Quantum speed limits for change of basis

Quantum speed limits provide ultimate bounds on the time required to transform one quantum state into another. Here, we introduce a novel notion of quantum speed limits for collections of quantum states, investigating the time for converting a basis of states into an unbiased one as well as basis permutation. Establishing an unbiased basis, we provide tight bounds for the systems of dimension smaller than 5, and general bounds for multi-qubit systems and the Hilbert space dimension d. For two-qubit systems, we show that the fastest transformation implements two Hadamards and a swap of the qubits simultaneously. We further prove that for qutrit systems the evolution time depends on the particular type of the unbiased basis. Permuting a basis, we obtain the exact expression for the Hilbert space of dimension d. We also investigate speed limits for coherence generation, providing the minimal time to establish a certain amount of coherence with a unitary evolution.

Introduction.Striving for quantum advantages, such as an increased speed of a computation, has become a competitive goal.However, nature has established a fundamental speed limit, via a minimal time that is necessary for the unitary evolution of an initial quantum state to a final quantum state, as pointed out in [1,2].In a geometric approach [3][4][5][6], the quantum speed limit is linked to the length of the shortest path between initial and final state, which can be quantified via a suitable distance measure.For a recent review of quantum speed limits, see [7].
The standard approach to quantum speed limits assumes that a quantum state |ψ is transformed into another state |φ via a unitary evolution U = e −iHt .The task is to determine the optimal evolution time for the transition |ψ → |φ , with respect to the energy scale of the Hamiltonian H. First results in this direction were presented for orthogonal states, and are known as Mandelstam-Tamm bound [1]: where (∆E ψ ) 2 = H 2 ψ − H 2 ψ is the energy variance.Another bound was derived later by Margolus and Levitin [2], giving with the mean energy E ψ = H ψ − E 0 , and E 0 is the ground state energy.Note that the speed limits (1) and (2) differ only by the different choice of the energy scale.
While the original approaches [1,2] studied the speed limit for unitary transitions between two quantum states, more general versions of the speed limit have been developed in the last years.This includes investigation of quantum speed limits for open system dynamics [10][11][12][13], as well as speed limits for the evolution of observables in the Heisenberg picture [14], and the study of speed limit for a bounded energy spectrum [15].A theoretical approach for measuring quantum speed limits in an ultracold gas has been proposed recently in [16].Speed limits for generating quantum resources have also been considered [17], allowing to determine optimal rates for generating quantum entanglement [18], quantum coherence [19], and quantum discord [20,21].
Note that the early approaches [1,2] studied the speed limit for transforming one quantum state into another one.However, many quantum technological applications require to transform a collection of states.An important example is quantum computation where a common operation is a change of basis, e.g. by applying the well-known Hadamard gate which transforms the computational qubit basis {|0 , |1 } into {|+ , |− }, with |± = (|0 ± |1 )/ √ 2. Which fundamental speed limits hold for such a basis transformation?We address this question in this Letter, investigating bounds on the time that is necessary to perform a basis change, i.e. a transformation of an ordered set of quantum states to another ordered set of quantum states, minimized over all Hamiltonians.In the spirit of the Margolus-Levitin bound (2), we aim for quantum speed limits of the form Here {|ψ j }, {|φ j } are two ordered sets of orthonormal ba-arXiv:2212.12352v1[quant-ph] 23 Dec 2022 sis states, with j = 1, ..., d, where d is the dimension of the Hilbert space, and g can in general depend on the sets {|ψ j } and {|φ j }.The quantity E in Eq. ( 4) denotes the mean energy of the Hamiltonian, which we define as naturally generalizing the mean energy E ψ appearing in the Margolus-Levitin bound (2).The mean energy ( 5) is equivalent to E = Tr[H/d] − E 0 , and thus independent on the particular choice of basis {|ψ j }.We also note that the mean energy is additive for non-interactive Hamiltonians of the form where E A and E B are the mean energies of H A and H B , respectively.
In addition to investigating speed limits for the change of basis, we also study speed limits for coherence generation.In particular, we consider the maximal coherence which can be established within a certain time, given some Hamiltonian with mean energy E. These results are highly relevant in the context of the resource theory of quantum coherence [19,22,23], taking into account that several recent works suggest that quantum coherence is more suitable than entanglement to capture the performance of certain quantum algorithms [24][25][26].
Speed limits for unbiased bases.In the following, we will determine speed limits for basis change from the computational basis {|n } into an unbiased basis {|n + } with | n|n + | 2 = 1/d, see also Fig. 1.For single-qubit systems we obtain the bound which is tight for any unbiased qubit basis.See Appendix A for more details on speed limits for single-qubit transitions.
It is now intuitive to assume that for d > 2 the evolution time into an unbiased basis increases, compared to the qubit setting.To support this intuition, consider a two-qubit system AB, and let H A and H B be qubit Hamiltonians which bring {|0 , |1 } into {|+ , |− } within minimal time π/(4E A ) and π/(4E B ), respectively.If we set , where E = 2E A is the mean energy of the total Hamiltonian H AB .From this argument, we see that for d = 4 an unbiased basis can be achieved within time π/(2E), which is longer compared to the single-qubit setup.
As we will see in the following, this intuition is not correct.For this, we will first focus on qutrit systems.As we show in Appendix B, a general unbiased qutrit basis can be obtained via a diagonal unitary from one of the following two bases (denoted by {|n + } and {|ñ + }, respectively): and Note that these two sets of basis states are odd permutations of each other.As discussed in Appendix C, this implies that speed limits for the transitions {|n } → {|n + } and {|n } → {|ñ + } will also lead to speed limits for general unbiased qutrit bases {|n } → {V |n + } and {|n } → {V |ñ + } with a diagonal unitary V. Equipped with these tools, we now present the first main result of this Letter.
Theorem 1.The time for converting a qutrit basis onto an unbiased basis is bounded below as We refer to Appendix D for the proof.
Having established a speed limit for basis change it is natural to ask whether this bound is tight, i.e., whether for any unbiased basis there exists a Hamiltonian H with mean energy E saturating the bound (12).Recalling the definition of the unbiased bases {|n + } and {|ñ + } in Eqs.(10) and ( 11), we answer this question in the following proposition.Proposition 2. The speed limit (12) is tight for the basis {|n + }, but not tight for basis {|ñ + }.
We refer to Appendix E for the proof.
The above results imply that there are two different classes of unbiased bases for qutrits: bases of the form {V |n + } can be obtained from the computational basis at time T = 2π/9E, while bases of the form {V |ñ + } require an evolution time T > 2π/9E, where V is an arbitrary diagonal unitary.For the second class {V |ñ + } we have numerical evidence that a tight speed limit is given as To see this, note that any unitary achieving the transformation |n → |ñ + must be of the form with some phases φ n (see also Appendix D).Let now λ j = e −iα j be the eigenvalues of U, such that the phases α j are in increasing order and −π ≤ α j ≤ π.For a given set of such phases {α j }, there exists a Hamiltonian implementing the unitary U = e −iHt such that where E j are the eigenvalues of H.The mean energy of the numerically obtained Hamiltonian then fulfills Using these results, we can test Eq.( 13), by numerically sampling random phases 0 ≤ φ n ≤ 2π and evaluating Et via Eq.( 16).The choice of E j t as in Eq. ( 15) guarantees that the numerical Hamiltonians obtained in this way contain Hamiltonians with the minimal value of Et.
In Fig. 2 we show the numerical probability for obtaining a certain value of Et for 10 6 samples.The numerical results suggest the following lower bound for Et: where ε is numerically upper bounded as ε ≤ 10 −5 , in good agreement with Eq. (13).A Hamiltonian saturating the bound (13) is given by H = − | α α| with A direct comparison of Theorem 1 with the corresponding qubit bound (7) shows that establishing an unbiased qutrit basis requires less time, compared to an unbiased qubit basis for the same mean energy E. In the  13).We sample 10 6 unitaries of the form (14) with random phases 0 ≤ φ n ≤ 2π and evaluate Et using Eq. ( 16).The plot shows the numerical probability as a function of Et.As a numerical bound, we obtain Et ≥ 4 9 π + ε with ε ≤ 10 −5 , in good agreement with Eq. ( 13).
following, we will discuss the main differences between the qubit and the qutrit setting.
If a single-qubit unitary U = e −iHt is optimal for rotating the basis {|0 , |1 } onto an unbiased basis, then the unitary U 2 = e −2iHt permutes the basis elements {|0 , |1 }.This is no longer the case in the qutrit setting.For this, note that an optimal Hamiltonian for the qutrit transition For the optimal Hamiltonian we can evaluate the fidelity between the initial state |0 and the time-evolved state e −iHt |0 : Note that the right-hand side of Eq. ( 20) is never zero, which means that the evolution never permutes |0 with another basis element, and the same can be shown for the states |1 and |2 .Moreover, if the single-qubit unitary U permutes the basis states {|0 , |1 }, then √ U always rotates the {|0 , |1 } basis onto an unbiased basis.This is no longer the case in the qutrit setting, as can be seen by inspection, with the permutation U = 2 n=0 |(n + 1) mod 3 n|.We further obtain and thus √ U |n is not a maximally coherent state for any 0 ≤ n ≤ 2. It can be verified by inspection that also U 1/3 does not transform any of the states |n into a maximally coherent state.
So far, we considered systems of dimension 2 and 3. We will now go one step further, giving the minimal evolution time for an unbiased basis for two-qubit systems.
Theorem 3. The time for establishing an unbiased twoqubit basis is bounded below as There exists a two-qubit Hamiltonian achieving this bound.
Remarkably, this bound is the same as for single-qubit systems, see Eq. ( 7).The Hamiltonian saturating Eq. ( 22) is given as The eigenvalues of this Hamiltonian are 3, −1, −1, −1, and the mean energy of H is given as E = 1.For t = π/4 we now define the unitary U = e −itH .The action of this unitary onto the computational basis of two qubits is as follows: This shows that the Hamiltonian in Eq. ( 23) indeed transforms a two-qubit basis onto an unbiased basis within time π/(4E).We refer to Appendix F for the proof of Theorem 3 and more details.
The results presented so far show that the optimal time for transformation onto an unbiased basis is the same for single-qubit and two-qubit systems, and in both cases given by π/(4E).For a qutrit system we have a shorter time 2π/(9E).We will now extend these results to manyqubit systems.As we will see, there exists a universal bound for n-qubit systems, allowing us to establish an unbiased basis within finite time.
Theorem 4. For systems with n qubits, the minimal time for estabishing an unbiased basis is bounded above as Proof.Consider the n qubit Hamiltonian where V is the Hadamard gate.Note that the mean energy of H n is given as E = 1.We now define the unitary U n (t) = e −iH n t .Using the fact that For t = π/2 we obtain This unitary transforms the computational basis of n qubits into an unbiased basis, and the proof is complete.
Theorem 4 shows that it is possible to establish an unbiased basis of n qubits within time π/(2E).We demonstrated this explicitly by presenting a Hamiltonian, which introduced interactions between all the qubits.Without interactions, i.e., if each of the qubits evolves independently, the optimal evolution time is given by nπ/(4E).
In the following, we present a general lower bound for the time required for establishing an unbiased basis for any d-dimensional system.Theorem 5.The time for establishing an unbiased basis for a system of dimension d is bounded below by As we see, for large Hilbert space dimension the lower bound converges to π/4E.We refer to Appendix I for the proof of the theorem.For systems of dimension 6 this bound can be improved slightly to T ≥ 0.227/E, see Appendix I for more details.Comparing this lower bound with the bound in the Theorem 4, we see that in the limit n → ∞ the minimal time T for establishing an unbiased basis of n qubits fulfills π/4E ≤ T ≤ π/2E.
Speed limits for basis permutation.It is instrumental to compare the above results to the speed limits for permuting the basis {|n }: for all 0 ≤ n ≤ d − 1.
Proposition 6.The time for permuting a basis is bounded below by Proof.As we discuss in the Appendix G, the eigenvalues of the permutation unitary (30) have the form where integer j is in the range 0 ≤ j ≤ d − 1.It follows that for any permutation unitary U = e −iHt it must hold that The proof of the proposition is complete by noting that Interestingly, for a given Hamiltonian H there are only two options: either the unitary U = e −iHt leads to permutation with t = π(d − 1)/(dE), or the Hamiltonian never leads to a basis permutation.We further note that our analysis applies only to permutations of the form (30).
Speed of evolution for coherence generation.We will now present speed limits for the creation of quantum coherence under unitary evolution.In particular, we are interested in the maximal value of coherence C max which can be achieved from a given state ρ within a fixed time t: and the maximization is performed over all Hamiltonians H with average energy E = Tr[H]/d − E 0 .As a quantifier of coherence we use the 1 -norm of coherence [19,22] which can be estimated efficiently in experiments by using collective measurements [27,28].We will first discuss the single-qubit setting.Recall that in this case the unitary U(t) = e −iHt can be interpreted as a rotation by an angle 2Et about the axis n of the Bloch sphere.As for single-qubit states the amount of coherence C corresponds to the Euclidean distance to the incoherent axis, C max (ρ, t) corresponds to the largest distance from the incoherent axis, maximized over all rotations with a fixed angle 2Et.The optimal rotation axis n is orthogonal to the Bloch vector r and the incoherent axis, and C max takes the following form: Note that C max cannot be larger than |r|, and this value is attained for the time in which case the final state is in the maximally coherent plane.If the initial state is pure, it can be parametrised as and the maximal amount of coherence achievable in a given time t takes the form In the next step we will consider systems of arbitrary dimension d ≥ 2 and evaluate the minimal time for converting a pure state |ψ into a maximally coherent state of the form with phases φ j .The following proposition gives a bound for the evolution time T (|ψ → |+ d ).
Proposition 7. The time for converting a state |ψ into a maximally coherent state |+ d via unitary evolution U = e −iHt is bounded as Proof.From Lemma 1 in Appendix H, it follows that the evolution time into a maximally coherent state is bounded as Thus, in order to obtain a bound which is valid for all maximally coherent states, we need to estimate the maximal overlap | ψ|+ d | over all states of the form (40). Expanding the initial state |ψ in the incoherent basis {|i } as with c j ≥ 0, it is straightforward to see that the overlap | ψ|+ d | 2 is maximized if we set φ j = α j , thus arriving at max Alternatively, this result can be obtained following [29,30], noting that max Conclusions and outlook.We have investigated speed limits for basis change via unitary evolutions, providing bounds on the evolution time which are optimal for several interesting scenarios.
For dimensions d ≤ 4 we found the optimal evolution time required to convert the computational basis into an unbiased, i.e., maximally coherent basis.Perhaps surprisingly, the minimal evolution times coincide for d = 2 and d = 4, when Hamiltonians with the same mean energy E are considered.Moreover, for d = 3 the saturation of the speed limit prefers a special ordering of the basis that is unbiased with respect to the computational basis.We also showed that an n-qubit Hadamard gate can be implemented within time π/2E.This proves that in multi-qubit systems, a maximally coherent basis can be established within a period of time which is independent on the number of qubits.These results further imply that in multi-qubit systems interactive Hamiltonians can significantly reduce the evolution time, compared to the time for establishing an unbiased basis by evolving each qubit independently.We further showed that in the limit d → ∞ the time for establishing an unbiased basis is at least π/4E.Speed limits for basis permutation are also discussed.
We have also investigates speed limits for generating a certain amount of quantum coherence, as well as minimal time to convert a pure state into a maximally coherent one.We expect that our methods can also be used to derive minimal transformation times for general bases and other quantum resources, such as quantum entanglement and imaginarity [31][32][33].
where the eigenvalues E ± and eigenstates |E ± can be parametrized as Here, G and E ≥ 0 are real numbers, n = (n x , n y , n z ) is a normalized vector, and σ = (σ x , σ y , σ z ) contains the three Pauli operators.The Hamiltonian (A1) can thus be equivalently expressed as Note that E corresponds to the mean energy of the Hamiltonian: Equipped with these tools, we will now present a bound for the evolution time between any two single-qubit states.
Proposition 8.The time for converting a single-qubit state ρ 0 into the state ρ 1 via unitary evolution U = e −iHt is bounded as where r i is the Bloch vector of the state ρ i .
Proof.Note that the unitary can be interpreted as a rotation by an angle 2Et about the axis n of the Bloch sphere.The minimal value for Et is achieved by choosing the rotation axis n to be orthogonal to both Bloch vectors r 0 and r 1 : This completes the proof of the proposition.
(A9) The proof of Proposition 8 implies that this bound is tight, i.e., for any two single qubit-states ρ 0 and ρ 1 , there exists a Hamiltonian with mean energy E saturating Eq. (A9).For pure qubit states this expression simplifies to the tight bound For single-qubit systems, any unitary transforming For a transition from the computational basis {|0 , |1 } to an unbiased unbiased qubit basis we thus obtain as claimed in the main text.
directly leads to a speed limit for any basis which can be obtained from {|φ j } via a unitary V = j e iα j |ψ j ψ j |: The speed limit (C2) is tight whenever Eq. ( C1) is tight.
To prove this, let H be a Hamiltonian such that e −iHt |ψ j = |φ j .

(C3)
Then the Hamiltonian H = V HV † achieves the transformation which can be seen by using the expression e −iH t = Ve −iHt V † .Noting that H and H have the same mean energy E, we see that Eq. (C1) implies the speed limit (C2) for any unitary V which is diagonal in the {|ψ j } basis.Moreover, the speed limit (C2) is tight for all diagonal unitaries V whenever Eq. (C1) is tight.
As we have seen in Appendix B, any unbiased basis of a qutrit can be created from the basis {|n + } or {|ñ + } [see Eqs.(B5) and (B6)] via a diagonal unitary V.In combination with the arguments mentioned above, this implies that speed limits for the transitions {|n } → {|n + } and {|n } → {|ñ + } will also lead to speed limits for general unbiased qutrit bases {|n } → {V |n + } and {|n } → {V |ñ + }.Noting that n|n + = e iγ n / √ d with some phases γ n we arrive at the inequality On the other hand, recalling that U = e −iHt with a Hamiltonian H we obtain where E i are the eigenvalues of the Hamiltonian.In summary, for any unitary transformation U = e −iHt leading to the transformation {|n } → {|n + } it must hold that We will now consider d = 3.In this case, we will show that any unitary U = e −iHt leading to the transformation {|n } → {|n + } fulfills Assuming that E i are in increasing order, we see that E ≥ (E 2 − E 0 )/3.Thus, for proving Eq. (D6) it is enough to prove that We will prove this by contradiction, assuming that the transformation is possible with a unitary violating Eq. (D7).Violation of Eq. (D7) implies that In the first case (E 1 − E 0 )t ≤ π/3, we can set (without loss of generality) E 0 t = −π/6, which implies the inequalities which is a contradiction to Eq. (D5).The remaining case (E 2 − E 1 )t ≤ π/3 can be treated similarly, by choosing (without loss of generality) E 2 t = π/6, thus obtaining the following inequalities: Also in this case we obtain the inequality (D10), in contradiction to Eq. (D5).This completes the proof of the bound (D6).Since the methods presented above apply for any qutrit basis which is unbiased with respect to the computational basis, this completes the proof of Theorem 1.
Appendix E: Proof of Proposition 2 According to Theorem 1, we have the following inequalities for transition into the bases (B5) and (B6): As can be checked by inspection, Eq. ( E1a) is saturated for the basis (B5) by the Hamiltonian H = |α α| with We will now prove that the inequality (E1b) is strict for the basis (B6), i.e., there is no evolution e −iHt leading to the transformation |n → |ñ + within the time t = 2π/(9E).Assume -by contradiction -that the bound is saturated for some unitary U = e −iHt : Recalling that E i are in decreasing order and following the arguments from the proof of Theorem 1, it must be that Without loss of generality we can choose Summarizing these arguments, there exists a unitary U = e −iHt fulfilling Eq. (E3) and having eigenvalues which implies that it fulfills On the other hand, the unitary also admits the form with some phases φ n .We find that Together with Eq. (E8) we obtain This equation has a unique solution in the range 0 ≤ φ i ≤ 2π, given by This implies that the eigenvalues of U must be which is a contradiction to Eq. (E7).This completes the proof of the proposition.
Appendix F: Proof of Theorem 3 We will now focus on the case d = 4.For this case we will prove the lower bound We will prove this by contradiction, assuming that there exists a unitary U = e −iHt transforming {|n } onto a maximally coherent basis with Without loss of generality we can assume that E 0 = 0, which implies It follows that cos(α 1 ) + cos(α 2 ) + cos(α 3 ) > cos(α 1 ) + cos(α 2 ) (F4) − cos(α 1 + α 2 ).
Collecting the above arguments, Eq. (F2) implies that there is a unitary U = e −iHt achieving the transformation {|n } → {|n + } with i cos(E i t) > 2, in contradiction to Eq. (D5).This completes the proof of the lower bound (F1).
As is explained in the main text, it is indeed possible to achieve the transformation {|n } → {|n + } within time t = π/(4E).This completes the proof of the theorem.
Noting that the coefficients c j fulfill the condition j |c j | 2 = 1, our figure of merit can be expressed as where the minimum on the right-hand side is taken over all probability distributions {p j }.Recalling that E gap t ≤ π, it is straightforward to see that the minimum is attained for the following choice of {p j }: for j = 0 and j = d − 1, 0 for 0 < j < d − 1. (H5) It follows that the optimal state |ψ min , minimizing the overlap | ψ|e −iHt |ψ |, can be chosen as as claimed.In the last step, it is straightforward to verify that which completes the proof of the proposition.
Remarkably, F min does not depend on the structure of the Hamiltonian, but only on the gap between the largest and the smallest eigenvalue E gap .In the following, we will use this result to bound the evolution time between pure states.
Noting that E gap ≤ dE, where E = Tr[H]/d − E 0 is the average energy of the Hamiltonian, we immediately obtain the following lemma.Moreover, for any two pure states |ψ 0 and |ψ 1 there exists a Hamiltonian H saturating Eq. (H11).To see this, recall that Eq. (H11) is tight for d = 2, see also Eq. (A10).Let now H = |φ φ| be a Hamiltonian which saturates the inequality for d = 2.Note that the mean energy in this case is given by E = 1/2.This implies that the Hamiltonian achieves the transformation |ψ 0 → |ψ 1 within the time Without loss of generality, we consider E 0 = 0 and E j ≥ 0 for all j.Also we define α j = E j T , therefore we have: By Eq. (D5) we must have − √ d ≤ j cos α j ≤ √ d.Minimizing the function f (α) = j cos α j , we show that f (α) is always greater than √ d in the region (I2), hence T cannot be smaller than T low .First, we find the critical points of the function f (α) inside the region (not on the boundary).Taking the first derivatives of the function in α i , we obtain the following equations: This shows that α i = K i π and K i ≥ 0. For these values, cos α i is either 1 or −1, thus the minimum of the function (among these critical points) occurs when we have maximum number of −1 which with respect to the constraint (I2), d−1 4 number of α i must be equal to π and the others be zero.Therefore the minimum is d 4 is not an integer.In the case d−1 4 is an integer, the point will be on the boundary of the region which we will consider it in the following.Now, we find the critical points on the boundary of the region (I2) where we have j α j = (d − 1) π 4 and α j ≥ 0. Generally, we assume that we are on the part of the boundary where x number of the {α i } d−1 i=1 are zero.Applying the Lagrange multipliers method, we end up with the equations below: where k is the Lagrange multiplier.Eqs.(I4) show that either α i = λ + 2K i π or α i = π − λ + 2K i π in which 0 ≤ λ ≤ π 2 and K i , K i are non-negative integers (because α i ≥ 0).Being on the part of the boundary with x number of α i to be zero and assuming that N number of them are of the form α i = π − λ + 2K i π, we must have (by We define K ≡ j K j + l K l .If we write λ in terms of K and N we obtain: and the function takes the form x + (d − x − 2N) cos λ.If we are in the domain N < d−x 2 then the function takes its minimum when λ is largest and it occurs for K = 0 (for any x and N) .If we are in the domain N > d−x 2 then we have: d which is a contradiction to Eq. (D5), and the proof is complete.
We will now present a lower bound for the speed limit in the Hilbert space of the dimension d = 6.We will show that the minimal time for transformation of the basis {|i } 5 i=0 to an unbiased basis via a Hamiltonian with fixed mean energy E is bounded below by 2 ).We define E i T = α i and without loss of generality we consider the minimum eigenenergy of the Hamiltonian E min = E 0 = 0.By Eq. (D5) we must have − √ 6 ≤ j cos α j ≤ √ 6.We show that the function f (α) = j cos α j is always greater than 2 ).We minimize the function f (α) = 5 i=0 cos α i in the region closure of R. First, we find all the critical points inside the region.By taking the derivatives of the function f (α) and equating them to zero, we obtain the critical points as α i = K i π, K i ≥ 0 and K i are integers.As cos (K i π) = ±1, the minimum of the function among these critical points occurs when we have the maximum number of −1 (with respect to our region R, we are allowed to have only one −1).Thus the minimum among

Figure 1 .
Figure 1.Generation of an unbiased basis {|n + } from the computational basis {|n } via a unitary evolution e −iHt min .
Appendix D: Proof of Theorem 1 Before we focus on the case d = 3 we will discuss the problem for general d.For this, let U = e −iHt be a unitary achieving the transformation {|n } → {|n + }, where {|n + } is now a maximally coherent basis of dimension d.Any unitary achieving the desired transformation must be of the form |n + n| (D1) with some phases φ n .We further obtain Tr[U + U † ] = d−1 n=0 e iφ n n|n + + e −iφ n n + |n .(D2)

Appendix I: Proof of Theorem 5 4 and d ≥ 2 .
We define T low = d−1 dE π Let us assume that T ≤ T low .Then there must exist a Hamiltonian such that:

√ d in the region R = { 5 i=0 α i < 2
H12) which is the shortest possible time for E = 1/2.For d > 2 we can use the same Hamiltonian H = |φ φ| to achieve the transformation within the same time as given in Eq. (H12).The mean energy is now given by E = 1/d, and we see that Eq. (H11) is saturated.