Mosca

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**Bit-Flip Quantum Error Correction**

Three-qubit quantum error correction with superconducting circuits Bit-Flip Quantum Error Correction Without decoding Nielsen & Chuang, Kaye, Laflamme & Mosca

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**Bit-Flip Errors in GHZ-Like States**

Three-qubit quantum error correction with superconducting circuits Bit-Flip Errors in GHZ-Like States Measured observable-pair is unique for each error – SYNDROME CHECK Nielsen & Chuang, Kaye, Laflamme & Mosca

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**Phase-Flip Quantum Error Correction**

Three-qubit quantum error correction with superconducting circuits Phase-Flip Quantum Error Correction Phase flips are equivalent to bit flips in the Hadamard basis Codewords are +1 eigenstates of the XiXj operators Nielsen & Chuang, Kaye, Laflamme & Mosca

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**Three-qubit quantum error correction with superconducting circuits**

Roadmap

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**Circuit QED Architecture**

Three-qubit quantum error correction with superconducting circuits Three-qubit quantum error correction with superconducting circuits Circuit QED Architecture ~8 GHz ~7 GHz ~6 GHz Strauch et al., PRL 91, (2003) Wallraff et al., Nature 431, 162 (2004) DiCarlo et al., Nature 460, 240 (2009) Reed et al., PRL 105, (2010) DiCarlo et al., Nature 467, 574 (2010) Strauch et al., PRL 91, (2003)

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**Three-qubit quantum error correction with superconducting circuits**

Roadmap

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**CPHASE Gate – Energy Levels**

Three-qubit quantum error correction with superconducting circuits CPHASE Gate – Energy Levels Experimental detuning parameter ϵ relates the normalized currents of two capacitively coupled Josephson junctions Strauch et al., PRL 91, (2003)

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**CPHASE Gate – Sudden Dynamics**

Three-qubit quantum error correction with superconducting circuits CPHASE Gate – Sudden Dynamics Suddenly move to and allow phase accumulation 11 02 → exp 𝑖 Δ 𝐸 𝑡 2ℏ exp −𝑖 𝛥 𝐸 𝑡 2ℏ − 11 → + + − ; 𝛥 𝐸 = 𝐸 5 𝜖 − − 𝐸 4 ( 𝜖 − ) After 𝑡= 2𝜋ℏ Δ 𝐸 →− + + − →−|11〉 CPHASE! Strauch et al., PRL 91, (2003)

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**CPHASE Gate – Experimental Results**

Three-qubit quantum error correction with superconducting circuits CPHASE Gate – Experimental Results Go back and measure 11 – Black 02 – White 12 ns Suddenly move to resonance Wait some time to accumulate phase 12 ns! DiCarlo et al., Nature 467, 574 (2010)

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**CPHASE Gate – Adiabatic Interaction**

Three-qubit quantum error correction with superconducting circuits CPHASE Gate – Adiabatic Interaction Single excitation manifold Two excitation manifold 00 ↦|00〉 11 ↦ 𝑒 𝑖( 𝜙 01 + 𝜙 10 + 𝜙 11 ) |11〉 01 ↦ 𝑒 𝑖 𝜙 01 |01〉 𝜙 11 =−2𝜋∫𝜁 𝑡 𝑑𝑡 10 ↦ 𝑒 𝑖 𝜙 10 |10〉 𝜙 01 =0 , 𝜙 10 =0 ,𝜙 11 =𝜋 𝜙 𝑙𝑟 =2𝜋∫𝛿 𝑓 𝑙𝑟 𝑡 𝑑𝑡 CPHASE! DiCarlo et al., Nature 460, 240 (2009)

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**Three-qubit quantum error correction with superconducting circuits**

Roadmap

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**Preparing GHZ States 〈𝑍𝑍𝐼〉 〈𝑍𝐼𝑍〉 〈𝐼𝑍𝑍〉 〈𝑋𝑋𝑋〉 〈𝑋𝑌𝑌〉 〈𝑌𝑋𝑌〉 〈𝑌𝑌𝑋〉**

Three-qubit quantum error correction with superconducting circuits Preparing GHZ States 〈𝑍𝑍𝐼〉 〈𝑍𝐼𝑍〉 〈𝐼𝑍𝑍〉 〈𝑋𝑋𝑋〉 〈𝑋𝑌𝑌〉 〈𝑌𝑋𝑌〉 〈𝑌𝑌𝑋〉 Stabilizers are being used as entanglement witnesses! 88% Fidelity DiCarlo et al., Nature 467, 574 (2010)

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**Three-qubit quantum error correction with superconducting circuits**

Roadmap

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**Bit-Flip Quantum Error Correction**

Three-qubit quantum error correction with superconducting circuits Bit-Flip Quantum Error Correction Encoding scheme produces entangled GHZ-like states Codewords are +1 eigenstates of the ZiZj operators We have all the ingredients except a CCNOT gate Can make CCNOT gate from five two-qubit gates, six CNOTs Nielsen & Chuang T.C. Ralph et al., PRA 75, (2007)

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** 𝟎𝟏𝟏 →|𝟎𝟎𝟐〉 𝟏𝟏𝟏 →|𝟏𝟎𝟐〉 𝜙 001 𝜙 010 𝜙 100 Sudden Transfer**

Three-qubit quantum error correction with superconducting circuits 𝜙 001 𝜙 010 𝜙 100 Sudden Transfer Adiabatic Interaction 𝟎𝟏𝟏 →|𝟎𝟎𝟐〉 Three excitation manifold 𝟏𝟎𝟐 →|𝟎𝟎𝟑〉 𝜙 011 𝝓 𝟏𝟎𝟏 𝜙 110 𝜙 011 𝜙 101 𝜙 110 𝟏𝟏𝟏 →|𝟏𝟎𝟐〉 Same as with a 6 GHz offset 3-QUBIT PHASE! 𝜙 111 M. Reed et al. Nature 482, 382 (2012)

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**Three-qubit quantum error correction with superconducting circuits**

Move to 111 → 102 site Move to cancel phase between 𝑄 2 and 𝑄 3 Return population to 111 𝜙 011 𝝓 𝟏𝟎𝟏 𝜙 110 𝜙 111 Move to 102 ↔ 003 site Move back to 102 M. Reed et al. Nature 482, 382 (2012)

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**State Tomography – Theory**

Three-qubit quantum error correction with superconducting circuits State Tomography – Theory J.M. Chow, Thesis (2010)

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**State Tomography – Classical Action**

Three-qubit quantum error correction with superconducting circuits State Tomography – Classical Action 85±1% M. Reed et al. Nature 482, 382 (2012)

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**Three-qubit quantum error correction with superconducting circuits**

Roadmap

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**Process Tomography – State Evolution**

Three-qubit quantum error correction with superconducting circuits Process Tomography – State Evolution M. Reed et al. Nature 482, 382 (2012)

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**Alternate Toffoli Gate**

Three-qubit quantum error correction with superconducting circuits Alternate Toffoli Gate 76%, 69% 90 ns vs 85%, 78% 63 ns Federov et al., Nature 481, 170 (2012)

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**Bit-Flip Error Correction**

Three-qubit quantum error correction with superconducting circuits Three-qubit quantum error correction with superconducting circuits M. Reed et al. Nature 482, 382 (2012) Bit-Flip Error Correction Error causes rotation by angle 𝜃 Step 1: encoding 𝛼 0 +𝛽 1 ↦𝛼 000 +𝛽|111〉 Step 2: error channel; 𝒑= 𝒔𝒊𝒏 𝟐 𝜽 𝟐 𝛼 000 +𝛽 111 ↦ 1−𝑝 𝛼 0 +𝛽 1 ⊗ 00 + 𝑝 𝛽 0 +𝛼 1 ⊗ 11 Step 3: recovery 𝑝 𝛽 0 +𝛼 1 ⊗ 11 ↦ 𝑝 𝛼 0 +𝛽 1 ⊗ 11 Result 𝛼 0 +𝛽 1 ⊗( 1−𝑝 |00〉+ 𝑝 11 ) 𝟖𝟏.𝟑% 𝟔𝟗.𝟕% 𝟕𝟑.𝟏% 𝟔𝟏.𝟐%

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**Phase-Flip Error Correction**

Three-qubit quantum error correction with superconducting circuits Phase-Flip Error Correction Single qubit error rate =𝑝 Probability of more than one error =3 p 2 (1−p)+ p 3 =3 𝑝 2 −2 𝑝 3 Fidelity of error correction = 1−3 𝑝 2 +2 𝑝 3 Fidelity = 𝑶 𝒑 𝟐 Single qubit error rate =𝑝 Probability of more than one error =3 p 2 (1−p)+ p 3 =3 𝑝 2 −2 𝑝 3 Fidelity of error correction = 1−3 𝑝 2 +2 𝑝 3 Fidelity = 𝑶 𝒑 𝟐 M. Reed et al. Nature 482, 382 (2012)

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**Three-qubit quantum error correction with superconducting circuits**

Future Directions Shor’s 9-Qubit Code Fault-tolerant error correction Introducing measurement-based error correction

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**Three-qubit quantum error correction with superconducting circuits**

Summary Questions? Implemented encoding into GHZ-like states using two CNOT gates CNOT gates from sudden excitations of 11 to 02 and waiting for phase accumulation in the two-excitation manifold (Strauch, Reed) Characterized error channel with at most one bit-flip and at most one phase-flip Implemented recovery using three-qubit CCPHASE from adiabatic interaction between 102 and 003 in the three-excitation manifold (Reed, also Federov) Used the circuit QED architecture with transmon qubits

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**Thank you for your attention!**

Three-qubit quantum error correction with superconducting circuits Three-qubit quantum error correction with superconducting circuits Thank you for your attention! Thanks to Matt Reed for helpful discussions.

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