What is possible to do with noisy quantum computers?

What is possible to do with noisy quantum computers? Decoherence, inaccuracy and errors in Quantum Information Processing Sara Felloni and Giuliano Strini sara.felloni@disco.unimib.it Dipartimento di Informatica Sistemistica e Comunicazione Milano Bicocca http://www.disco.unimib.it Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. /58

Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 2/58

Decoherence, Errors and s Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Studies of the impact of noise on the stability of quantum computation and communication protocols are of primary importance for practical implementations. Qubits unavoidably interact with the external world (decoherence). Under realistic conditions, in every experimental implementation, quantum operations are unavoidably affected by errors. Noise thresholds for computations on sufficiently small - nonetheless useful - quantum computers will benefit experimental research. Theoretical studies in literature offer very much lower noise thresholds which are necessary for longer quantum simulations. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 3/58

Quantum Error Models Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Single-qubit error models that can be currently found in literature only consider 2 well known parameters: t Decay (displacement along the z axis) t 2 Decoherence (bit flip channel) These 2 parameters are far from being sufficient to give a complete and realistic description of a general experimental situation. The general single-qubit error model presented here provides a description of all physically possible single-qubit errors. This description is based on density matrices and Kraus operators. Its efficiency allows to include all the single-qubit errors, thus describing the quantum systems under experimentally realistic conditions. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 4/58

Evolution of a Single-qubit Density Matrix Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation According to the Kraus representation, the most general evolution of a single-qubit density matrix is: F i ρf i F i F i = I ρ = i Density matrices can be expressed by means of Pauli operators σ: ρ [I + λ σ] 2 This leads to an affine transformation: λ = M λ + c i of a Quantum Entanglement Purification Protocol Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 5/58

2 Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol 2 parameters are needed to characterize a generic quantum noise operation acting on a single qubit. Decomposing M in diagonal and orthogonal matrices: M = O DO T a geometric interpretation of the 2 parameters can be given by associating each parameter to a transformation of the Bloch sphere representing the qubit s state: 3 : Rotations of the Bloch sphere about the axes x, y or z; 3 : Displacements of the Bloch sphere along the same axes; 3 : Deformations of the Bloch sphere into an ellipsoid, with x, y or z as symmetry axes; 3 : Deformations of the Bloch sphere into an ellipsoid with symmetry axis along an arbitrary direction. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 6/58

Measurement of the Error Parameters /5 Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 Single-qubit density matrix tranformation: X a a 2 a 3 X Y = a 2 a 22 a 23 Y Z a 3 a 32 a 33 Z + Measurements of the final density matrix with suitable choice of the initial state allow to measure each of the 2 parameters. c c 2 c 3 of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 7/58

Measurement of the Error Parameters 2/5 Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Since [ C C ] [ C C the following equations hold: ] = [ C 2 C C C C C 2 C C = 2 (X iy ) C 2 = 2 ( + Z) C 2 = ( Z) 2 The initial state: { C = C = leads to the system of equations: { X = a 3 + c Y = a 23 + c 2 Z = a 33 + c 3 X = Y = Z = ] Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 8/58

Measurement of the Error Parameters 3/5 Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation A second initial state: { C = C = leads to a different system of equations: { X = a 3 + c Y = a 23 + c 2 Z = a 33 + c 3 X = Y = Z = These 6 equations determine 6 out of the 2 parameters. of a Quantum Entanglement Purification Protocol Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 9/58

Measurement of the Error Parameters 4/5 Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation A third initial state: C = C = 2 { leads to the system of equations: X = a + c Y = a 2 + c 2 Z = a 3 + c 3 X = Y = Z = and 3 further measurements determine the coefficients a i ; i =, 2, 3. of a Quantum Entanglement Purification Protocol Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. /58

Measurement of the Error Parameters 5/5 Decoherence, Errors and s Quantum Error Models Evolution of a Single-qubit Density Matrix 2 Single-qubit Noise Channels Parameters /5 Parameters 2/5 Parameters 3/5 Parameters 4/5 Parameters 5/5 of a Quantum Simulator for the Schrödinger Equation Finally, the initial state: { C = 2 C = i 2 leads to the system of equations: X = a 2 + c Y = a 22 + c 2 Z = a 32 + c 3 { Y = X = Z = and 3 further measurements determine the coefficients a i2 ; i =, 2, 3. of a Quantum Entanglement Purification Protocol Thus all the 2 parameters are determined by means of 2 measurements. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. /58

Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 2/58 Minimum Deformation for Displacement

Kraus Representation and Quantum Circuits Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 3/58 Minimum Deformation for Displacement For each noise channel it is possible to provide: the Kraus representation the transformation of the Bloch sphere coordinates an equivalent quantum circuit leading to a unitary representation in an extended Hilbert space. A great advantage of these equivalent quantum circuits is that the evolution of the reduced density matrix describing the single-qubit system is automatically guaranteed to be completely positive.

Rotations of the Bloch /2 Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 4/58 Minimum Deformation for Displacement Rotations through an angle θ about an arbitrary axis directed along the unit vector n are given by the operator: ( R n (θ) = cos θ ) ( I i sin θ ) nσ 2 2 Any generic rotation can be obtained by composing rotations about the axes x, y and z.

Rotations of the Bloch Rotations through an angle θ and corresponding transformations of the Bloch sphere coordinates: R x = R y = R z = [ [ [ cos θ 2 isin θ 2 isin θ 2 cos θ 2 cos θ 2 sin θ 2 sin θ 2 cos θ 2 cos θ 2 isin θ 2 cos θ 2 + isin θ 2 ] ] ] x = x y = cosθy sinθz z = sinθy + cos θz x = cosθx sinθz y = y z = sinθx + cosθz x = cosθ x sinθ y y = sinθ x + cosθ y z = z Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 5/58

Deformations of the Bloch /2 Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 6/58 Minimum Deformation for Displacement The well known bit flip, phase flip and bit-phase flip channels correspond to deformations of the Bloch sphere into an ellipsoid. Bit flip x maps the Bloch sphere Phase flip into an ellipsoid z with symmetry axis Bit-phase flip y A single auxiliary qubit, initially prepared in the state ψ = cos θ 2 + sin θ 2 θ π is sufficient to obtain unitary representations for these noise channels.

Deformations of the Bloch Quantum circuits for deformations of the Bloch sphere: Bit flip Bit-phase flip Phase flip Corresponding transformations of the Bloch sphere coordinates: x = x y = cosθy z = cosθz x = cosθx y = y z = cosθz x = cos θx y = cosθy z = z Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 7/58

Bit flip: Graphic Action of the bit flip channel: θ = π 4, π 3..5.5.5 -.5 -.5 -.5.5.5.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 8/58

Phase flip: Graphic Action of the phase flip channel: θ = π 4, π 3..5.5.5 -.5 -.5 -.5.5.5.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 9/58

Bit-phase flip: Graphic Action of the bit-phase flip channel: θ = π 4, π 3..5.5.5 -.5 -.5 -.5.5.5.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 2/58

Displacements of the Bloch Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 2/58 Minimum Deformation for Displacement A displacement of the center of the Bloch sphere must go with a deformation of the sphere. This is necessary for the resulting ρ to still represent a density matrix: the associated Bloch radius r must be r. This condition can be easily fulfilled by describing displacement channels by means of quantum circuits. Displacements of the center of the Bloch sphere along the +z axis can be seen as representative of zero temperature dissipation. Displacements of the center of the Bloch sphere along the -z axis can be seen as representative of thermal excitations. Displacements of the center of the Bloch sphere along the x,y axes are also considered.

Displacements along +z semi axis Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 22/58 Minimum Deformation for Displacement This single-qubit quantum operation is also known as the amplitude damping channel. The Kraus operators and the corresponding transformation of the Bloch sphere coordinates are: [ ] [ ] F = cos θ F = sinθ x = cosθx y = cosθy z = sin 2 θ + cos 2 θz

Displacements along -z semi axis Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 23/58 Minimum Deformation for Displacement The Kraus operators and the corresponding transformation of the Bloch sphere coordinates are: F = [ cosθ ] F = [ sinθ ] x = cosθx y = cosθy z = sin 2 θ + cos 2 θz

z Displacements: Graphic Displacement along +z and -z semi axes: θ = π 6, π 4, π 3..5.5.5.5 -.5 -.5 -.5 -.5.5.5.5.5 -.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 -.5.5.5.5.5.5 -.5 -.5 -.5 -.5.5.5.5.5 -.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 -.5.5 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 24/58

Displacements along ± x semi axes [ The unitary transformation U is described by U = 2 ± Kraus operators and transformation of the Bloch sphere coordinates: [ ] [ F = + cosθ ±( cos θ) 2 F = sinθ sin θ 2 ±( cosθ) + cosθ sinθ ± sinθ x = ± sin 2 θ + cos 2 θx y = cosθy z = cosθz ] ] Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 25/58

x Displacements: Graphic Displacement along +x and -x semi axes: θ = π 6, π 4, π 3..5.5.5.5 -.5 -.5 -.5 -.5.5.5.5.5 -.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 -.5.5.5.5.5.5 -.5 -.5 -.5 -.5.5.5.5.5 -.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 -.5.5 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 26/58

Displacements along ± y semi axes [ The unitary transformation U is described by U = 2 ±i ±i Kraus operators and transformation of the Bloch sphere coordinates: [ ] [ F = + cosθ ±i( cos θ) 2 F = ±isinθ sin θ 2 i( cosθ) + cosθ sinθ isin θ x = cosθx y = ± sin 2 θ + cos 2 θy z = cosθz ] ] Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 27/58

y Displacements: Graphic Displacement along +y and -y semi axes: θ = π 6, π 4, π 3..5.5.5.5 -.5 -.5 -.5 -.5.5.5.5.5 -.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 -.5.5.5.5.5.5 -.5 -.5 -.5 -.5.5.5.5.5 -.5 -.5 -.5 -.5 -.5.5 -.5.5 -.5.5 -.5.5 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 28/58

Minimum Deformation for Displacement Applying a displacement of the center of the Bloch sphere along the +z direction, the new center becomes Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 29/58 Minimum Deformation for Displacement (,, b) < b < The Bloch sphere x 2 + y 2 + z 2 = is deformed into an ellipsoid with z as symmetry axis: x 2 + y 2 a 2 + [z ( b)]2 b 2 = Imposing a higher order tangency to the Bloch sphere: b = a 2 and defining a = cosθ ( < θ < π/2), the ellipse becomes: x 2 + y 2 cos 2 θ + (z sin2 θ) 2 cos 4 θ =

Minimum Deformation for Displacement Kraus Representation and Quantum Circuits /2 /2 Bit flip: Graphic Phase flip: Graphic Bit-phase flip: Graphic Displacements along +z semi axis Displacements along -z semi axis z Displacements: Graphic Displacements along ± x semi axes x Displacements: Graphic Displacements along ± y semi axes y Displacements: Graphic Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 3/58 Minimum Deformation for Displacement This equation corresponds to the minimum deformation required to the Bloch sphere in order to displace its center along the z-axis by b = sin 2 θ.

Error Model Applications A geometric interpretation of 9 out of the 2 parameters describing a generic single-qubit quantum operation has been given. 3 parameters correspond to x, y, z rotations; 3 to x, y, z displacements; 3 to x, y, z deformations of the Bloch sphere into an ellipsoid; the remaining 3 parameters correspond to deformations of the Bloch sphere into an ellipsoid with arbitrary symmetry axis, that can be obtained by combining the 9 previous quantum operations. This single-qubit error model can be applied to study quantum protocols stability under quantum noise effect:. A quantum simulator for the resolution of the Schrödinger equation 2. A quantum iterative entanglement purification protocol Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 3/58

of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 32/58 of a Quantum Simulator for the Schrödinger Equation

A Quantum Simulator A quantum simulator for resolution of basic problems of quantum mechanics. of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 33/58. How many qubits are necessary to simulate interesting dynamics? 2. How does the system work under quantum noise effects? Only 6 qubits are sufficient to build a useful quantum simulator. The exceptionally small number of qubits allows to include the single-qubit decoherence model and to describe the system by means of the density matrix representation. The tolerable noise threshold is much higher in respect to the theoretical noise threshold necessary for much longer quantum simulations in fault-tolerant quantum computation.

Ideal Quantum Harmonic Oscillator and Error Distribution Consider the quantum mechanical motion of a particle in one dimension, governed by the Schrödinger equation: of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 34/58 i d ψ(x, t) = Hψ(x, t) dt where the Hamiltonian H is given by: H = H + V (x) = 2 2m d 2 dx 2 + V (x) Initial condition Gaussian wave packet of given width and center. Error distribution Uniform random distribution with maximum noise strenght fixed by the parameter θ. All kind of errors are applied to every single qubit, one by one.

Wave Packet Graphic Representation Horizontal axis: time-steps iterations Vertical axis: 64 states of the 6 qubit quantum simulator Gray shading: probability of the system being in the considered state Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 35/58

Some Meaningful Examples of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 36/58 In order to illustrate the working of the 6 qubits quantum simulator, other significant examples have been considered: Harmonic oscillator A particle in a box Transmission through a barrier Notice that not all these examples are analytically solvable. 6 qubits are sufficient to treat complex and meaningful cases.

Rotations of the Bloch /2 Rotations and corresponding fidelities, relative to random errors with maximum noise strenght θ =.5, about: x axis y axis z axis Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 37/58

Rotations of the Bloch Table of the final fidelities, relative to random errors with maximum noise strenght θ =.5. Rotation q q q 2 q 3 q 4 q 5 U x.95.83.48.3.7.33 U y.48.5.68.62.6.66 U z.23.38.37.2.3.9 The quality of the simulator s states is measured by the fidelity F : F = ψ ψ ψ ψ = ψ ψ 2 which measures the probability that an imperfect state would pass a test for being in the exact state. Fidelity decays are obtained from simulations of 4 time-step iterations. Notice that even with such high maximum noise strenght, θ =.5, the solution can be clearly detected by the plots. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 38/58

Deformations of the Bloch /2 Deformations and corresponding fidelities, relative to random errors with maximum noise strenght θ: x axis θ =.2 y axis θ =.5 z axis θ =.3 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 39/58

Deformations of the Bloch Table of the final fidelities, relative to random errors with maximum noise strenght θ. of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 4/58 Deformation θ q q q 2 q 3 q 4 q 5 Bit flip.2.9.75.43.42.33.29 Bit-phase flip.5.56.66.75.54.52.49 Phase flip.3.53.57.52.5.6.86

Displacements of the Bloch /3 Displacements and corresponding fidelities, relative to random errors with maximum noise strenght θ =.2, along: +x semi axis +y semi axis +z semi axis Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 4/58

Displacements of the Bloch 2/3 Displacements and corresponding fidelities, relative to random errors with maximum noise strength θ =.2, along: -x semi axis -y semi axis -z semi axis Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 42/58

Displacements of the Bloch 3/3 Table of the final fidelities, relative to random errors with maximum noise strenght θ =.2. of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 43/58 Displacement q q q 2 q 3 q 4 q 5 +x semi axis.99.97.82.47.59.67 x semi axis.32.42.52.67.59.68 +y semi axis.73.7.58.58.59.67 y semi axis.76.76.62.59.59.67 +z semi axis.76.78.7.59.57.52 z semi axis.76.78.7.6.55.53

Conclusions of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 44/58 The quantum system can be successfully treated even if affected by strong perturbation. Iterating the system for a longer time-interval causes error to gradually accumulate; but, in order to study periodical behaviors, it is sufficient to consider a few periods. The algorithm used is so efficient to successfully simulate interesting quantum dynamics by means of only 6 qubits. The number of qubits required by the quantum simulator is sufficiently small to successfully simulate all the 2 single-qubit quantum noise channels. The number of gates required by the algorithm is sufficiently small to study quantum noise effects to simulate imperfect gates implementation.

Future Research of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 45/58 Simulations will be extended to more complex systems. Some meaningful two-qubit errors will also be considered. An exhaustive study of a general transformation of a two-qubit density matrix appears hardly feasible, since there are 24 parameters involved. Thus special cases of realistic physical errors in the main experimental implementations will be addressed. The general error model will be applied to study the phenomenon known as sudden de-entanglement (or sudden death of entanglement), to discover which noise channels originates this rapid decay of the initial entangled states.

Thanks! of a Quantum Simulator for the Schrödinger Equation A Quantum Simulator Ideal Quantum Harmonic Oscillator and Error Distribution Wave Packet Graphic Representation Some Meaningful Examples /2 /2 /3 2/3 3/3 Conclusions Future Research Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 46/58 References [] G. Benenti, G. Casati and G. Strini, Principles of Quantum Computation and Information, Vol. : Basic Concepts (World Scientific, Singapore, 24). [2] G. Benenti, G. Casati and G. Strini, Principles of Quantum Computation and Information, Vol. 2: Basic Tools and Special Topics (World Scientific, Singapore, 27). [3] G. Benenti, S. Felloni and G. Strini, Effects of single-qubit quantum noise on entanglement purification, Eur. Phys. J. D 38, 389 (26); quant-ph/5577. [4] G. Strini, A. Carati, S. Vicari, Algoritmi quantistici per la risoluzione dell equazione di Schroedinger, dipartimento di Matematica, facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi di Milano, a.a. 24/25.

of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! of a Quantum Entanglement Purification Protocol Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 47/58

Entanglement Purification and Cryptography of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! Entanglement-based quantum cryptography If a member of a maximally entangled EPR pair is transmitted from a sender (Alice) to a receiver (Bob) through a quantum channel, then noise in the channel, as well as eavesdropping (Eve) can degrade the amount of entanglement of the pair. Quantum privacy amplification (QPA) iterative protocol It eliminates entanglement with an eavesdropper by creating a small number of nearly perfect (pure) EPR states out of a large number of partially entangled states. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 48/58

Entanglement Purification Protocol of a Quantum Simulator for the Schrödinger Equation LOCC local quantum operations (quantum gates and measurements performed by Alice and Bob on their own qubits) supplemented by classical communication. of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 49/58

Ideal Entanglement Purification: Fidelity of a Quantum Simulator for the Schrödinger Equation Fidelity F measures the quality of the purified EPR pair: F = φ + ρ AB φ + and it measures the probability that the control qubits would pass a test for being in the pure state. of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! solid line: weak intrusion (perturbation) dash-line: middle intrusion dot-line: strong intrusion Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 5/58

Ideal Entanglement Purification: Survival Probability of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! EPR pairs must be discarded when Alice and Bob obtain different measurement outcomes. Survival probability P(n) measures the probability that a n-step QPA protocol is successful: if p i is the probability that Alice and Bob obtain coinciding outcomes at step i, P(n) = n i= p i. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 5/58

Noisy Entanglement Purification of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! A systematic numerical study of the impact of all possible single-qubit noise channels on the quantum privacy amplification protocol shows that both:. the qualitative behavior of the fidelity of the purified state as a function of the number of purification steps 2. the maximum level of noise tolerable by the protocol strongly depend on the specific noise channel applied. These results provide valuable information for experimental implementations of this protocol. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 52/58

Rotations of the Bloch Fidelity decays and survival probabilities of rotations relative to random errors with maximum noise strenght θ, about: x axis θ =. y axis θ =. z axis θ =.5 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 53/58

Deformations of the Bloch Fidelity decays and survival probabilities of deformations relative to random errors with maximum noise strenght θ: x axis θ =. y axis θ =.3 z axis θ =. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 54/58

Displacements of the Bloch Fidelity decays and survival probabilities of displacements relative to random errors with maximum noise strenght θ, along: x axis θ =. y axis θ =. z axis θ =.3 Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 55/58

Conclusions of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! Even though all noise channels degrade the performance of the protocol, the level of noise that can be safely tolerated strongly depends on the specific channel. Two main distinct behaviors are observed:. The fidelity is continuously improved by increasing the number of purification steps (e.g. bit flip) 2. The fidelity saturates to a value F < after a finite number of steps, so that any further iteration is useless (e.g. x displacement) Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 56/58

Fidelity Table Noise channel θ max x rotation y rotation z rotation.6 x deformation. y deformation.3 z deformation. x displacement. y displacement. y displacement.3 The QPA protocol is much less sensitive to displacements along z than along x or y. This suggests that the z-axis should be chosen along the direction of noise. The axes x and y should then be chosen to minimize other noise effects. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 57/58

Thanks! of a Quantum Simulator for the Schrödinger Equation of a Quantum Entanglement Purification Protocol Entanglement Purification and Cryptography Entanglement Purification Protocol Ideal Entanglement Purification: Fidelity Ideal Entanglement Purification: Survival Probability Noisy Entanglement Purification Conclusions Fidelity Table Thanks! References [] G. Benenti, G. Casati and G. Strini, Principles of Quantum Computation and Information, Vol. : Basic Concepts (World Scientific, Singapore, 24). [2] G. Benenti, G. Casati and G. Strini, Principles of Quantum Computation and Information, Vol. 2: Basic Tools and Special Topics (World Scientific, Singapore, 27). [3] G. Benenti, S. Felloni and G. Strini, Effects of single-qubit quantum noise on entanglement purification, Eur. Phys. J. D 38, 389 (26); quant-ph/5577. [4] G. Strini, A. Carati, S. Vicari, Algoritmi quantistici per la risoluzione dell equazione di Schroedinger, dipartimento di Matematica, facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi di Milano, a.a. 24/25. Sara Felloni and Giuliano Strini, May 28, 27 Noisy Quantum Computation - p. 58/58