Multi-bit quantum random number generation from a single qubit quantum walk

We present a scheme for multi-bit quantum random number generation using a single qubit discrete-time quantum walk in one-dimensional space. Irrespective of the initial state of the qubit, quantum interference and entanglement of particle with the position space in the walk dynamics certifies high randomness in the system. Quantum walk in a position space of dimension 2l + 1 ensures string of (l + 2)-bits of random numbers from a single measurement. Bit commitment with the position space and control over the spread of the probability distribution in position space enable us with options to extract multi-bit random numbers. This highlights the power of one qubit, its practical importance in generating multi-bit string in single measurement and the role it can play in quantum communication and cryptographic protocols. This can be further extended with quantum walks in higher dimensions.


I. INTRODUCTION
Random numbers play an important role in many applications where unpredictability is a key [1,2], especially in cryptography protocol [3][4][5] where security is assured by unpredictability.There are some statistical tests [6][7][8] which can assure us the randomness of the observed sequence, but it is almost impossible to discriminate between a predetermined random string of bits that comes from a dishonest provider or malicious random number generator (RNG) and a true random sequence.In the first case the sequence may pass all the statistical tests but still can be completely predictable to the provider or anyone else who wants to eavesdrop.Therefore, generation of genuine randomness and its certification is generally considered impossible with only classical methods.Quantum physics comes by as a saviour, quantum system are intrinsically random and outcome is highly unpredictable and probabilistic in nature [9].Certification of true randomness in quantum systems comes purely from the principles of quantum physics.
The random nature of quantum mechanics [10] has gained a lot of interest from the time of it's inception.Though the description of quantum system is probabilistic, the probabilistic prediction of a theory does not imply that it is intrinsically random.There can be some limitation to the formalism and a more complete theory can describe it in a completely deterministic way [11,12].However, previous works [13][14][15] suggests that using the nonlocal correlation between two particles one can generate the randomness which is truly intrinsic.For example, like measuring entangled particles one can assess the randomness of the process independent of it's quantum description which cannot be described deterministically within the framework of any no-signalling theories.Nonlocality has been proved as an important resource in many information processing tasks like random number generation protocols [16,17], randomness expansion [16,18,19] and * Electronic address: chandru@imsc.res.inamplification [20,21] protocols, and quantum key distribution [15,22,23].Though there is no direct connection between nonlocality and entanglement [24,25], but it is known that any pure entangled states are nonlocal.Using this nonlocality of observed statistics in bipartite Bell scenario a device independent Quantum Random Number Generator (QRNG) has been suggested [16].Other than that, various other approaches to built an efficient QRNG have been developed and all of them can be classified into three categories trusted device, self-testing and semi self-testing [17].Though the device-independent or self-testing QRNG is more secure compared to two other protocols, it is unsuitable in some cases because of it's slow generation of random numbers with time.
In our work we propose a QRNG solely based on the superposition and entanglement property of the quantum walk and we use pure states which associates the nonlocal behaviour as described before.The motivation for using quantum walk is propelled by multiple advantages it can offer along with ability to generate multi-bit from a single qubit.The practical limit is bounded by the experimentally implementable number of steps of quantum walks in any system like, NMR [26],trapped ions [27,28], cold atoms [29], and photonic systems [30][31][32][33].Our analytical and numerical analysis shows that the randomness of an initial state of the particle is being enhanced using the quantum walk dynamics.The result suggests that it's dependency on the initial state is very weak and this ensures that a significantly high randomness is seen even when randomness in initial state is zero.
Using the pure bipartite entangled state of the walker (coin space) and position space we can generate the true randomness and can be quantified in the way suggested in Ref. [34].One can extract a single bit random number by measuring only on the coin space and a multi-bit random number can be extracted by measuring only on the position space or on both, coin and position space.The underlying idea here is the QRNG using a quantum dynamics where quantum interference guarantee generation of true multi-bit random numbers irrespective of the initial state.We analyse different quantum walk evolution schemes and identify that the split-step quantum walk ensures maximum randomness in the system.Control over the dynamics of the quantum walk also ensures us the control over the probability distribution from which random numbers can be extracted.This highlights the power of one qubit and its practical importance in generating multi-bit string in single measurement.

II. RANDOMNESS OF THE INITIAL STATE OF QUANTUM WALKER
Discrete-time quantum walk : The Discrete Time Quantum Walk (DTQW) is defined on the Hilbert space H = H c ⊗ H p [35][36][37][38][39][40][41][42].In this paper we will consider onedimensional quantum walk with the particle having two internal degrees of freedom.Therefore, H c is spanned by the basis states {|↑ , |↓ } and for the position space the basis states will be {|i : i ∈ Z}.Each step of DTQW comprises of quantum coin operation, followed by a position shift operator defined as S x ≡ x |↑ ↑|⊗|x − 1 x|+|↓ ↓|⊗|x + 1 x| .The resulting operation W(θ) = [S(C(θ) ⊗ 1)] evolves the particle in superposition in position space which has no classical analogue and quite advantageous for many information processing tasks and an integral part of quantum simulation schemes.The state of the walker after t-steps will be |ψ t = W(θ) t |ψ in , where |ψ in is the initial state of the walker or the particle.In our consideration |ψ in = cos δ |↑ +e iη sin δ |↓ ⊗|x = 0 .Using this initial state we will study the behaviour of the randomness under quantum walk dynamics.
Randomness quantification : The intrinsic randomness of a quantum system is related to the random outcomes of the measurement on the system [34].If we measure a pure state ρ = |ψ ψ|, (where |ψ = i a i |i ) in the basis |i considering projective measurement, then measurement outcomes are intrinsically random.According to Born's rule will be the probability of obtaining the i'th outcome.P i are the rank one projectors on the basis states.Then randomness of the output random variable is defined as Which is the Shannon entropy function of the probability distribution {p i }.In another way it can be written as R i (ρ = |ψ ψ|) = S(ρ diag ), where ρ diag is the density matrix that has only diagonal terms of ρ in the computational basis {i}.If we think ρ as a n × n matrix, having only diagonal terms ρ ii in the computational basis then the randomness inherited by state ρ can be quantified as Randomness in coin space : Let us consider the initial state of the walker as |ψ in = (cos δ |↑ +e iη sin δ |↓ )⊗|0 .
Then density matrix for the pure state will be By tracing out the position Hilbert space we will get the density matrix for the coin state which is denoted by, Therefore, which can take any value between 0 and 1.
Randomness in position space : Considering the same initial state of the walker as before, we will calculate the randomness associated with the position space by tracing out the coin space, It implies that randomness of the initial state in position space is zero.Which is expected due to the certainty of the particle's position and no corresponding inherent randomness should be evident here.

III. RANDOMNESS AFTER t-STEPS
We have already mentioned in previous section how an initial state evolves under DTQW dynamics.After t-steps, the generic form of the walker can be written as FIG. 1: Intrinsic randomness in the coin space (particle) as a function of initial state parameter δ before implementing quantum walk and after implementing 50 step of quantum walk using difference coin operation parameter θ.Though we see some dependency on δ a significant enhancement of randomness is seen even when the randomness in the initial state is zero.
density matrix denoted by ρ c in expressed in the form, In Fig. 1 we show the randomness in the coin space as function of δ which fixes the initial state before the walk and after 50 step of walk using different coin operation parameter θ.Irrespective of the initial state, that is, even when initial states randomness is zero, we can see an high value of randomness after 50 steps of DTQW.From the analytical results presented in the appendix and the numerical results we can say that the same behaviour will be seen even after small number of steps.Randomness in position space : We can use the randomness of the state contained in the position space part alone.To extract intrinsically random classical bit string out of it, which we will discuss in extracting randomness section following this.For quantification of randomness in position space we will follow the same recipe as we used in quantifying randomness in coin space.If the dynamics of the walker involves t number of steps of walk then we know that generic state can be written as |ψ t = x (a x,t |↑ + b x,t |↓ ) ⊗ |x , and ρ t = |ψ t ψ t |.By tracing out the coin space we will get the form as To calculate the randomness inherited by the reduced state, we need the diagonal entries in computational basis that is, ρ i,i .The density matrix after time t can be written in the form,  tion space after 25 step of quantum walk as a function of initial state parameter δ is shown.Before the walk, randomness in position space is zero therefore, what we note after 25 steps of walk is a significant quantity of randomness in the system even though it varies a bit as function of δ and for different value of θ.In the inset of Fig. 2, randomness in complete system, coin and position space together is shown.We see an overall boost in the randomness when compared to randomness in position space alone.

IV. EXTRACTING RANDOMNESS
The working principle of a QRNG is using quantum states and making measurement to obtain classical output string which is desirable to be random enough to pass any statistical tests.It is known before measuring that, a two-level system can acquire random classical bit string from the random outcome of the measurement.We will see how a qubit quantum walk can generate single random bit and a multi-random bit string and it's advantage's over using a single copy of qubit system.From coin space : We have already discussed the procedure to quantify randomness in coin space and our analysis shows how it is less dependent of the initial state of the walker.A clear enhancement of the randomness in comparison to the initial state is also seen.Now after tsteps the particle evolves according to the dynamics and probability amplitude corresponding |↑ and |↓ states will keep changing after each step.To get the random classical bit out of it we have to use a detector to detect the state of the walker after any arbitrary number of steps.Here we will trust both, the device implementing quantum walk and detector to get intrinsically random classical bit.Walker will be in the superposition state a x,t |↑ + b x,t |↓ before the detection and it would collapse on either of these two states after detection and we will code a classical bit 0 or 1 for detecting |↑ and |↓ state, respectively.Here, the bit commitment is arbitrary and we could use the opposite commitment too.Since quantum mechanics assures us the outcome being inherently random and we can't have a prior knowledge about the outcome before detection so that we can expect a perfectly random classical string as output, repeating this scheme for several times.From position space : We will use the same kind of extraction process in the position space as in the coin space.The advantage of using the position space is its ability to generate a multiple-random bit string rather than a single bit after each round of extraction process like it is in coin space.More explicitly, after t-steps and ignoring the internal degrees of freedom, mathematically speaking tracing out the coin space, we can write the generic state in position space of the form |ψ t = t x=−t a x,t |x , which is in the superposition of all possible states corresponding to each position.To make a measurement we have to place a position resolving detectors (or a multiple -detector at each position) where the particle position can be detected.If we use the standard version of DTQW discussed earlier in this paper, We can define the state of the detector using a simple mathematical formula, 2 n = 2t if n is an integer then we will use (n + 1) number of quantum bits to denote the state of the detectors and if n is not an integer then the maximum number of quantum bits needed to specify all detectors is {n| min n 2 n ≥ 2t}.If 2t = 2 n then the detector states will be defined as follows, Therefore, if the particle is being detected at position t then we will note the n + 1 bit string associated with the detector is measured.However, in DTQW evolution we know that after odd (even) number of steps of walk positions identified with even (odd) number will have zero probability of finding a particle.This will eliminate the occurrence of half of the configuration of multi-bit random number.To address this concern we can use splitstep quantum walk [43,44] or directed quantum walk [45,46].Split-step quantum walk : In a one-dimensional split-step quantum walk (SS-QW) the shift operator is divided into two parts denoted by S − and S + .These operations are defined as Unlike standard form of DTQW (SQW), here two different coin operators dependent on two different parameters θ 1 and θ 2 are used.Therefore, the resulting operation will be defined as, , where for the coin operator we used the same form as above.After t steps of walk the state will be |ψ t = W(θ 1 , θ 2 ) t |ψ in and it can be written in the form |ψ t = t x=−t (a x,t |↑ + b x,t |↓ ) ⊗ |x here the probability amplitude will be nonzero for each position, We can calculate the randomness using the pure state density matrix and by tracing out the coin space as we did in SQW case.
Directed quantum walk: To define the directed quantum walk (D-QW) in one dimension, we will be using one directed edge connecting two vertices of the graph and (n − 1) self looping edges at each vertex and we will assign a basis vector to each edge.Then every state at each edge can be expressed as linear In Fig. 3, randomness in the system as function of number of steps is shown when measurement are made only in the position space and in both, position and coin space (inset).The increase in randomness shows the way number of equivalent quantum bit the system mimics using a single qubit.The plot shows the randomness for all three types of quantum walk evolution.Due to non-zero probability at all position space, the randomness is more (maximizes) for the SS-QW than the randomness obtained for SQW (Fig. 3).In D-QW the number of positions on which probability amplitude is non-zero is equivalent to number of position space with non-zero probability in SQW, the randomness measure is also identical (Fig. 3).
In the inset we have sown the randomness in both, coin and position space together.Inclusion of coin space enhances the randomness in the system by a small amount.Advantages of using both space : If we take into account both, the coin and position degrees of freedom, we can generate an extra bit compared to the position space.FIG.3: Randomness with number of steps in standard QW, SS-QW, and DQW.Due to non-zero probability al all position space in SS-QW we can see a maximum randomness compared to other two.Amount of randomness measure corresponds to equivalent number of qubit it can mimic in the process.Inset is the randomness when both coin and position space are taken together.
The reason is very obvious, we have seen already how internal degrees of freedom of the particle is able to generate one classical random bit after a single round of evolution.Therefore, when we use both spaces we will get random classical bit coming from both space.The extraction process is same as discussed before just only difference is that here we have to place the detectors at each possible positions and it should be capable of capturing the internal state of the walker properly.
In this process, from a given position space we can generate a uniform width classical multi-bit string rather than a single bit from a standard single particle QRNG after of the particle.To conclude the fact, it is remarkable in the sense that using a single qubit we can generate multiple number of classical random bits whose random-ness only depends on the devices trustedness.Controlled distribution of the quantum walk can be further engineered to pick the string desired distribution of strings.

V. CONCLUSIONS
The main idea of our work was to construct a protocol for generating multiple random bits using quantum walk of a single particle.We have shown the advantages of using split-step quantum walk over directed or standard quantum walk in extracting higher randomness and all combination of multi-bit string.This behaviour is kind of expected from physical intuition too because for split-step quantum walk the degrees of freedom in position space is double, roughly speaking in comparison with other two and probability amplitude for all the possible positions contribute to the expression of randomness.Other than that another interesting result is that for split step quantum walk the amount of randomness contained in the state after 2 t number of steps exactly maps to the t number of random classical bits.Now the question can arise about the probability distribution of the position space, where uniformity of the distribution is desired for the security purpose and here we can answer it affirmatively by using some engineered coin operations and phase operation to produce the desired distribution and uniform distribution [47,48] therefore the randomness of the output string can be guaranteed with more confidence.

5 FIG. 2 :
FIG.2: Intrinsic randomness in position space as a function of initial state parameter δ after 25 step of walk using different coin operation parameter θ.It is evident from the figure that randomness shows some dependency on the initial state the variation is vary small when compared to the zero randomness in the initial stage.

1 n
combination of the states |x, → , |x, 1 , |x, 2 • • • |x, n − 1 where x is non-negative integer and → indicates edge along the line and each number comes from distinct self loop.The action of shift operation is defined as S x |x, → = |x + 1, → and S x |x, i = |x, i , i ∈ [1, n − 1].Therefore the shift operator takes the form S x = x,i |→ →| ⊗ |x + 1 x| + |i i|⊗|x x|.Coin operation has the form C ≡ α β β −α where α = 1/ √ n and β = n−.Therefore, one step of D-QW comprises of two operation W d = S x [C ⊗ 1].After t steps of D-QW the state will be |ψ which is the outcome of the operation W(θ) t on the initial state.Therefore, the pure state density matrixρ t = |ψ t ψ t | = x,y (a x.t a * y,t |↑ ↑| + a x,t b * y,t |↑ ↓| + b x,t a * y,t |↓ ↑| + b x,t b * y,t|↓↓|) ⊗ |x y|.Now using this expression we can compute the randomness associated with the position and coin space individually and both together.Randomness in coin space : The walker has two internal degrees of freedom and total randomness is being distributed in the form of probability amplitude associated with the |↑ and |↓ states.By tracing out the position space, we will be remained with the reduced y (a x.t a * y,t |↑ ↑| + a x,t b * y,t |↑ ↓| + b x,t a * y,t |↓ ↑| + b x,t b * y,t |↓ ↓|) ⊗ |x y| therefore ρ p t = t x,y=−t (a x,t a * y,t +b x,t b * y,t )⊗|x y|.Comparing the two expression for ρ p t we can write down the form of the randomness as R i (ρ p t ) = x (|a x,t | 2 + |b x,t | 2 ) ln(|a x,t | 2 + |b x,t | 2 ).In Fig. 2, randomness in posi-