31St Jerusalem Winter School in Theoretical Physics: Problem Set 1 Contents

Home , Phase qubit

31st Jerusalem Winter School in Theoretical Physics: Problem Set 1

Contents

Scott Aaronson: Quantum Complexity and Quantum Optics 2

Renato Renner: Quantum Foundations and Thermodynamics 4

Michal Horodecki: Quantum Information and Thermodynamics 7

David DiVincenzo: Quantum Computing with Superconducting Circuits 8

Barbara Terhal: Quantum Error Correction 10

1 Scott Aaronson Quantum Complexity and Quantum Optics

Scott Aaronson: Quantum Complexity and Quantum Optics

Lecture 1 - Extended Church-Turing Thesis and Basic Complexity

SA1.1. Give a physics argument for why a computer that exceeded 1043 Hertz would exceed the ∼ Schwarzschild bound and collapse to a black hole.

SA1.2. Suppose we wanted to use special-relativistic time dilation to obtain a factor-of-T computational speedup, using a spaceship with some ﬁxed (nonzero) mass. Calculate how the amount of energy needed to do this scales with T .

SA1.3. Explain why, if P=NP, then we could actually *ﬁnd* solutions to NP problems in polynomial time (rather than just deciding whether they exist).

SA1.4. Recall, look up, or work out for yourself Turing’s proof of the unsolvability of the halting problem. By ”scaling down” that proof, show that P = EXP. 6 SA1.5. In the deﬁnition of BQP, suppose we let the ”control algorithm,” the thing that outputs the descrip- tion of the quantum circuit to be applied, *itself* be a polynomial-time quantum algorithm. Explain why this would just give us BQP again, and not a more powerful complexity class.

SA1.6. Explain why, if NP=coNP, then all of PH collapses to NP; and why, if some level of PH equals P#P, then all of PH collapses to that level.

Lecture 2 - Linear Optics, the Permanent, and BosonSampling

SA2.1. Prove that the number of basis states, for linear optics with n photons and m modes, is exactly m+n−1 n . SA2.2. Prove the result of Reck et al. 1994: that any m m unitary transformation can be implemented × using O(m2) beamsplitters and phaseshifters.

SA2.3. Prove the Glynn formula for the permanent. Closely related to that, prove the permanent formula for linear optics (i.e., S φ(U) T = Per(US,T )/√s1! sm!t1! tm!) using bosonic creation operators. h | | i ··· ··· SA2.4. Using the Glynn formula, prove that Per(U) 1 for any unitary matrix U, and more generally | | ≤ that Per (A) A n for any A (where A denotes the largest singular value). Using this result, explain | | | ≤ k k k k why Gurvits’s randomized approximation algorithm for the permanent works.

SA2.5. Given any n n complex matrix A, show how to embed A/ A as an n n submatrix of a 2n 2n × k k × × unitary matrix.

SA2.6. Prove that, given an n n complex matrix A, estimating Per(A) 2 to within a multiplicative factor × | | is already a #P-complete problem.

2 Scott Aaronson Quantum Complexity and Quantum Optics

Lecture 3 - Postselection, KLM, and Hardness of Approximate BosonSampling

SA3.1. Show that PPP = P#P.

SA3.2. Explain how to use ”Feynman path integrals” to prove the containment BQP PP, and indeed ∈ PostBQP PP (improving BQP P#P from class). ∈ ∈ SA3.3. [Challenge] Without consulting the KLM paper, show how to implement a Controlled-SIGN gate (and hence, universal quantum computation) in the dual-rail representation, using postselected linear optics.

0 SA3.4. Prove that the variation distance D D equals maxf ( Ex∈D[f(x)] Ex∈D0 [f(x)] ) over all 0, 1 - k − k | − | { } valued functions f.

SA3.5. If X is a matrix of i.i.d. entries with mean 0 and variance 1, show that E[ Per(X) 2] = n!. | |

m SA3.6. Prove that, if we consider the ﬁrst k entries x1, . . . , xk of a Haar-random unit vector x C (where ∈ k m), the result is close in variation distance to a vector of k independent Gaussians. Estimate the error in this approximation.

Lecture 4 - Implementation, Scaling, and Veriﬁcation of BosonSampling Devices

No problems.

3 Renato Renner Quantum Foundations and Thermodynamics

Renato Renner: Quantum Foundations and Thermodynamics

RR1. Bell-type experiment

A ψ+ B | i + Consider a two-dimensional Hilbert space with basis v1, v2 and the Bell-state ψ , which is H { } ∈ H ⊗ H deﬁned by

+ 1 ψ = ( v1 v1 + v2 v2 ) . | i √2 | i| i | i| i

Furthermore, consider the measurement1

α ⊥ ⊥ ⊥ α α ⊥ α α = α α α, α α α , where α = cos v1 + sin v2 and α = sin v1 + cos v2 . M | ih | | ih | | i 2 | i 2 | i | i − 2 | i 2 | i Assume that the two parts of a bipartite system prepared in state ψ+ propagate towards two distant observers A and B. Observer A then performs a measurement α on his part of the system. MA a) Give the expressions for the partial state at B after the measurement at A, depending on whether the result of the measurement at A is known or not.

b) Determine the probability distribution of a measurement 0 at B conditioned/not conditioned on MB the outcome of the measurement α at A. MA c) Convince yourself with a) and b) that the subjective assignment of states at B does not contradict the objective measurement results of B.

RR2. No-signalling correlations Consider again the Bell-type experiment of the previous exercise, with a bipartite system prepared in state ψ+. Assume that observer A chooses between two possible measurements, labelled 0 and 2. Similarly, observer B chooses between measurements labelled 1 and 3. These are speciﬁed by the following table, where the two possible measurement outcomes are denoted by “+” and “ ”. (The states α are deﬁned as − | i in the previous exercise. The pictures show the measurement bases in the Bloch sphere representation.)

+ + +

- +

- - - label 0 1 2 3 π 0 + π 2 + A α = 0, | i → α = 2 , | −iπ → ( π ( 2 | i → − π | i → − 3π π 4 + 3π 4 + B α = 4 , | 5πi → α = 4 , | −πi → (| 4 i → − (| 4 i → −

a) Verify that the joint probabilities PXY |ab(x, y) of the measurement outcomes X = x and Y = y conditioned on the measurement choices A = a and B = b are given by

1The factor 1/2 is motivated by the Bloch sphere notation.

4 Renato Renner Quantum Foundations and Thermodynamics

a = 0 a = 2 + + 1− − 1− − + 2 2 2 2 2 b = 1 1− 1− where = sin (π/8) 0.146. 2 2 2 2 ≈ − 1− 1− + 2 2 2 2 b = 3 1− 1− − 2 2 2 2 0 b) Verify that this distribution is no-signalling, i.e., PX|ab = PX|ab0 for all a, b, b . c) A “PR-box” is a hypothetical system that takes as input values A and B and generates outcomes X PR and Y , with probabilities PXY |ab(x, y) given by the following table. a = 0 a = 2 + + − − + 1 0 1 0 b = 1 2 2 0 1 0 1 − 2 2 + 0 1 1 0 b = 3 2 2 1 0 0 1 − 2 2 Show that the PR-box is no-signalling. d) Show that the PR-box is non-local, i.e., its behaviour cannot be reproduced by any local classical mechanism sitting at A and B’s position (without communication). e) Show that the same is true for the correlations obtained by the quantum measurements above. f) How could a PR-box be used to emulate the quantum correlations?

RR3. Smooth min-entropy in the i.i.d. limit The smooth min-entropy of a random variable X with distribution P over is deﬁned as X Hmin(X)P = maxQ∈B(P ) Hmin(X)Q, where the maximum is taken over the set (P ) of all probability distributions Q that are -close to P . B ×n n Furthermore, we deﬁne an i.i.d. random variable X~ = X1,X2,...,Xn on with distribution P (~x) = n { } X i=1 P (xi). Use the weak law of large numbers to show that the smooth min-entropy converges to the Shannon entropy,Q H, in the i.i.d. limit:

1 ~ lim lim Hmin(X)P n = H(X)P . →0 n→∞ n

RR4. Information-to-work conversion In this exercise, we will analyse one possible way to extract work kT ln 2 from one bit of information, in an environment of temperature T . There have been many attempts to model both the interaction with a heat bath and the storage of energy as information. Here, we use a very simple model — but it contains already the main ideas behind the information-to-work conversion. Our bit of information is stored in a system S, which is just a qubit with a degenerate Hamiltonian H = 0 that is initialised in a pure state 0 . We make the following assumptions: | i a) We are allowed to change the Hamiltonian of the system (for example, by tuning a magnetic ﬁeld). If we raise or lower an occupied energy level by ∆E, we lose or gain that energy in the form of work (similarly to raising or lowering a weight).

5 Renato Renner Quantum Foundations and Thermodynamics

b) We may put the system in contact with a heat bath at temperature T . This takes the system to a thermal state, ρ = e−H/kT /Z, where k is Boltzmann’s constant and Z = Tr(e−H/kT ) is the partition function. This models a thermalisation of the system. c) We may apply any unitary on S for free (i.e., at no work cost) when its Hamiltonian is degenerate. Our protocol will take the system S to a fully mixed state, while extracting work kT ln 2. It has two steps: First we raise the energy of state 1 quickly, such that H = Emax 1 1 1, with Emax very large. Since | i | ih | the system was in state 0 , this does not cost us any work. Now we bring in the heat bath and slowly lower | i the energy of 1 , while letting the system thermalise, until the Hamiltonian is degenerate again. | i a) In our model, when the system has Hamiltonian H = E 1 1 1 and is thermalised, the state 1 is | ih | | i occupied with probability p(E) = e−E/kT /Z. If we lower the energy of 1 by δE, we have that | i probability of extracting work δE. Compute the average work extracted in the quasi-static limit, in which we change the Hamiltonian very slowly,

0 W = p(E)dE . ZEmax

For simplicity, you can take the limit Emax . What is the ﬁnal state of the system? → ∞ b) How would you reverse the protocol so that we erase one bit of information (that is, start with a fully mixed state and take it to 0 ) at work cost kT ln 2? | i c) Now imagine that you want to take S from 0 to a mixed state ρ = (1 p) 0 0 0 + p 1 1 1. How | i − | ih | | ih | would you proceed? What is the average work gain of the transformation, and how does it relate to the entropy of the ﬁnal state?

d) To complicate things, try to take an arbitrary state σ to a new state ρ, using the operations allowed in our model. How would you proceed? What is the work cost of the transformation? The states may not be diagonal in the energy basis, so you might need to take care of that.

6 Michal Horodecki Quantum Information and Thermodynamics

Michal Horodecki: Quantum Information and Thermodynamics

MH1. Show how to obtain Shannon entropy from the formalism of noisy operations.

MH2. Show that a state ρ⊗n that is many copies of a Gibbs state ρ has degeneracy of energy levels satisfying g(E +∆E) = g((E) exp(β∆E) for sufﬁciently small ∆E and energies E in a typical set. (i.e. in the set of levels that are occupied with probability tending to 1, while number of copies tends to inﬁnity). For simplicity, you may assume that ρ describes a system with just 2 energy levels and that they are nondegenerate.

MH3. Prove that work of formation is determined by max-free energy (using the thermomajorization cri- terion), and construct the unitary transformation that does the job.

7 Problem set 12

Problem 1: Modiﬁed Cooper Pair Box

FIG. 1: Modiﬁed Cooper Pair Box

DeriveDavid a HamiltonianDiVincenzo for the superconducting Quantum circuit Computing in Fig. with 1. Superconducting Qubits Hint: See equation (36) in [1]. You need to ﬁnd the capacitance matrices deﬁned by (22) David DiVincenzo: Quantum Computing with Superconducting Cir- and (23).cuits

DD1. Phase qubit. ProblemThe circuit 2: Phase diagram qubit of the phase qubit is given below.

The phase qubit Hamiltonian is given by:FIG. 2: Phase qubit

Qˆ2 Φˆ 2 H = EJ cos 2πΦˆ/Φ0 + , (1) 2CJ − 2L The circuit diagram of the phase qubit is given in Fig. 2. The phase qubit hamiltonian where Qˆ is the charge on the junction capacitance and Φˆ the ﬂux across the junction with Josephson energy is given by: 2 4 Φ0 −1 2π 2 EJ = L . Assume that CJ = 100 ~ L. The phase qubit potential has the form given in Figure 2π J ˆ2 Φ0 ˆ 2 1. Q H = EJ cos 2⇡ˆ/0 + , (1) 2CJ 2L ⇣ ⌘ where Qˆ is the charge on junction capacitance and ˆ the ﬂux across the junction with 4 2 1 2⇡ 2 Josephson energy E = 0 L . Assume that C =100 L. The phase qubit J 2⇡ J J 0 ~ potential has the form given in Fig. 3. ⇣ ⌘

FIG. 3: Phase qubit potential Figure 1: Phase qubit potential

1. Choose LJ (in units of L )sothatthereare4(metastable)levelsinthequbitwell. a) Choose LJ (in units of L) so that there are 4 (metastable) levels in the qubit well. Hint: Make a Hint: Make a harmonic approximation in the qubit well. You should have LJ = ↵L harmonic approximation in the qubit well. You should have LJ = αL for some α (0, 1). ∈ for some ↵ (0, 1). 2 8 2. Estimate the tunneling lifetime of these levels. (Hint: Use a WKB approximation with a harmonic approximation in the barrier region. It is OK to ﬁnd rates up to the factors of O (1)).

3. Estimate the number of energy levels in the measurement well up to E⇤ (see Fig. 3).

4. What should the lifetime of these states be so that the measurement well can be considered a continuum?

[1] Circuit theory for decoherence in superconducting charge qubits, Guido Burkard, Phys. Rev. B 71, 144511 (2005), arXiv:cond-mat/0408588v2.

2 David DiVincenzo Quantum Computing with Superconducting Qubits

b) Estimate the tunneling lifetime of these levels. Hint: Use a WKB approximation with a harmonic approximation in the barrier region. It is OK to ﬁnd rates up to the factors of O(1). c) Estimate the number of energy levels in the measurement well up to E∗ (see Figure 1). d) What should the lifetime of these states be so that the measurement well can be considered a continuum?

DD2. The gyrator. Please analyze the classical and quantum dynamics of the following electric circuit:

The four-terminal object in the middle is a gyrator. You can learn all the basics that you need about it by consulting http://en.wikipedia.org/wiki/Gyrator. In particular, use the current-voltage relations given there to derive the equations of motion of the two Josephson phases, and thus also the Lagrangian or Hamiltonian of the system. Try ﬁrst analyzing the linear circuit that you get by removing the two Josephson elements – the dynamics is still well deﬁned in terms of the phases of the two capacitors. Hint: For the case with the junctions present, you should end up with something very much like in Phys. Rev. B 14, 2239-2249 (1976).

DD3. Do we need fermions? This is a very open ended problem: it is more the posing of a research direction. It is motivated by the objection that is often made to the theory of superconducting qubits, that one should really derive things from the start, using electrons, BCS theory, tunneling Hamiltonians, etc. To see that one really shouldn’t need to do all this, please design a “super... qubit” using superﬂuid He-4. Read up on what replaces the Josephson junction, the capacitor, and maybe the inductor in this system. A good place where one begins reading about this is around p. 97 of Tony Leggett’s book, that is http://www.amazon.com/Quantum-Liquids-Condensation-Condensed-Matter-Graduate/dp/0198526431/. The experimental achievement of the Josephson effect in He-4 is fairly recent, see http://www.physics.berkeley.edu/research/packard/past_research/Emile/emile_h_files/Hoskinson_ Nature,433,376,2005.pdf A worthy goal of this problem would be to get an order-of-magnitude estimate of the qubit frequency of a ”helium transmon” or a ”helium ﬂux qubit”. Describing the most relevant decoherence mechanisms for these qubits would also be worth several PhD degrees.

9 Barbara Terhal Quantum Error Correction

Barbara Terhal: Quantum Error Correction

BT1. The smallest non-trivial quantum code is the [[4, 2, 2]] error-detecting code. Its linearly independent parity checks are X1X2X3X4 and Z1Z2Z3Z4: the code encodes 4 2 = 2 qubits. You can verify that one can − choose X1 = X1X2, Z1 = Z1Z3 and X2 = X2X4, Z2 = Z3Z4 as the logical operators which commute with the parity checks. The code distance is 2 which means that the code cannot correct a single qubit error. The code can however still detect any single qubit error as any single qubit error anti-commutes with at least one of the parity checks which leads to a nontrivial 1 syndrome. −

Exercise: Consider the stabilizer code C6 (deﬁned in http://arxiv.org/abs/quant-ph/0410199) with parity checks X1X4X5X6, X1X2X3X6, Z1Z4Z5Z6 and Z1Z2Z3Z6 acting on 6 qubits. How many qubits does it encode, what are its logical operators and distance? One can concatenate the code C6 with the code [[4, 2, 2]] (called C4 in the linked article) by replacing the three pairs of qubits, i.e. the pairs 12, 34 and 56, in C6 by three pairs of C4-encoded qubits, to obtain a new code. What are the parameters d, k, n of this concatenated code?

BT2.

MZZ 0 MX MXX

Figure 2: CNOT via 2-qubit quantum measurements. Here MXX measures the operator X X etc. The ⊗ ancilla qubit in the middle is discarded after the last measurement MX disentangles it from the other two input qubits. Each measurement has equal probability for outcome 1 and Pauli corrections depending on ± these measurement outcomes are done on the two output qubits.

Exercise: Verify the identity in Fig. 2 by either following the input states c 1 0 2 t 3 (where 1 is the top | i | i | i and 3 the bottom qubit in the circuit) or following the stabilizer checks and logical operators of such input state, using the Knill-Gottesman simulation technique, see Section 7 in http://arxiv.org/abs/quant-ph/ 9807006. What are the Pauli corrections on the output qubits?