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Improved Simulation of Stabilizer Circuits Scott Aaronson (UC Berkeley) Joint work with Daniel Gottesman (Perimeter) 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 +ZI +IX +XI +IZ

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Quantum Computing: New Challenges for Computer Architecture Cant speculate on a measurement and then roll back Cache coherence protocols violate no- cloning theorem How do you design and debug circuits that you cant even simulate efficiently with existing tools?

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Our Approach: Start With A Subset of Quantum Computations Stabilizers (Gottesman 1996): Beautiful formalism that captures much (but not all) of quantum weirdness – Linear error-correcting codes – Teleportation – Dense quantum coding – GHZ (Greenberger-Horne-Zeilinger) paradox

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Gates Allowed In Stabilizer Circuits 1. Controlled-NOT |00 |00, |01 |01, |10 |11, |11 |10 2. Hadamard |0 (|0 +|1 )/ 2 |1 (|0 -|1 )/ 2 H 3. Phase = |0 |0, |1 i|1 P 100i100i 4. Measurement of a single qubit AMAZING FACT These gates are NOT universalGottesman & Knill showed how to simulate them quickly on a classical computer To see why we need some group theory…

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X 2 =Y 2 =Z 2 =IXY=iZYZ=iXZX=iY XZ=-iYZY=-iXYX=-iZ Unitary matrix U stabilizes a quantum state | if U| = |. Stabilizers of | form a group X stabilizes |0 +|1 -X stabilizes |0 +|1 Y stabilizes |0 +i|1 -Y stabilizes |0 -i|1 Z stabilizes |0 -Z stabilizes |1 10011001 10 0-1 I =Z = 01100110 X = 0-i i0 Y = Pauli Matrices: Collect Em All

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If | can be produced from the all-0 state by just CNOT, Hadamard, and phase gates, then | is stabilized by 2 n tensor products of Pauli matrices or their opposites (where n = number of qubits) So the stabilizer group is generated by log(2 n )=n such tensor products Indeed, | is then uniquely determined by these generators, so we call | a stabilizer state Gottesman-Knill Theorem

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Goal: Using a classical computer, simulate an n- qubit CNOT/Hadamard/Phase computer. Gottesman & Knills solution: Keep track of n generators of the stabilizer group Each generator uses 2n+1 bits: 2 for each Pauli matrix and 1 for the sign. So n(2n+1) bits total Example: But measurement takes O(n 3 ) steps by Gaussian elimination +X X -ZZ |01 +|11 |01 +|10 CNOT(1 2) +XI -IZ Updating stabilizers takes only O(n) steps OUCH!

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Our Faster, Easier-To-Implement Tableau Algorithm Idea: Instead of n(2n+1) bits, store 2n(2n+1) bits n stabilizers S 1,…,S n, 2n+1 bits each n destabilizers D 1,…,D n Together generate full Pauli group Maintain the following invariants: D i s commute with each other S i anticommutes with D i S i commutes with D j for i j

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0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Destabilizers Stabilizers State: |00 +XI +IX +ZI +IZ x ij bitsz ij bitsr i bits I: x ij =0, z ij =0+ phase: r i =0 X: x ij =1, z ij =0- phase: r i =1 Y: x ij =1, z ij =1 Z: x ij =0, z ij =1 S1S2S1S2 D1D2D1D2

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0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Destabilizers Stabilizers State: |00 +XI +IX +ZI +IZ Hadamard on qubit a: For all i {1,…,2n}, swap x ia with z ia, and set r i := r i x ia z ia

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0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 Destabilizers Stabilizers State: |00 +|10 +ZI +IX +XI +IZ

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0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 Destabilizers Stabilizers State: |00 +|10 +ZI +IX +XI +IZ CNOT from qubit a to qubit b: For all i {1,…,2n}, set x ib := x ib x ia and z ia := z ia z ib

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0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 Destabilizers Stabilizers State: |00 +|11 +ZI +IX +X X +Z Z

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0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 Destabilizers Stabilizers State: |00 +|11 +ZI +IX +X X +Z Z Phase on qubit a: For all i {1,…,2n}, set r i := r i x ia z ia, then set z ia := z ia x ia

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0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 1 Destabilizers Stabilizers State: |00 +i|11 +ZI +IY +X Y +Z Z

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0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 1 Destabilizers Stabilizers State: |00 +i|11 +ZI +IY +X Y +Z Z Measurement of qubit a: If x ia =0 for all i {n+1,…,2n}, then outcome will be deterministic. Otherwise 0 with ½ probability and 1 with ½ probability.

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1010 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 Destabilizers Stabilizers State: |11 +X Y +IY -ZI +Z Z Random outcome: Pick a stabilizer S i such that x ia =1 and set D i :=S i. Then set S i :=Z a and output 0 with ½ probability, and set S i :=-Z a and output with ½ probability, where Z a is Z on a th qubit and I elsewhere. Finally, left-multiply whatever rows dont commute with S i by D i

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Novel part: How to obtain deterministic measurement outcomes in only O(n 2 ) steps, without using Gaussian elimination? Z a must commute with stabilizer, so for a unique choice of c 1,…,c n {0,1}. If we can determine c i s, then by summing corresponding S h s we learn sign of Z a. Now So just have to check if D i commutes with Z a, or equivalently if x ia =1

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CHP: An interpreter for quantum assembly language programs that implements our scoreboard algorithm Example: Quantum Teleportation H H HH | |0 | Alices Qubits Bobs Qubits Prepare EPR pair Alices partBobs part h 1 c 1 2 c 0 1 h 0 m 0 m 1 c 0 3 c 1 4 c 4 2 h 2 c 3 2 h 2 CHP Code

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Performance of CHP Average time needed to simulate a measurement after applying βnlogn random unitary gates to n qubits, on a 650MHz Pentium III with 256MB RAM

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Simulating Stabilizer Circuits is L-Complete L = class of problems reducible in logarithmic space to evaluating a circuit of CNOT gates Natural, well-studied subclass of P Conjectured that L P. So our result means stabilizer circuits probably arent even universal for classical computation! Simulating L with stabilizer circuits: Obvious Simulating stabilizer circuits with L : Harder

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How many n-qubit stabilizer states are there? Fun With Stabilizers Can we represent mixed states in the stabilizer formalism? YES Can we efficiently compute the inner product between two stabilizer states? YES

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Future Directions Measurements (at least some) in O(n) steps? Apply CHP to quantum error-correction, studying conjectures about entanglement in many-qubit systems… Efficient minimization of stabilizer circuits? Superlinear lower bounds on stabilizer circuit size? Other quantum computations with efficient classical simulations: bounded entanglement (Vidal 2003), matchgates (Valiant 2001)…

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