Simulation and Design of Stabilizer Quantum Circuits Scott Aaronson and Boriska Toth CS252 Project December 10, X X +Z Z +ZI +IX 

Quantum Computing: New Challenges for Architecture If you speculate on a measurement, rollback will not happen Cache coherence protocols violate no- cloning theorem How do you design and debug circuits that you can’t even simulate efficiently?

Our Approach: Start With A Subset of Quantum Computations Stabilizers (Gottesman 1996): Beautiful formalism that captures much (but not all) of quantum weirdness – Quantum linear error-correcting codes – Teleportation – Dense quantum coding – GHZ (Greenberger-Horne-Zeilinger) paradox What We Did: Invented new algorithms for simulating and designing quantum circuits described by the stabilizer formalism. Implemented and tested an efficient simulator with possible practical value.

Quantum Gates We Allow 1. Controlled-NOT (CNOT): Replaces a,b by a,b  a |00   |00 , |01   |01 , |10   |11 , |11   |10  2. Hadamard: Applies /  2 to single qubit |0   (|0  +|1  )/  2 |1   (|0  -|1  )/  2 H Phase: Applies to single qubit |0   |0 , |1   i|1  P 100i100i 4. Measurement of a single qubit AMAZING FACT These gates are NOT universal—Gottesman & Knill showed how to simulate them quickly on a classical computer To see why we need some group theory…

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 an abelian group Theorem: |  can be produced from the all-0 state by just CNOT, Hadamard, and phase gates, iff |  is stabilized by 2 n tensor products of Pauli matrices or their opposites (where n = number of qubits) In that case, |  is uniquely determined by these stabilizers I =Z = X = 0-i i0 Y = Pauli Matrices: Collect ‘Em All

Goal: Using a classical computer, simulate an n- qubit CNOT/Hadamard/Phase computer. Gottesman & Knill’s 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 as we discovered when we tried to implement, 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!

Our Faster, Easier-to-Implement Solution: “Scoreboarding” Idea: Instead of n(2n+1) bits, store 2n(4n+1) bits (1) n stabilizers, 2n+1 bits each (2) n “destabilizers” (3) A 2n  2n scoreboard, that stores how to write XIIII,…,IIIIX, ZIIII,…,IIIIZ as products of the stabilizers and destabilizers Together generate full Pauli group +ZI +IZ +XI +IX XI IX ZI IZ Destabilizers Stabilizers Scoreboard Initial State: |00 

Our Faster, Easier-to-Implement Solution: “Scoreboarding” Idea: Instead of n(2n+1) bits, store 2n(4n+1) bits (1) n stabilizers, 2n+1 bits each (2) n “destabilizers” (3) A 2n  2n scoreboard, that stores how to write XIIII,…,IIIIX, ZIIII,…,IIIIZ as products of the stabilizers and destabilizers Together generate full Pauli group +XI +IZ +ZI +IX XI IX ZI IZ Destabilizers Stabilizers Scoreboard Hadamard the 1 st qubit: |00  +|10  Swap

Our Faster, Easier-to-Implement Solution: “Scoreboarding” Idea: Instead of n(2n+1) bits, store 2n(4n+1) bits (1) n stabilizers, 2n+1 bits each (2) n “destabilizers” (3) A 2n  2n scoreboard, that stores how to write XIIII,…,IIIIX, ZIIII,…,IIIIZ as products of the stabilizers and destabilizers Together generate full Pauli group +X X +Z Z +ZI +IX XI IX ZI IZ Destabilizers Stabilizers Scoreboard CNOT into the 2 nd qubit: |00  +|11  

Advantages Because we force each instruction to “tell the scoreboard” what it did, measuring a state (and updating it after the measurement) can be done in only O(n 2 ) steps. No Gaussian elimination needed! Recently measured observables are automatically “cached”—measuring them again takes only O(n) steps.

CHP: An interpreter for “quantum assembly language” programs that implements our scoreboard algorithm Example: Quantum Teleportation H H HH |  |0  |  Alice’s Qubits Bob’s Qubits Prepare EPR pair Alice’s partBob’s 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

Performance of CHP Randomly-generated circuits with equal mix of CNOT, Hadamard, phase, and measurement gates Updating the state after measurements with random outcomes dominates the running time. Amdahl’s Law suggests this is what we should optimize—and we have ideas! 650MHz Pentium III, 256MB RAM Compiler optimizations made it 50% slower! gates gates 5000 gates gates

Other Stuff We Did Proved that any stabilizer quantum circuit can be simulated using only CNOT gates In theory jargon: Simulating stabilizer circuits is “  L- complete” Proved that any stabilizer circuit has an equivalent circuit with at most O(n 2 /log n) gates, saturating the Shannon lower bound Builds on work by an architecture group at U. Michigan—K. Patel, I. Markov, and J. Hayes (quant- ph/ )—who showed this for CNOT circuits

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)…